Cantini - Logical framework

LOGICAL FRAMEWORKS
FOR TRUTH
AND ABSTRACTION
An Axiomatic Study
STUDIES IN LOGIC
AND
THE
FOUNDATIONS
OF
VOLUME
MATHEMATICS
135
Honorary Editor:
E SUPPES
Editors:
S. ABRAMSKY, London
S. ARTEMOV, Moscow
J. BARWISE, Stanford
H.J. KEISLER, Madison
A.S. TROELSTRA, Amsterdam
ELSEVIER
A M S T E R D A M 9L A U S A N N E 9N E W Y O R K 9O X F O R D 9S H A N N O N 9T O K Y O
LOGICAL FRAMEWORKS
FOR TRUTH
AND AB STRACTION
An Axiomatic Study
Andrea CANTINI
Department of Philosophy
Universityof Florence
Florence, Italy
1996
ELSEVIER
AMSTERDAM
9L A U S A N N E
9N E W Y O R K
9O X F O R D
9S H A N N O N
9T O K Y O
ELSEVIER SCIENCE B.V.
Sara Burgerhartstraat 25
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN: 0 444 82306 9
9 1996 Elsevier Science B.V. All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by
any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written
permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, EO. Box 521,
1000 AM Amsterdam, The Netherlands.
Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright
Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained
from the CCC about conditions under which photocopies of parts of this publication may be made in the
U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred
to the Publisher.
No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a
matter of products liability, negligence or otherwise, or from any use or operation of any methods,
products, instructions or ideas contained in the material herein.
This book is printed on acid-flee paper.
Printed in The Netherlands
PREFACE
This book is concerned with logical systems, which are usually termed typefree or self-referential and emerge from the traditional discussion on logical
and semantical paradoxes. We will consider non-set-theoretic frameworks,
where forms of type-free abstraction and self-referential truth can
consistently live together with an underlying theory of combinatory logic.
However, this is not a book on paradoxes; nor we aim at a grand logic
la Frege-Russell, inspired by a foundational program. We shall rather
investigate type-free systems, in order to show that"
(i) there are rich theories of self-application, involving both operations and
truth, which can serve as foundations for property theory and formal
semantics;
(ii) these theories give new outlooks on classical topics, such as inductive
definitions and predicative mathematics;
(iii) they are promising as far as applications are concerned.
This way of looking is justified by the history of the antinomies in our
century. In spite of isolated foundational and philosophical traditions, the
research arising from paradoxes has become progressively closer to the
mainstream of mathematical logic and it has received substantial impulse
during the last twenty years: a number of significant developments,
techniques and results have been cropping up through the work of several
logicians (see below for our main debts). Therefore a major aim of this book
is to attempt a unifying view of relevant research in the field, by dwelling
on connections with well-established logical knowledge and on applicable
theories and concepts.
However, the present work is far from being comprehensive. We do not
treat illative combinatory logic (with the exception of a system of Ch.VI,
investigated by Flagg and Myhill 1987), nor we deal with the BarwiseEtchemendy approach to self-reference via non-well-founded sets. Another
significant direction, which is only touched upon in two sections of chapter
XIII, is the systematic development of the general theory of semi-inductive
definitions (in the sense of Herzberger, Gupta and others).
vI
Preface
The project started some years ago, when Prof. A. S. Troelstra kindly
suggested an English translation of the author's monograph (Cantini 1983a)
about theories of partial operations and classifications in the sense of
Feferman (1974). The attempted translation soon shifted towards a
thorough expanded revision of the old text, and eventually gave rise to an
entirely new set of notes at the end of 1988. After a stop of almost two
years, these notes were taken up again, fully rewritten and reorganized. The
manuscript was submitted to the editor for final refereeing in October 1993.
The content and the results of the present version are disjoint from the
1983 monograph; they partly overlap with the 1988 notes, except for a
different choice of primitive notions and for the addition of Ch.VI, parts of
Ch.XIII and the epilogue. Ch.VIII offers a development of topics, contained
in the author's paper "Levels of Truth" (to appear in the Notre Dame
Journal of Formal Logic, 1995): we thank the Editors for granting the
permission of using parts of that paper in Ch.VIII of this book.
Acknowledgments. The present work owes a great deal to the writings of
several logicians, and even if I tried hard to make a complete list of my
debts in the text and in the reference list, I am sure that there are
omissions: I apologize for them.
As to the proper content of the book, pertaining to type-free abstraction
and self-referential truth, I would like to underline my intellectual debt
with the following papers (listed in alphabetical order): Aczel(1980),
Feferman (1974), (1984), (1991), Fitch(1948), (1967), Friedman and Sheard
(1987), Kripke (1975), Myhill (1984), Scott (1975).
Profl W. Buchholz offered an invaluable help in correcting errors of any
kind and in proposing technical improvements. I owe a special thank to
him, also because the topics I dealt with were not touching his main
research interests.
I am grateful to Prof. S. Feferman and to Profi G. Js
for keeping me
informed over the years about their own research on type-free systems and
proof theory, and for important advice. J~iger's Ph.D. student, T. Strahm
made useful critical comments on the first chapter.
Dr. R. Giuntini and Dr. P. Minari undertook the final proof-reading of
chapters I-VIII and XII-XIV; I warmly thank them for a host of useful
remarks and corrections.
I am deeply indebted to Dr. A. P. Tonarelli for proof-reading the
remaining chapters and for eagle-eyed assistance in the unrewarding task of
preparing the final manuscript.
Preface
vii
Of course, I must stress that I am fully responsible for all errors, to be
found in the whole work.
I am grateful to the Alexander-von-Humboldt Stiftung (Germany) for
granting me a "Wiederaufnahme" of a research fellowship at the LudwigMaximilians-Universit~it M/inchen in Sommer Semester 1991, when the
present work was at a difficult stage.
Partial support to the present project was granted by the Italian
National Research Council (CNR)-and the Italian Ministry for University,
Scientific Research and Technology (MURST).
Last but not least, this work is dedicated to my children Giulia and
Francesco.
Firenze, April 1995
This Page Intentionally Left Blank
CONTENTS
Preface
Contents
IX
Introduction
PART A: COMBINATORS AND TRUTH
Introducing operations
The basic language
2.
Operations I: general facts
Operations Ih elementary recursion theory
3.
4A. The Church-Rosser theorem
4B. Term models
The graph model
5.
An effective version of the extensional model D co
6.
Appendix
13
14
15
18
22
26
28
34
39
Extending operations with reflective truth
7.
Extending combinatory algebras with truth
8.
The theory of operations and reflective truth:
simple consequences
9A. Type-free abstraction, predicates and classes
9B. Operations on predicates and classes
10A. The fixed point theorem for predicates
10B. Applications to semantScs and recursion theory
11. Non-extensionality
Appendix I: a property theoretic definition of the
fixed point operator for predicates
Appendix Ih on the explicit abstraction theorem
Appendix III: independence of truth predicates
from the encoding of logical operators
43
45
o
II
11
51
55
59
63
68
73
76
77
80
Contents
x
PART B: TRUTH AND RECURSION THEORY
III
IV
V
Inductive models and definability theory
12. Inductive models and the induction theorem
13. The envelope of an inductive model
14. The uniform ordinal comparison theorem for inductive
models
15. Applications of the uniform ordinal comparison theorem
VII
85
86
88
91
97
Type-free abstraction with approximation operator
16. Approximating properties by classes
17. The approximation theorem for extensional operations
and the fixed point theorem for monotone operations
18. Topology displayed: basic definitions
19. The representation theorem for explicitly
CL-continuous operators
Appendix: alternative proofs
103
104
Type-free abstraction, choice and sets
20. Choice principles and the distinction between operations
and functions
21. Admissible hulls: elementary facts
22. A model of admissible set theory
23. The boundedness theorem
125
PART C: SELECTED TOPICS
VI
83
109
113
117
122
126
131
137
144
149
Levels of implication and intensional logical equivalence
24. Myhill's levels of implication
25. Formal deducibility based on levels of implication
and its proof-theoretic strength
26. Introducing an intensional equivalence relation
27. The infinitary reduction relation :=~
28. The Church-Rosser theorem for ==~
29. A model of type-free logic based on intensional
equivalence
151
152
On the global structure of models for reflective truth
30. The lattice of fixed point models for the neutral
minimal theory
31. The sublattice of intrinsic fixed point models
and the cardinality theorem
177
158
162
165
169
174
179
186
Contents
32.
33.
34.
35.
Variations on the encoding technique: non-modularity
and other oddities
A model for an impredicative extension of reflective truth
On Kripke's classification of self-referential sentences
On the consistency of coinduction principles
Appendix: a variant to the basic operator F
and the restriction axiom
XI
192
198
203
207
209
PART D: LEVELS OF TRUTH AND PROOF THEORY
213
VIII Levels of reflective truth
36. A language and axioms for reflective truth with levels
37. Simple consequences
38. Universes and the Weyl extended iteration principle
39. A recursion-theoretic model
40. Levels of truth and predicatively reducible subsystems
of second-order arithmetic
41. Consistency of a reducibility principle for classes
42. Levels of truth and impredicative subsystems of
second-order arithmetic
Appendix: on projectibility and stronger reflection
215
218
220
225
230
IX
Levels of truth and predicative well-orderings
43. On well-orderings
44. Ramified hierarchies
45. Predicative well-orderings I
46. Predicative well-orderings II
257
258
261
269
277
X
Reducing reflective truth with levels to finitely iterated
reflective truth
47. A sequent calculus STLR for a theory of reflective
truth with levels
48. Basic properties of STLR
49A. Elimination of the full level induction schema
49B. Elimination of unbounded level quantifiers
50. The infinitary sequent calculus I T ~ of n-iterated
reflective truth
51. Embedding STLR n into I T ~
XI
Proof-theoretic investigation of finitely iterated reflective truth
52. The ramified system RS n
53. Cut elimination
54. Some derivable sequents of RS n
238
244
248
253
285
286
291
293
297
303
305
311
312
316
320
XII
Contents
55.
56.
57.
58.
Embedding ITn~ into RS n
The upper bound theorem for I T ~
Upper bound theorems for TLR and its subsystems
Conclusion: the conservation theorems
Appendix: primitive recursive cut elimination for RS n
PART E: ALTERNATIVE VIEWS
XII
Non-reductive systems for type-free abstraction and truth
59. The core system V F - and transfinite induction
60. Supervaluation models of V F 61. An abstract sequent calculus and truth
62. Cut elimination and related properties
63. A provability interpretation and the upper bound theorem
64. Reconciling supervaluation models with provability
interpretation
XIII The
65.
66.
67.
68.
69.
variety of non-reductive approaches
An inconsistency
On a truth theory of Friedman and Sheard
Fitch's models
Introducing semi-inductive definitions
Semi-inductive models for reflective truth
324
327
329
335
338
349
351
352
357
358
364
369
375
379
380
383
386
390
394
XIV Epilogue: applications and perspectives
70A. A logical theory of constructions: informal motivations
70B. A logical theory of constructions: basic syntax
71. Axioms for the computation relations
72. Extending the logical theory of constructions with higher
reflection
73. Proof-theoretic reduction
74. Perspectives: related work in Artificial Intelligence
and Theoretical Linguistics
75. Sense and denotation as algorithm and value: subsuming
theories of reflective truth under abstract recursion theory
401
402
403
407
Bibliography
425
Index
441
List of Symbols
453
411
416
419
422
INTRODUCTION
"There never were set-theoretic paradoxes, but the property-theoretic
paradoxes are still unresolved" (K. Gbdel, as reported by J. Myhill 1984).
"... the theory of types brings in a new idea for the solution of the
paradoxes, especially suited to their intensional form. It consists in blaming
the paradoxes not on the axioms that every propositional function defines a
concept or a class, but on the assumption that every concept gives a
meaningful proposition, if asserted for any arbitrary object or objects as
arguments" (K. G6del 1944)
1. Informal ideas. The starting point of our investigation is the idea that
the notions of predicate application and property are susceptible of
independent study; in particular, these intuitive notions should be kept
distinct from their counterparts of set-theoretic membership and set, as it is
readily seen through a brief comparison.
According to the iterative conception, a set is always a collection of
mathematical entities of a given type (possibly, sets of lower rank); thus it
has its being in its members, and equality among sets is ruled by the
extensionality principle. Sets are conceived as completed totalities,
generated by language independent operations and iterations thereof. The
membership relation is a standard mathematical relation: this means that
a C b is a well-defined proposition, whenever a and b are sets. Moreover, if
we reflect upon the intuitive picture of the cumulative hierarchy, we come
to know that C is well-founded and does not allow self-application.
By contrast, a property is an abstract object, which is grounded in a
concept, i.e. a function, not in the objects which fall under it (Frege 1984,
p.199); it has no a priori bound on its extension, and it usually depends on
some sort of explicit or implicit finite specification. Properties satisfy the
so-called unrestricted abstraction or comprehension principle (AP):
every condition A(x) determines a property {x:A}, which applies to all
and only those things of which A(x) holds true.
Of course, on the face of the well-known paradoxes, A P introduces elemcnts
which are open to dispute and to multiform solutions; for instance, as
GSdel's citation suggests, the predication r e l a t i o n - henceforth 7/- cannot be
always meaningful, and therefore the laws of standard (classical) logic
cannot be valid.
2
Introduction
The present approach, to be developed in various forms in this book,
tries to keep the regimentation for predication and abstraction at a
minimum; we maintain that {x" A} is an individual term and that r/applies
to statements possibly involving 7/ itself. Thus we are looking for flexible,
type-free theories of predication. More specifically, we are influenced by the
tradition of illative combinatory logic in the sense of Curry and Fitch, by
the work of Feferman (1975) on partial classifications and of Aczel (1980)
on Frege structures. The inspiring idea is that properties and predication
can be adequately explained in terms of the primitive notions of function
and truth.
As to the notion of function, we cannot expect to deal with functions in
set-theoretic sense. In fact properties, given in intension, may apply to
anything in a given realm, without type restrictions; and the same must
hold of the functions underlying the properties themselves. Thus we are
driven to understand functions essentially as rules of constructions (or, in
short, operations) in the sense of combinatory logic. In contrast to the
set-theoretic conception, operations are prior to their graphs and have no a
priori bound on their domain; in particular, they do support non-trivial
forms of self-application. On this view, it is natural to identify properties
with object-correlates of functions, and to reduce the abstraction operation
to familiar )~-abstraction; formally, {x:A} simply becomes a h-term of the
form )~x[a], where [A] is a term of combinatory logic, canonically
representing the function defined by the condition A (of any given
language).
The second point concerns the reduction of predication to a primitive
notion of reflective (or self-applicable) truth. Indeed, the expression
yq{x : A}
is analyzed as: " the result of applying the function represented by {x:A}
to the argument y turns out to be true". Therefore, if we let T stand for the
truth predicate, yq{z: A} is defined as T({x: A}y) (with juxtaposition of
{x: A} and y as application), and the abstraction principle AP becomes
obviously derivable from the basic law of h-abstraction (i.e. we convert
{x: A}y to the term [A[x := y]], the result of replacing x with y in [A]).
Of course, these preliminary considerations do not solve the main
problem of specifying the basic features of the truth predicate T.
Nevertheless, they direct our attention towards the study of simple
mathematical objects, namely expansions of combinatory algebras by
reasonably closed truth sets. The typical structure (essentially) consists of a
pair
where (i) 3t~ is a combinatory algebra, i.e. a model of Curry's combinatory
Introduction
3
logic; (ii) ff is a subset of M ( - t h e domain of 31,), which assigns a
semantical structure to Jtt~.
These expansions are uniformly described by means of operators F from
the power-set of M into itself, acting as abstract valuation schemata.
Informally, if X C_ M, F(X) represents the set of "truths" we come to know
by means of the semantic rules of F on the basis of X. A central role in this
book is played by an operator F, which essentially embodies Kleene's three
valued non-strict interpretation of logical constants. In general, if F is
monotone and reflects a cumulative conception of knowledge, the natural
candidates for o-j-will be those subsets of M, that cannot be further extended
with new information by means of F, i.e. the so-called fixed points of F,
satisfying F(X) - X. Among these sets, a special role will be played by the
minimal ones: they are technically the most interesting objects for the
recursion-theoretic and proof-theoretic investigations. Conceptually, they
reflect the idea that abstraction is not the mere description of an
independent logical realm, but rather a process with its own logic implicit
in F.
In order to provide a few intuitions behind the construction of the first
part of the book, it may be suggestive to regard 31~ as an abstract syntax,
in which formal languages can be processed and defined. In particular,
elements of M may be thought of as symbolic expressions, to be combined
and identified according to the operations and laws of combinatory logic. M
will typically include (notations for) natural numbers or any other chosen
ground type, but also and most important, objects representing functions.
The objects associated to computable functions can be seen as (functional)
programs, implementing effective algorithms. On the other hand, still
pursuing the computational analogy, properties-as representatives of
(generally non-computable) propositional functions-can be considered as
programs implementing a sort of generalized algorithms. While application
of an effective algorithm to an input produces a computation, possibly
converging to a value, a property { x ' P ( x ) } is ultimately applied to an
object, in order to produce a verification that the object itself satisfies the
given condition, according to the rules specified by the truth set ~.
We wish to conclude by raising three conceptual points. First of all, the
notion of truth T is not understood as a formalized truth predicate in the
usual metamathematical sense: T classifies the objects of a combinatory
algebra, and not an inductively defined collection of sentences ! In this sense,
T, like set-theoretic membership, does not depend upon a specific language.
As it should be clear from the sketched schema of interpretation ~ + , the
predicate T is a primitive concept, which is prior to the specification of any
formalism and is the source of the abstract notion of proposition. The
underlying philosophy is that there are certain objects in our universe AI~,
4
Introduction
which carry information and can be called propositions; they can be
partitioned into atomic or complex. Atomic propositions are simply grasped
and reflect implicit (synthetic) knowledge, to be accepted as given. On the
other side, complex propositions correspond to some sort of construction via
logical operators; thus they require a(n analytic) process, in order to be
understood (think of the search for verification), and they are controlled by
the truth predicate T.
As a second point, we like to stress the importance of having operations
acting on classifications. Indeed, the fact that operations and classifications
live together has the consequence of a symmetry, lacking to set theory: not
only we can classify operations, but we can operate on classifications. So we
can treat classifications, which depend on parameters, as operations; this is
generally impossible in set theory. It follows that many constructions and
statements acquire an "explicit character" and greater uniformity.
A final comment is left for the choice of non-extensional basic notions.
In general, even if we make use of intensional data (like definitions or
enumerations), we never appeal to specific features of them, and thus we
obtain results with an intrinsic character. Moreover, we find that the nonextensional language helps to avoid "strong logical principles" and to carry
out proofs in rather weak systems (just as remarked in Kreisel 1971, p. 170);
it often permits uniform and explicit statements of the results obtained,
which do not obscure the appreciation of proper extensional aspects. On the
contrary, non-extensional and extensional features are free to interact in a
unified framework. As it will be clarified by the introduction of the
approximation structure in chapter III and its applications in the
subsequent chapters, the essential interplay of these aspects leads to rather
smooth generalizations of the Myhill-Shepherdson theorem (w
to the
appreciation of extensional choice principles (w
and to a satisfactory
"internal" treatment of inductive definitions (boundedness and covering;
w
2. Organization and contents. As we previously explained, the starting point
of the book is the need for an independent logical approach to the notions of
predicate application, property, abstraction, truth. The arrangement of the
material reflects the increasing logical complexity of the truth notions that
are met in the text. The different proposals, though generally motivated by
model-theoretic constructions, are developed in axiomatic style. This is
mainly because we wish to emphasize the connections with standard
concepts of mathematical logic and deductive systems for (substantial parts
of) mathematics.
Proof-theoretic considerations and conservative extension results play an
important role in classifying the various systems: very loosely, we tend to
stress the importance of frameworks not stronger than Peano arithmetic and
Introduction
5
to distinguish predicative from impredicative systems. We also underline
that type-free systems should not be opposed to type theories; we regard the
former as a sort of generalized type assignment systems, in which types are
left implicit and emerge from the theory itself.
More concretely, the book is divided into five parts, which group
together relatively homogeneous topics. The read thread can be described as
follows. By and large, the first three parts form a sort of independent essay
on a first-order theory of reflective truth over combinatory logic, whose
truth axioms essentially stem from Fitch's extended basic logic (Fitch 1948)
through Scott (1975) and Aczel (1977). The notion of reflective truth
explicitly refers to Feferman (1991). After the general results of Part A, the
theory is motivated and enriched by means of recursion-theoretic
investigations (part B), by showing its unifying power and studying its
semantics (part C). Parts D and E explore alternative routes. Part D
deepens the intuitions underlying the systems of parts A-C by use of prooftheoretic techniques and by relativizing the concept of truth. Part E is
experimental in character and scans over a variety of approaches, which are
still subject of investigation.
To give the reader a closer idea of what is in the book, we shall survey
the content of the single chapters. A more detailed account can be found in
the introductory section to each chapter.
Part A: it offers a general introduction to the basic notions of operation and
reflective truth. The basic aim is to illustrate, both axiomatically and
semantically, a consistent notion of type-free logical structure, which will be
fundamental in the whole book.
The opening chapter contains the necessary preliminaries on (expanded)
combinatory logic, which is here taken as the core of a classical first-order
theory of operations OP. There is an introduction to concrete models of OP,
as they form the ground structures in the entire book.
In the second chapter, we inductively expand combinatory algebras with
a notion of self-referential truth, which naturally generalizes the familiar
Tarskian semantical clauses, in order to cope with a situation of partiality.
The given expansions only depend on the isomorphism type of the
underlying combinatory algebras. By inspection of the model-theoretic
construction, we are led to a minimal axiomatic first-order system MF-,
which contains a version of the Kripke-Feferman axioms for reflective truth
and yields a theory of partial and total properties ( = classes henceforth),
satisfying natural closure conditions. For instance, classes are provably
closed under Feferman's join and elementary comprehension principles;
moreover, MF- is provably closed under inductive definitions (though not
capable of showing the corresponding induction schemata). We also consider
6
Introduction
extensions of MF- with various number-theoretic induction principles.
Part B: we show that there is a two-sided link between generalized recursion
theory and languages with operations and self-referential truth. Not only
inductive definitions are crucial for building up models of self-referential
languages, but these languages offer smooth formulations of non-trivial
definability results.
In chapter III we prove that classes (properties) in the inductive model
over a given combinatory algebra a~ exactly define the hyperelementary
(inductive) subsets of dtt, in the sense of Moschovakis (1974). The recursiontheoretic approach suggests to extend the minimal system by simple
approximation conditions on properties. The new axioms, together with
MF-, a powerful generalized induction schema GID and number-theoretic
induction for classes, form an axiomatic system PWc+GID , which is still
conservative (actually proof-theoretically reducible to) over the theory of
operations and hence over Peano arithmetic.
In chapter IV we show that PWc+GID proves a number of interesting
consequences (separation and reduction principles) and, above all, an analog
of the Myhill-Shepherdson theorem for operations which are y-extensional
(i.e. extensional with respect to the predication relation). The results can be
restated in topological terms via a natural generalization of the positive
information topology.
In chapter V, we produce models for admissible set theory and a
boundedness theorem for inductive sets, again provably in PWc+GID.
Part C: it is a natural complement to the previous parts. In chapter VI, the
reader will find two alternative type-free logics. The first system, due to
Myhill (1972, 1980), relies on a logic with levels of implication. The second
system, inspired by Aczel-Feferman (1980), offers a type-free logic with a
definitional equivalence relation on formulas, which is inspired by
conversion in combinatory logic. Both systems are formally interpreted in
the theory PWc+GID of chapters IV-V.
Chapter VII offers a general outlook on the global structure FIX(art,) of
fixed point models of N M F - ( = M F - without a consistency axiom) over
arbitrary combinatory algebras art,. We prove that FIX(.Jt) only depends
on the isomorphism type of art, and that the set of sentences A such that
TA holds in every structure of FIX(..~), for arbitrary d~, is axiomatizable.
It is shown that FIX(art) is a very rich and intricate non-distributive
complete lattice; a few applications to consistency results and to formal
semantics are thereby outlined (see connection with Kripke 1975).
Part D: it focuses on proof theory and the foundations of mathematics. We
investigate a type-free logic TLR, which is able to internalize to a certain
Introduction
7
extent quantification on classes and negative semantic information. The
intuitive idea is that truth is the (direct) limit of local self-referential truth
predicates, which are related one another by a directed pre-order of levels.
Formally, we present TLR and its variants in chapter VIII. Among its
consequences, it is possible to introduce a notion of "mathematical
universe" with nice closure properties and interpret non-trivial subsystems
of second-order arithmetic (ranging from versions of predicative analysis,
like Friedman's ATR0, to the so-called A12-CA).
In chapter IX we develop the prerequisites for a proof-theoretic analysis
of TLR: in particular, we describe a well-ordering proof of the so-called
Feferman-Sch/itte ordinal. Chapter X proves that the theory of truth with
levels is proof-theoretically reducible to (infinitary) theories of finitely
iterated self-referential truth ITS; on the other hand, each I T ~ is shown to
be reducible to fragments of predicative analysis in chapter XI. The
methods used include cut elimination for ramified systems in w-logic and
asymmetrical interpretations d la Girard.
Part E: we are concerned with logics of truth and type-free abstraction,
which are based upon non-reductive, non-truth functional semantical
valuation schemata. In contrast to the reductive schema underlying the
semantics of chapter II, we study systems which are well-behaved with
respect to logical consequence (e.g. a tautology is always classified as true;
this does not work under a partial semantics d la Kleene).
Chapter XII investigates a minimal system VF endowed with a simple
supervaluation monotone semantics; VF naturally justifies principles of
generalized inductive definitions (in contrast to what happens with the
theories of parts A-C, it yields a model of the theory of elementary
inductive definitions ID1). We also develop an alternative interpretation for
VF by means of proof-theoretic infinitary methods.
Chapter XIII addresses the problem of extending the logic of truth, as
codified in VF. We discuss a refinement of supervaluation methods; but the
new point is the introduction of semi-inductive definitions (in the sense of
Herzberger 1982) and the application of the related notion of stable truth.
We also consider consistent though w-inconsistent logics of truth, due to
Friedman, Sheard and Mc Gee.
The epilogue (chapter XIV) discusses prospective applications of typefree systems, as they result from the literature. In particular, we consider a
logical theory of constructions, that has been investigated in Theoretical
Computer Science and is strictly linked with the theories of part D. We
conclude with a short survey of applications in other fields.
Introduction
8
3. How to use the book. The interdependence of the chapters is roughly
indicated in the diagram below:
I
II
1
III
IV
VII
V
VIII
~
XlI
IX
VI
1
x
1
1
XI
~
XlII
XIV
Certain parts of the book, once suitably combined, offer a non-conventional
approach to:
1) generalized recursion theory and inductive definability (part A + part B);
2) predicative mathematics and subsystems of analysis (part A + part B +
+ part D).
If we disregard the recursion-theoretic and proof-theoretic parts, the book
can serve as an introduction to"
3) formal semantics (part A + III + part C + VIII (w167
36-39) 4- part E).
If the reader has in mind possible connections with logics for Artificial
Intelligence, Theoretical Computer Science or semantics for natural
languages, the suggestion 3) can be profitably modified to:
4) part A + part B + VIII (w167
36-39) + part E.
Some chapters have appendices, containing additional details for proofs
or suggestions for alternative routes: they can be always skipped without
prejudice of understanding the later developments.
4. Prerequisites. The text is intended for readers who are familiar with the
topics usually covered in an advanced undergraduate or basic graduate logic
course. Thus we assume acquaintance with the elements of first-order logic
Introduction
9
and model theory, recursion theory, set theory and proof theory, as they are
developed in good standard textbooks, or in the corresponding chapters of
the Handbook of Mathematical Logic (Barwise 1977).
In particular, it is useful to have a preliminary knowledge of the basic facts
of hyperarithmetic and inductive definability (see Aczel 1977a, Moschovakis
1974). For the proof theory of Chapters VIII-XI, a previous exposure to
sequent calculi and w-logic would be helpful (e.g. see Schwichtenberg 1977
or the textbooks of Takeuti 1975, Schfitte 1977, Girard 1987, Pohlers 1989).
The simple topological notions of Ch. IV can be obtained from any standard
textbook in general topology. Ch. VII presupposes a few elementary facts
about partially ordered sets and lattices, usually met in logic courses
(consider the classical reference of Birkhoff 1967). In Ch. VIII we hinge upon
some advanced results of admissible set theory, to be found in Barwise
(1975), Hinman (1977); however, the basic definitions and results are briefly
recalled there.
5. General notations and conventions. A number of notations are adopted
in the whole text. We summarize them below.
5.1. Internal and bibliographical references. The book is structured in five
parts from A to E; each part is subdivided into chapters; the chapters are
organized in sections, which are numbered in progressive order. Within each
section, each specific item (subsection, definition, remark, axiom, rule,
theorem, lemma or corollary) is usually assigned a pair "m.n" of numbers:
"m.n" refers to the nth item of the ruth section. Sometimes, for finer
classifications and reference, we allow the use of three (and exceptionally
four) numbers (e.g. 37.4.1 locates the first sub-item of the 4th-item of
section 37). In some cases, we specify the class, which the referred item
belongs to (e.g. we may speak of theorem 3.2 or definition 34.5).
References to publications are given by means of the author's name followed
by the year of publication, possibly followed by a letter in the case of more
publications by the same author in the same year.
5.2. Definitions. := is used as the definition symbol (definiendum on the
left of : = , definiens on the right), while - stands for literal identity,
unless it is specified otherwise.
5.3. Variables and substitution. We adopt the standard notions of free and
bound variable; FV(E) is the set of free variables of the expression E.
E[x := t] denotes the substitution of x with t in E. E(E') means that E'
possibly occurs as a subexpression of E.
5.4. Logical Symbols. As usual we use V, 3, -1, ~ , A, V, ~ . For
bracketing, we adopt the usual conventions; V, 3, -~ bind stronger than the
other symbols, while A, V bind more than ---, and ~ . To enhance
Introduction
10
readability, dots may be used instead of brackets as separating symbols.
A A B.---,C, A---,.B V C, 3x.A stand for (A A B)---,C, A ~ ( B V C) 3xA
(respectively); ~x.ts shortens ~x(ts), etc... Sometimes, we make use of
bounded quantifiers as abbreviations: if R : = r / , E, VxRa.A, 3xRa.A
shorten Vx(xRa~A), 3x(xRa A A). If bounded quantifiers are iterated, we
write: (VxRa)(VyRb)(...), (VxRa)(3yRb)(...), or even VxRa.VyRb.(...),
VxRa.3yRb.(...), for the proper Vx(xRa---,Vy(yRb...)), Vx3y(xRa A yRb...)
(respectively).
We shorten logical equivalence on the metalevel (i.e. "if and only if") by the
standard "iff" . "3!x" stands for "there is exactly one x". Sometimes, we
adopt :=~ as implication on the metalevel.
5.5. Logical Complexity. The logical complexity of any given formula A is
the number of distinct occurrences of logical symbols in A.
5.6. Set-theoretic symbols. We use the standard
E , ~ (negation of E ), w (the set of natural
infinite ordinal), 0, U, A, ~P(X) (power set of
(Cartesian product), f" X-~Y (to be read as
Y"), cz (characteristic function for the set Z).
notations:
numbers, but also the first
X), X - Y ,
C, C_, D, D_, •
"f is a function from X to
If k,m E w,
[k, m] "- {i E w " k _< i _< m}; ( k , m ] ' - { i e w ' k < i _ < m } ;
(k,m) . - {i
k < i < m}; [ k , m ) . - {i
k < i < m}.
{ x : . . . } is the set of objects satisfying the condition (...); {al,... , an} is the
set containing exactly the elements a l , . . . , a n. (...) denotes set-theoretic ntuple operation, unless otherwise specified.
We warn the reader that set-theoretic symbols will be sometimes adopted as
abbreviations for corresponding non-extensional operations on properties
and predicates. But possible ambiguities will be spared by the context. The
arithmetical symbols are the standard ones.
5.7. Provability and standard Tarskian semantics.
~ I = A stands for "A holds in the structure Eft,".
S F A means that A is derivable from S by means of classical logic (unless
otherwise specified).
5.8. Inductive proofs. We often carry out proofs by induction (either in the
metatheory or within axiomatic theories). As a rule, we adopt the acronym
IH as a shortening for "induction hypothesis".
PART A
COMBINATORS AND TRUTH
~r v a s t & "V r ~ v a l r o p ~ v
" M//~'~ #c7' c~lrrls: cvL ")'~p, ~ #a~c&pLc, carL,
'"
~1 #cT"Larrl ~a't 7rpJorrI. 7rcp'~ ~/&p avrrlv a v r o v riTv apxrlv o ~ a rvTx~vet"
(Plato, Soph.238a)
This Page Intentionally Left Blank
CHAPTER 1
INTRODUCING OPERATIONS
w
w
w
w
w
w
w
The basic language
Operations I: general facts
Operations II: elementary recursion theory
The Church-Rosser theorem
Term models
The graph model
An effective version of the extensional model D oo
Appendix
This chapter contains an elementary introduction to combinatory logic. The
topic is highly developed, but the chapter has quite a limited aim: that of
yielding all the necessary prerequisites and making the book self-contained.
According to the informal ideas outlined in the general introduction, we
aim at investigating an axiomatic notion of abstract logical system, whose
ground structure (the abstract syntax) is a combinatory algebra, extended
with suitable built-in operations and with a primitive notion N of natural
number. The choice of N is largely a matter of convenience and tradition;
the basic constructions do not depend on the initial stock of built-in
predicates and operations.
The central aim of this chapter is to clarify what we understand by
ground structure. We begin in the axiomatic style and we describe a formal
system OP for a type-free theory of operations; we then outline three basic
semantic constructions. We underline that the basic constructions can be
carried out in OP itself.
After the description of the formal language (w we define OP and we
discuss its general features (w closure under /?-conversion, fixed point
theorem, relation with )~-calculus), while w reviews some basic facts on
recursion theory. We then present the term models of OP, which are based
on the fundamental Church-Rosser theorem (w
In w we give an
elementary description of the Plotkin-Scott graph model Pw, together with
its recursive submodel R E and Engeler's construction. Finally, following an
elegant procedure, due to Scott (1976, 1980), we show how to isolate an
extensional submodel D oo of RE.
Introducing Operations
14
[Ch.1
w The basic language
We describe an axiomatic theory of operations OP, which is a first-order
extension of pure combinatory logic by simple number-theoretic notions. OP
is proof-theoretically equivalent to PA, the elementary system of Peano
arithmetic, s
it will constitute the basis of all systems to be investigated
in this book.
The basic language 2, contains:
(i) countably many individual variables Xl, x2, x3, ... ;
(ii) the logical constants -1, A, V;
(iii) the individual constants K (constant function combinator), S
(composition combinator), SUC (successor), P R E D (predecessor), P A I R
(ordered pair operation), L E F T (left projection), R I G H T (right projection),
0 (zero), D (definition by cases on numbers);
(iv) the binary function symbol Ap (application operation) and the
predicate symbols N (natural numbers), T (truth), = (equality).
Terms are inductively defined from variables and constants via
application of Ap. We use x, y, z, u, v, w, f, g as metavariables; while t, t',
t ' , s, s ~, r, r ~, etc., are metavariables for terms. We write (ts) instead of
Ap(t,s), and outer brackets are usually omitted, while the missing ones are
restored by associating to the left; for instance, xyz stands for ((xy)z).
We adopt familiar shorthands for special terms: t + 1 : - SUCt ( - the
successor of t); (t,s):-- P A I R t s ( = the ordered pair composed by t and s);
(t)i := L E F T t ( - t h e left projection of t ) a n d (t)2 : - R I G H T t
( = the
right projection of t).
Formulas are inductively generated by means of the logical operations
from atomic formulas (atoms, in short) of the form t = s, Nt, Tt.
A, B, C are syntactical variables for formulas of 2,. As to the
syntactical notions of free and bound variable, substitution, etc., we follow
the standard conventions and terminology (Shoenfield 1967). In particular,
if E is an expression (term or formula), E(x) means that x may occur free
in E, while E[x := t] stands for the result of substituting t for the free
occurrences of x (provided t is substitutable for x in E). FV(E) is the set of
free variables of the expression E; x E FV(E) means that "x occurs free in
E~
9
The remaining logical symbols are defined classically:
3xA := -~Vx~A; A V B := -~(-~A A -~B); A - , B := ~A V B ;
A + B := (A---,B) A (B---,A).
We stick to the usual convention that --1 and quantifiers bind more than the
remaining connectives, while A, V bind more than --, and ~-,; sometimes
dots are used in place of parentheses (see w of the introduction).
Basic language
1.1]
15
As usual, a numeral is any term obtained from the constant zero by
means of a finite number of successor applications; if n E w (w - the set of
natural numbers), ~ stands for the n-th numeral, i.e. the term built up
from 0 with n applications of S U C .
We now recall the standard definition of A-abstraction in combinatory
logic.
1.1. D E F I N I T I O N . If t is an arbitrary term of s
induction on the notion of s
(i) Ax.x "- S K K ;
(ii) Ax.t : - K t if x it FV(t);
(iii) A x . ( t s ) ' - S ( A x . t ) ( A x . s ) , if x E FV(ts)
A x . t is introduced by
Of course Ax.t has exactly the same free variables as t, minus
Coding of n-tuples can be obviously defined by iteration of ( , ).
x.
1.2. We inductively put 9( t ) " - t and ( t l , . . . , tk+i} "-- ((ti, . . . , t k ) , tk+i).
If 1 _<i _<k, we define projections for k-tuples:
if/>
(t)kl "-- L E F T ( k - l ) t ;
1, (t) k "- R I G H T ( L E F T ( k - i ) t ) ,
where L E F T ( ~
" - t and L E F T ( k + I ) t " - L E F T ( L E F T ( k ) t ) .
index is clear, e. g. if t - ( t l , . . .
, tk) , we write (t)i for (t) k.
If the upper
w Operations I " general facts
OP, our basic theory of operations, is based on PC, classical predicate
calculus with equality in Hilbert style presentation (the equality axioms
being given by the formula
wvyvz(~
A
-
9 A (~ -
y ~
y -
~) A (~ -- y A y -- z ~
9 -- z )) A
V x V y V z ( x -- y ~ (zx -- zy A xz -- yz) A ( N x ----,N y ) A ( T x ~ T y ) ) .
The non-logical axioms of OP are listed in 2.1 below.
2.1. A x i o m s f o r operations and natural numbers
COMB
W, V y V z ( K x y
- x A
PAIR
W, V y ( ( ( x , Y>)i -- x A (<~,Y))2 -- Y);
NAT.1
NO A V x ( N x -~ ( N ( x + 1) A --(x + 1) -- 0 A P R E D ( x + I )
NAT.2
V x V y V u V v ( N x A N y A -~x -- y --, D x x u v - u A D x y u v - v);
NIND
A(O) A V x ( A ( x ) - - ~ A ( x + l ) ) - - , V x ( N x --, A ( x ) ) ( A arbitrary).
S~yz
--
~(yz));
- x));
Introducing Operations
16
[Ch. !
Clearly OP states that the universe is a combinatory algebra, which is
suitably extended with natural numbers. It is here important to mention a
natural extension of OP, which includes the extensionality axiom for
operations"
Ext op
Vx(zx - yx) ~ z - y.
CONVENTION. If S is any system containing NIND, S - - S minus NIND.
Similarly, if S- is any system not containing NIND, S - S- plus NIND.
We now recall the basic elementary facts underlying the pure theory of
combinators and A-calculus. Of course, the results below are quite standard
and are quickly surveyed for the reader's sake. To this aim, we first restrict
our attention to the induction free fragments O P - and O P - + E x t o p of OP.
A-abstraction, as defined in 1.1, commutes with substitution and satisfies aand fl-conversion. It also satisfies 7/- and ~-conversion, in presence of
extensionality; projections of k-tuples, as defined in 1.2, verify the due
equations.
2.2. P R O P O S I T I O N
1. The following formulas are provable in OP-, for arbitrary t,s"
(i)
a-conversion
Ax.t = Ay.t[x := y];
(ii)
fl-conversion:
(Ax.t)s = t[x := s];
(iii)
(Ay.t)[x := s] = Ay.(t[x := s]).
Proviso: in (i) y ~ FV(t) ; in (iii)y ~ F V ( s ) U {x}.
2. The following formulas are provable in O P - + Extop , for arbitrary t, s"
(i) 5-conversion:
(ii)
~-conversion:
Vx(t-
s)---, A x . t - Ax.s;
Ax.tx-t
(x ~ FV(t)).
3. For each k, i C ~ such that 1 <_i <_k, O P - proves:
((Xl,...,Xk))i--X
i.
PROOF. 1. (i)-(iii) follow by straightforward induction on the build up of t.
2. (i)-(ii)" by/%conversion and Extop.
3: argue by induction on k with the axiom P A I R . D
In (i) and ( i i i ) - can be replaced by literal identity - . Closed term are
sometimes called combinators. In combinatory logic every operation has a
fixed point; indeed, a stronger result holds:
2.3. T H E O R E M (Fixed-point).
o e - t- V f ( f ( F P f )
- FPf).
We can find a combinator F P
such that
Operations
1.2]
PROOF: define
fl-conversion:
17
FP:=Af.(Ax.f(xx))(Ax.f(xx)).
Then
we
have
by
F P f = (Ax. f(xx))(Ax, f(xx)) = f((Ax, f(xx))(Ax, f(xx))) = f ( F P f ) .
0
FP is Curry's paradoxical combinator Y; for other fixed point combinators,
see Barendregt (1984), pp.131-32.
We now introduce, following Scott (1980a) and Meyer (1982), certain
combinators, which characterize (the first-order theory of) A-calculus as a
fragment of OP-.
2.4. DEFINITION
(i) I := Ax.x and 1 := S(KI);
(ii) 10 := I and lk+ 1 := S(Klk).
2.5. LEMMA (provable in O P - + E x t o p )
MS.1
MS.2
MS.3
MS.4
Vx(yx = zx)-~ ly = lz;
12K = K;
13S = S;
1 -- I.
The proof is a routine application of extensionality and the axioms for
K and S. The reason for these esoteric equations is that MS.1-MS.3
axiomatize (-conversion over O P - and yield all together the full strength
of Extop.
2.6. Let O P A - : = O P - + MS.1-MS.3 .
2.7. PROPOSITION
(i) oP -
(2.2.2 (i)).
ch ma of
(ii) O P A - + ( 1 - I) proves Extop.
PROOF. (ii)is immediate by MS.1. As to (i), assume Vx(t = s); then by flconversion
=
which yields with MS.1
l(Ax.t) = l(Ax.s).
(1)
But MS.2-MS.3 imply K x = l ( K x ) and Sxy = l(Sxy); for instance, we
have;
Sxy = S(K12)Sxy = ((K12)x(Sx))y =
18
Introducing Operations
-- (12(Sx))y - S ( K 1 ) ( S x ) y - ( K 1 ) y ( S x y )
[Ch.1
- l(Sxy).
Hence by definition of A, we have A x . t - l(Ax.t) and the conclusion follows
by (1). 13
Of course, we can assume A-abstraction as primitive, and define:
s
:= (s {A})-{K,S}. In the new language s
terms are inductively
generated by the new clause: if t is a term, Ax.t is a term and
FV(Ax.t)= F V ( t ) - { x } ( = x is bound in Ax.t). We also identify terms
differing only by the names of their bound variables (e.g. Ax.x is identified
with Ay.y).
2.8. DEFINITION. A is the first-order theory in the language s
which
is obtained from OP by omitting COMB and by assuming the schemata of
fl- and ~-conversion (in fl-conversion it is understood that s is substitutable
for x in t).
2.9. THEOREM. We can define maps C" s163
and L" s163
that, if ~g - OPA- (OP-+Extop), Y - A- (A-+Extop), then for every
L-formula A and every s
B,
(i)
~Y F A ~ C ( L ( A ) )
such
and Y F B ~ L(C(B));
(ii)
~Y F A iff Y F- L(A); Y F B iff ~Y F C(B).
(The exponent - m e a n s that the number-theoretic induction axioms are
omitted).
As to the verification of 2.9, C ( B ) is the L-formula which results by
replacing in A each term of the form Ax.t with the corresponding term,
built up from K , S and defined in w L(A) is the s
which is
obtained from A by replacing K, S by AxAy.x and AxAyAz.xz(yz) in the
given order (A primitive symbol). The proof requires some work and
computations (apply 2.2, 2.5, 2.7 or see Barendregt 1984).
After 2.9, we henceforth identify OPA (OP+Extop) with h (A+Exto.p).
It is well-known that the inclusions OP C OPA C OP+Extop are proper, 1.e.
there are models of OP which refute MS.1-MS.3 and there are models of
OPA falsifying MS.4 (see w167
w Operations H " elementary recursion theory
In this section we review a few basic results of elementary recursion theory
on w, formalized in the setting of combinatory logic. Since this topic is
thoroughly developed in the literature (see Barendregt 1984, Hindley-Seldin
1.3]
Elementary Recursion Theory
19
1986), we only outline the main steps.
CONVENTION. In formal contexts m, n,k (also with indices or apices) are
syntactical variables for bound number variables; thus we write Vm, 3m
instead of Vx(gx---~...), 3 x ( g x A . . . ) . Informally n, m, k, p, q range over
natural numbers in contexts of the form n E w, k E w; n, m, k, etc., denote
numerals.
3.1. LEMMA. OP proves the following sentences:
(i) A(0) A Vn(A(n)---, A ( n § 1))---, VnA(n) , A arbitrary;
(ii)
Vn(n - 0 V 3 m ( m + l - n));
(iii)
V n V m ( n + l - m + l ~ n - m);
(iv)
Vn(-,n - 0 ~ N ( P R E D n ) A ( P R E D n ) + I - n);
(v)
Vn(n+lCn)
Agr
P R O O F . (i)-(ii): apply N-induction, identity logic and the consequence of
the axiom NAT.1 N0 A v x ( g x - - - , g ( x + l ) ) to the formulas
A'(x) "- g x A A ( x ) and A ( x ) "- (x - 0 V 3 y ( g y A y + l -- x)).
(iii)" if n + l - m + l , also P R E D ( n + I ) P R E D ( m + I ) , whence n - m
by
use of V n ( P R E D ( n §
- n).
(iv): i f - - n - 0, then there exists an m such that n - m + l , by (ii); hence
m - F R E D ( m - t - l ) - F R E D ( n ) and n - m + 1 - P R E D ( n ) T 1.
Since N m , also N ( P R E D n ) .
(v): the first conjunct follows by N-induction, NAT.1 and (iii); it also
implies 3x3y(x ys y); but K - S yields V x V y ( x - y) by a well-known trick.
O
3.2. LEMMA (Existence of recursor on natural numbers). We can find a
closed term R N such that, provably in OP"
VmVyVz(R gOyz - y A RN(m+l)yz
- zm(RNmYZ)).
P R O O F : choose t - )~w)~x)~y)~z.DxOy(z(PREDx)(w(PREDx)yz)).
Define R N " - F P t (by 2.3); then apply ~-conversion and definition by
cases on N.D
Clearly, we can define the standard primitive recursive functions and
predicates by means of R N and then verify by NIND that the usual
definitions are correct; in particular, we have a formula x < y which
represents the standard less-relation on numbers and satisfies its elementary
properties, provably in OF.
20
Introducing Operations
[Ch.1
3.3. DEFINITION
(i)
(ii)
f " N - ~ N "- V n . N ( f n ) ( - f is a number-theoretic operation);
Vn < m . A "- Vn(n < m ~ A) and 3n < m . A " - 3 n ( n < m A A).
3.4. LEMMA. We can find a closed term #, such that, provably in OP:
gk - O A Vm < k. gm > O-~ #g - k.
PROOF. By fixed point theorem, we find a term h such that
h - ) ~ g A z . D O ( g z ) z ( h g ( z + l ) ) , and we choose # ' - ) ~ g . ( h g O ) . Assume that
gk -- O A V m < k. gm > O. Then hgk - k and h g m - hg(m + l ), if m < k .
By induction we verify h g k - h g O - k. []
3.5. DEFINITION. A partial number-theoretic function F" w - ~ w is
representable in a theory ~T (in the language L) iff there is a closed term f
such that:
F ( n l , . . . , nk) ~_ n iff ~ F f n l ' - " nk -- n;
( n l , . . . , n k are arbitrary natural numbers; ___ is Kleene's notation and
F ( n a , . . . , nk) "~ n means that F ( n l , . . . , nk) is defined and has value n).
3.6. THEOREM. The partial recursive functions are representable in OP.
PROOF. S U C , KO, )~Xl...)~Xn.X i represent the initial functions successor,
constant-zero, projections (respectively). The recursor and minimalization
operators exist by 3.3-3.4; the substitution operator is immediately available
by )~-abstraction. F1
3.6.1. REMARK. The representing combinators in 3.6 can always be chosen
in normal form and such that, if F ( n l , . . . ,nk) diverges, then
f ~ i . . . ~ k - f 2 - ()~x.xx)()~x. xx); cf. w below and narendregt, cit. ,p.179.
By 3.6, we denote the standard primitive recursive number-theoretic
predicates (e.g. the ordering relation on w) by their customary symbols.
Strictly speaking, if P is a primitive recursive predicate, P x stands for the
quantifier-free formula f x - O ,
where f is a term representing the
characteristic function of P. It is also clear that OP can provably formalize
the standard facts of elementary recursion theory s la Kleene. In the
following, we shall adopt the bracket notation { a } ( x ) ~ _ y without
distinguishing it from its formal presentation in OP. We conclude with a
few observations.
3.7. First of all, the distinction between operations and functions in the
set-theoretic sense has interesting conceptual consequences. Let Church's
thesis be the statement"
CT
Vf(f " N ~ i. ~ qnVm({n}(m) ~ fm)).
1.3]
E l e m e n t a r y Recursion Theory
21
Then CT is consistent with the basic theories we consider in this book, even
if full classical logic is used (see 4.11 below).
3.8. It is well-known (see Barendregt 1984, Hindley-Seldin 1986) that
numerals, successor, predecessor, definition by cases on numerals and
pairing are representable in the theory CL of pure combinatory logic. By CL
we here understand the subsystem of O P - , formalized in the sublanguage
of s which only contains the function symbols Ap, K, S, the predicate - ,
variables and logical operators. The only non-logical axioms of CL are
COMB and (-~K - S). Here follow the basic steps.
(a)
Let T " - g and F " - K I ( I is the identity combinator).
Pairing: P A I R x y " - )~u.uxy; L E F T x " - x T ; R I G H T x
"- xF.
Then CL ~- T x y - x A F x y - y A ( P A I R x l X 2 ) i - x i (i - 1, 2).
(b)
Numerals:
0 " - I; S U C - ~ " - P A I R - ~ K ;
PRED-~ "- LEFT-~;
Z~ "- (RIGHT~)FT.
Then, for arbitrary n, m E w,
CL F- Z0 - T A Z ( S U C - ~ ) - F
and CL F- -~ 0 - S U C - ~ A ( S U C ~ - S U C - ~ ~ ~ - -~) (apply -~K - S). By
fixed point, choose
G-
and let D else G ~ -
AxAy.(Zx)(Zy)((Zy)(Zx)(G(LEFTx)(LEFTy)))
A x A y A a A b . ( G x y ) a b . Then CL proves that, if ~ - ~ , G ~ F. Hence by the properties of T, F we are done.
T,
Notice that, once we choose to enlarge combinatory logic by standard
numerals 0, S U C O , etc., and we assume S U C as a primitive constant, we
are forced to introduce P R E D and D: without them, it would be impossible
to define a number-theoretic recursion operator (see Curry et al. 1972, vol.II
w 13.A.3, theorem 2).
3.9. On f u l l d e f i n i t i o n by cases.
Let DIS be a new constant satisfying the axiom
3.9.1
V x V y ( ( x - y A DISxy - 0) V (-~x - y A DISxy - 1)).
Then 3.9.1 is inconsistent with CL.
P R O O F (folklore). Let N e g ( x ) - DISxl. By 2.3 we can find an e such that
e - i e g ( e ) . Hence we have that e - T implies e - i e g ( 1 ) - O, and - ~ e - T
implies e - Neg(e) - 1. F1
It follows:
3.9.2. CL plus the statement " e v e r y t h i n g is a n u m b e r " is inconsistent.
Introducing Operations
22
[Ch.1
Indeed, if we apply the above trick to D, we get an e such that -~Ne.
3.9.3. There cannot exist an injective operation f from the universe into the
natural numbers (define by D an operation h such that h x - 1 , if f x - fO
and hx - O, if -,fx - f0; any fixed point of h leads to a contradiction).
g4A. The Church-Rosser t h e o r e m
We are going to construct term models for the non-extensional theory OP of
operations. The strategy is well-known and it relies upon a fundamental
result of Church and Rosser (1936). In order to ensure that the given theory
of rules is consistent, we prove that c o m p u t a t i o n s - regardless of the various
patterns we may follow- give unambiguous results. This technique regards
the equality relation, as inductively generated by an asymmetric reduction
relation, which splits the computation process into basic atomic steps. For a
thorough treatment of the subject, we send the reader to Barendregt (1984),
Hindley-Seldin (1986).
In the following we deal with the term fragment of the basic language s
which contains individual variables and the individual constants K, S,
SUC, P R E D , O, D, P A I R , L E F T , R I G H T . Terms are inductively
generated by application from variables and individual constants.
(w
4.1. DEFINITION OF REDUCTION. (a) The reduction relation > is the
smallest binary relation among terms, which satisfies the clauses below"
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
> is reflexive ( R ) a n d transitive (T);
> is preserved by application, namely:
t > s, t ' > s' imply t t ' > ss';
K t s >_ t;
Strs > ts(rs);
L E F T ( P A I R t s ) > t and R I G H T ( P A I R t s )
PRED(SUC-~) > -if;
n f i f i t r >_ t;
D-~-~tr > r, if n # m;
> s;
(A)
(K)
(S)
(P)
(SUC)
(n.1)
(D.2)
(b) We then read t > r as "t reduces to r" and the clauses (iii)-(vii) are
called proper reductions.
(c) The terms which are on the left (right) of the proper reductions are
called redexes ( contracta).
(d) A term t is normal, or in normal form (in short t E NF), iff no
subterm of t (t included) is a redex. For instance, every numeral or basic
constant is normal; however, there are terms without normal form, the most
1.4A]
23
Church-Rosser Theorem
typical a m o n g t h e m being gt -()~x.xx)()~x.xx).
A reduction s t a t e m e n t t >_ s can be conveniently regarded as a derivable
formula of a formal system, where (R) and proper reductions play the role
of axioms, while the inference rules are (T) and (A). Hence t>_ s holds iff
there exists a derivation in tree-form which is locally correct with respect to
the axioms and the rules, and which has t _ s at its root; thus we can
recursively assign a length to reductions:
4.2. D E F I N I T I O N (reduction with length), t >_ n s iff
(i) 0 - n and t _ s is a proper reduction or an application of (R), or
(ii) there are k, rn < u E w and either t >_ k r, r >__m s, for some r, or
t' _ k r', t" --> m r" and t - t' t", s - r' r", for some t', t", r', r".
(In the following sections of this chapter - is literal identity).
4.3. T H E C H U R C H - R O S S E R P R O P E R T Y CR
CR states that the reduction relation _ is directed (confluent)" for all t, t',
t" if t > t' and t > t", there exists a term s such that t' > s and t" > s
We verify that C R holds for a relation R E D , whose transitive closure is >_.
4.4. (i) t R E D n s is defined by replacing everywhere _ by R E D in 4.2 and
by omitting the transitivity rule in 4.2 (ii);
(ii) t R E D s " - " t R E D I r s, for some k G w".
Obviously, t __ s holds iff t >_ k s holds for some k.
4.5. LEMMA. t ~ s iff there
t 1 R E D t 2 , . . . , t k _ 1 R E D t k.
are t l , . . . ,t k such that t l - t ,
t k - s and
The proof is i m m e d i a t e by induction on the length of the derivation. If we
replace >_ by R E D in 4.3, we have a statement of the Church-Rosser
property for R E D . It is also clear that we can obtain, by simple diagram
chasing:
4.6. LEMMA. I f C R holds f o r R E D , C R holds f o r > .
4.7. L E M M A ( A n a l y s i s of R E D )
(i)
(ii)
(iii)
I f C is an arbitrary constant and C R E D t,
then t -
C.
I f C - K , S , D and C s R E D t , then there is a t' such that
s R E D t' and t - C t ' .
I f C - S , D , P A I R and C t l t 2 R E D t
t i R E D t ~ ( = 1, 2) and t - C t l t ' 2.
, there are t], t'2 such that
Introducing Operations
24
(iv)
[Ch.1
I f C - D and C t l t 2 t 3 R E D t, there are t'l, t'2, t'3 such that
tiREDt~(i-1,
2, 3) a n d C t ' i t ' 2 t'3 - t .
P R O O F . (i) C R E D t can hold only by (R).
(ii) Let C s R E D O t: since C ~ P R E D , L E F T , R I G H T , we must have
applied (R) and we are done. Let C s R E D m t be derived by (A): we have
C R E D a t ' a n d s P E D kt'', for some n, k, t', t" such that m > k , n and
t - t't". One application of (i) yields the conclusion.
(iii) If C t l t 2 R E D o t, t must have the form C t l t 2 , because C ~ K.
If m, k < n and C t 1 R E D ks1, t 2 R E D ms2, we can find by (ii) a t~ such
that t l R E D t ' 1 and s I - C t ' 1" hence we have t~, t ~ - s 2 ,
such that
t i R E D t~, ( i - 1, 2) and t - Ct'lt'2.
(iv): by similar arguments resting on (iii). 13
4.8. T H E O R E M .
CR holds f o r R E D .
P R O O F . Assume t R E D n s l , t R E D ms2: we produce a term r such that
s 1 R E D r, s 2 R E D r. We argue by induction on l = n + m .
If 1 = 0 and (R) is applied on both sides, choose r - s 1 - s 2. If there is a
proper reduction, choose r as the result of the contraction. Let l > 0.
Case 1. One of the given derivations has length 0. By symmetry, it is
not restrictive to assume t R E D o s 1. We analyse its derivation ~1"
1.1 (R) is the inference applied in ~1: choose r - s 2.
1.2. (R) is not applied. We have to distinguish a few subcases.
1.2.1. Let (K) be the inference applied in ~1" Then t - K t l t
2 and
s I = t 1. Since m > 0, the last inference of the derivation ~2 (of t R E D m s2)
must be (A): K t 1 R E D k rl, t 2 R E D n r2, where f i r 2 = s 2 and k, n < m.
By 4.7 (ii), there is a t~ such that r I - K t ' 1 and t 1 R E D t"l. t'l is the right
choice.
1.2.2. Let (P) be the inference applied in ~1" for definiteness, assume
that we apply left projection. Then t - L E F T ( P A I R
tit2); in ~2 we have,
for some rl, r2,
L E F T R E D rl, ( P A I R t l t 2 ) R E D r 2.
Hence by 4.7 (i) and (iii), r I - L E F T and we find t~, t~ such that
t i R E D t~ ( i - 1, 2 ) a n d r 2 - P A I R t'1 t'2. Choose r - t~.
1.2.3 Let (S) be applied in ~1" Then t - S t l t 2 t 3 and s I - tlt3(t2t3); by
4.7 (iii), and noting that ( A ) i s applied in ~2, we find t~, t~, t~ such that
,
t i R E D t i, (1 _< i _< 3) and s 2 - S t l, t 2, t 3.
Choose r _ tl,t,(t,t,3)"
3~ 2
1.2.4 Let
(D.1)
be the
last
inference
of ~1" then
t R E D t I and
Church-Rosser Theorem
1.4A]
25
I
I
I
t-D-ff-fftlt2,
t1
s 1. By 4.7 (iv), we find t3,
t4,
tl,
t 2I such that
I
I
I
I
I
s 2 -- D t 3 t 4 t i t 2 and ~ R E D t'3, ~ R E D t'4, t I R E D t'a, t 2 R E D t 2. Since n is
I
~
I
I
normal, t 3
t 4 -- ~: hence we can choose r - t 1.
If D.2 is applied, t - D ~ t l t
2 with ~ distinct from ~ and s I - t 2. By
normality of ~, ~ and 4.7(iv), we find a t~ such that t 2 R E D t ' 2 and
I
s 2 R E D t 2.
1.2.5. Assume t - P R E D ( S U C ~ )
s2-PREDs'
, where ( S U C - f f ) R E D s ' :
s ' - S U C ~ and we can choose r - ~.
and s 1 - - n .
Then for some s',
but S U C ~
is normal, hence
Case 2. Assume n, m > 0: hence (A) is the last inference in both
derivations. We m a y suppose
2.1 t I R E D k t'x, t2 R E D m
t'l' and t - txt2, s I - t'lt'{, where k, m < n;
2.2 t i R E D p t'2, t 2 R E D q t'2' and s 2 - t~ t~', where p, q < m.
But k + p, m + q < n + m and by IH, there are t', t" such that
t '1 R E D t', t 2' R E D t ' a n d t 1" R E D
By
r -
(A) ,
t't".
81 - -
t'lt'1' R E D t't"
and
"'"~2~2
t"
, t 2" R E D
s 2 [ t E D t't":
t"
.
hence
we
choose
13
4.8.1. R E M A R K . It is essential to the proof that D be restricted to
numerals and, more generally, to closed normal terms (see Church's calculus
of A&conversion in Church
1941, Barendregt
1984). There are
counterexamples to C R in the case, where P A I R
satisfies surjectivity,
namely P A I R ( ( L E F T t ) ( R I G H T t ) ) > t ( s e e
Barendregt 1984, p.403).
However, this last reduction gives rise to a consistent convertibility relation
by w
If we apply 4.8, 4.7 and the definition of normal form, we get
4.9. C O R O L L A R Y . (i) C R holds f o r > .
(ii) I f t ,I t l l E NF and r _>_ t', r _>_ t" then t' and t" coincide ( u n i q u e n e s s of
normal form).
4.9.1. R E M A R K . (a) A notion of reduction which enjoys CR, can be defined
for the A-formalism, plus ~- and ~-conversion (see w
(b)
There exist several reduction strategies. However, for combinatory
logic, there is a standard reduction procedure SR, whose main virtue is
condensed in the Standardization Theorem (Curry-Feys 1958). If t has a
normal form, then SR terminates and yields the normal form of t.
(c)
The property of having a normal form is recursively enumerable,
but not recursive, by a classical result of Church.
Introducing Operations
26
[Ch.1
~4B. T e r m Models
We are ready
operations.
to
introduce
the
syntactical
models
of the
theory
of
4.10. D E F I N I T I O N
(i) C T M := { t : t is a closed term};
T M " - {t 9t is an arbitrary term}.
(ii) The open term model T M is the structure {TM, 9 , - ,
where the e m p t y set interprets the truth predicate T and
C,N,T),
m
1.
2.
.
4.
e - {K, S, D, P A I R , L E F T , R I G H T , SUC, P R E D , 0};
9 9T M 2 ~ TM is the operation of juxtaposition of terms (i.e.
application);
= C T M 2 and t - s holds iff t > r and s > r, for some term r;
N _C T M and t E N iff t > g, for some n C w.
(iii) The closed term model C T M is the substructure of TM, whose
support is TM.
4.11. T H E O R E M . Let ~ " - C T M or T M . Then ~ is a non-trivial model
of O P + C T , but ~ is not a model of OPA (for notations 2.6, 3.7).
P R O O F (sketch)..At, is non-trivial, because K, S are normal and ~ falsifies
K - S (by unicity of normal form). Again the Church-Rosser theorem
ensures that - is transitive and that ( N t A t - - s ~ N s )
holds in RAt,. In
.At,, -- preserves application, simply because > is closed under (A). The
axioms on the special constants and N are true by definition of > and
since the numerals of ~ are isomorphic to the standard numbers. Consider
the closed term 12K - S ( K 1 ) K where 1 - S ( K ( S K K ) ) . Then -~K - 12K
holds in ,A1, because the given terms are different normal terms; thus the
axiom MS.2 of OP,~ (see 2.5) fails in ,At,.
Assume that f " N--+N (see 3.3) is true in All,; we define F" w--+w by
F(n)-m
iff f g > ~ . By hypothesis, F is total and its graph is trivially
recursively enumerable" hence F is recursive. T h i s - t o g e t h e r with the fact
that ,A1, is a model of OP and OP provably formalizes the standard results
of elementary recursion t h e o r y - yields the soundness of CT. [3
4.12. C O R O L L A R Y . Let ~ " - C T M
(i)
(ii)
- ~ I- t - ~
or TM. If t and s are closed terms,
iff O P F t - s ;
~l, l= N t iff OP F t - g ,
for some numeral g.
P R O O F . F r o m right to left, apply 4.11. As to the opposite direction, it
suffices to verify that t _ r implies OP F t - r. 13
Term Models
1.4B]
27
The proof of CR is constructive and it can be carried out in the
system PA of first-order Peano arithmetic (PA is described in the
appendix). Hence we have"
4.13. THEOREM.
Arithmetic.
OP
is
interpretable
in
the
system
PA
of Peano
4.13 can be sharpened, once we realize that CR is (at least) provable in
PRA, the system of primitive recursive arithmetic. This remark naturally
leads to a subsystem of OP, which is tailored for PRA. Let 3(+) be the
smallest class of L-formulas which is generated from atoms of the form Nt,
t - s by means of A, V and existential quantification; 3(+)-NIND is Ninduction schema restricted to 3(+)-conditions. If we define
OP 1 := O P - + 3 ( + ) - N I N D ,
we can prove:
4.14. THEOREM. If f is a combinator such that OP 1 F f : N ~ N ,
defines a primitive recursive function.
then f
Of course, the proof depends upon a careful formalization of 4.11 (for
more details, see Troelstra 1973, Troelstra and van Dalen 1988, J~iger and
Strahm 1994, and the Appendix). 4.13-4.14 can be further strengthened by
adding to OP and OP 1 some truths of the closed term model CTM. An
important example (to be applied in Ch.VI) is the enumeration axiom:
EA
3 f V x 3 y ( N y A f y -- x).
EA holds in CTM, because there is a closed term which enumerates CTM
(cf. Appendix).
4.15. THEOREM. Theorem 4.13 (4.14) remains true if we replace OP
(OP1) respectively by the system O P + E A + C T (OPI+EA).
We do not know whether Church's thesis can be conservatively added to
O P I + E A . We also mention that there are consistency results involving
continuity of extensional operations, encoding type-2-functionals (see Beeson
1985, Troelstra and van Dalen 1988).
Introducing Operations
28
[Ch.1
w The graph model
In this section we describe the classical graph model Pw, due to Plotkin and
Scott, its recursive submodel R E and Engeler's DM-models. Pw, D M and
R E verify OPA, i.e. OP plus the Meyer-Scott axioms (cf. 2.5) and hence
they are models of (the extended) A-calculus with ~-conversion, but without
full extensionality (cf. 2.6). The construction of R E can be carried out in
OP and this fact yields another method to interpret OP plus the MeyerScott axioms into Peano arithmetic (see w While R E validates Church's
thesis, Pw is a model of a strong choice principle AC N on natural numbers"
O P + A C N yields a model of full second-order arithmetic.
Let us fix a few preliminaries. First of all, we let a, b, c, d, x, y, z, u
range over elements of P w - { x ' x
C_ w}, where w is the set of natural
numbers. Of course, if a, b E Pw, a - b stands for extensional equality; n,
m, k, p, i, j range over w. We also adopt the lambda notation informally,
i.e. to name (set-theoretically defined) functions.
If T is the Kleene predicate, W k - { n " 3 m T ( k , n , m ) } - the k-th r.e. set
(r.e -- recursively enumerable) and R E - {x" x E Pw A 3n( W n - x)}.
We also put ( n , m ) - l ( n + m ) ( n + m + l ) + m ;
$x$y.(x, y) is a primitive
recursive bijection of w • w onto w. We also define a canonical enumeration
of the finite subsets of w:
e n - { n o , . . . , n p _ i } , provided n 0 < ... < up_ 1 and n -
~
2 hi, and e0 - 0 .
i<p
F, G, H range over the set of operators ( - total functions) from Pw to Pw.
We set
F C_ G "- F(a) C_ G(a), for every a E Pw.
5.1. D E F I N I T I O N
(i)
GRAPH(F)"-
{(n, m) " m E F(en)};
F U N ( a ) := .~b.{n: 3k(e k _C b A (k, n) E a)} (for a E Pw);
(ii)
a. b := (FUN(a))(b);
(iii)
F is continuous iff F(a) = U {F(ek): e k C_ a} (for every a E Pw);
(iv)
(v)
F is r.e. iff G R A P H ( F )
is r.e;
F is effectively continuous iff F is continuous and r.e.
Pw is a topological space under the so-called positive information topology,
where the basic open sets have the form 0 k - { a ' a C P w A e k C a}, for
some k E w (cf. Scott 1976 or Barendregt 1984).
The Graph Model
1.5]
29
5.2. LEMMA
(i) Continuous operators are monotone, i.e. a C_ b implies F(a) C_ F(b);
(iN) Pw, R E are closed under the application operation .;
(iii) a . x - U {a . e k" e k C_ x} and a . x - tO {ek . x" e k C_ a};
x C y implies a. x C a . y and x . a C y . a ;
(iv) F U N ( a ) is continuous; if ais r.e, so is F U N ( a ) ;
(v) R E is closed under effectively continuous operators;
(vi) I f F C_ G, then G R A P H ( F ) C_ G R A P H ( G ) ;
a C_ b implies F U N ( a ) C_ F U N ( b ) .
The verification of 5.2 is straightforward by definitions of continuity,
application, F U N , G R A P H and the closure properties of R E .
5.3. LEMMA
(i) I f F is continuous, F U N ( G R A P H ( F ) ) F.
(iN) Let H ( a ) - G R A P H ( F U N ( a ) ) : then H is a closure operator ( H is
monotone and it satisfies a C_ H(a) and H ( H ( a ) ) C_ H ( a ) ) .
F U N ( a ) , for some r.e set a.
(iii) F is effectively continuous iff F -
PROOF. (i) By continuity and definition of F U N and G R A P H , we have
m E F(a)r
m E F(en), for some e n C_ a (continuity);
r (n, m) E G R A P H ( F ) ,
for some en C_ a;
r m E FUN(GRAPH(F))(a)
(iN): by (i), 5.2 (iv)-(vi).
(iii)" choose a - G R A P H ( F )
(by 5.1 (i)).
and apply (i). V1
By theorem 2.9, it is enough to interpret in Pw the elementary theory
A of 2.8, whose language L()~) contains the constants 0, S U C , D, P R E D ,
P A I R , L E F T , R I G H T , the binary symbol Ap, the )~-operator and N
(unary predicate constant). If C is a basic constant, [C] is the
interpretation of C in Pw , indeed an r.e subset of w. We freely use the
improper notation (a, n), whenever a - {hi,... , nk} instead of (p,n), p being
the canonical code of a.
5.4 Realization of s
in Pw a n d ' R E .
(i) [ 0 ] - {0};
(iN) [ S U C ] - { ( { n } , n + l ) " n E w };
(iii) [ P R E D ] - {({n + 1}, n)" n E w} ;
(iv) [ P A I R ] - {({n}, (0, 2n))" n E w} U {(0, ({m}, 2m + 1 ) ) ' m E w};
(v) [ L E F T ] -
{({2k}, k)" k E w};
Introducing Operations
30
(vi)
(vii)
[Ch.1
[ [ R I G H T ] = {({2k + 1}, k): k C w};
[D] = {({m},({n},(l,(k,i)))): k,l,m,n,i E ~ A
^ [(m = ~ ^ i e ~ ) v (-~m = ~ ^ i e ~k)]};
(viii) The unary predicate N is interpreted by IN] = {{n}: n E w},
while identity is interpreted by extensional equality.
5.5. A s s i g n m e n t s . An assignment is a map p 9w ~ Pw. If P w is replaced by
R E , p is called RE-assignment. Once p is given, we define, as usual, the
assignment p(i := a) such that p(i := a)(i) = a; p(i := a)(j) = p ( j ) if j :/: i.
If x := xi, we simply write p(x := a) instead of p(i := a).
5.6. INDUCTIVE DEFINITION OF THE VALUE It] M (t arbitrary term
of s
p assignment, M = Pw, R E ) .
(i) If C is one of the basic constants, then [ C ] y - ~C];
(ii)
Wxi] M - p(i);
(iii)
[rs] M - ~r] M . ~s]y;
(iv)
lAx.sly
- GRAPH()~a.[[S]p(Mx := a)) - { ( k , m ) " m E [S]p(x := ek)}.
N.B. In (iv))~x is the syntactical operator applied to the variable x; Aa is
ranging over D - Pw, R E .
Clearly, [[t]y e P w by definition of G R A P H
and 5.2 (ii).
5.7 LEMMA. For every term t and every valuation p,
( i ) ) ~ a [9t ] M
p(x
(ii)
: = ~)
is a continuous operator;,
if y is free f o r x
in t a n d s is free f o r x
[t],(~ := o ) = It Ix .- y]],~, :=
It Ix .- ~]]M _ [timp(~ := l ~ ] i )
(iii)
if p is an R E - a s s i g n m e n t ,
in t,
o);
'
[ t ~ y is r.e
and hence Aa.it~p(
M ._ ,) is
~Z~c.v~ly co~..uou~; mor~ov~ ~tl~ ~ - i t ~ E.
PROOF: by induction on the definition of t. We only check that
Aa'[t]p(Mx . - a ) continuously depends on the interpretation of the variables.
First note that Aa.[t]Mp(z ._ a ) i s monotone (apply 5.2 (iii)).
As to continuity, if t is atomic, the conclusion is trivial. Assume t pick any m e [t]p(Mx ._ a): then for some e k C
- Is] M
p(x := a)' we have
rs;
The Graph Model
1.5]
31
(k,m) E ~r]Mp(x :=a). By Iit, for every i E e k, there is a k(i) such that
ek(i) C_ a and i E ~s ~M~p(x:= %(i))" Since e k is finite, we can find a j such that,
for every i E e k, ek(i) C_ ej C_ a; by monotonicity, e k C_ ~s~Mp(x:= ej)"
by
IH,
llM : = en)
(k,m) E [r .Up(x
for
some
n
such
that
Again
en C a. Choose
p = max(j,n): by monotonicity
C Is] M
p(x
en
-
-
and (k, m) E [r]Mp(x :-- ep) ~
:-- e p )
m E [ t ] ~ . := ev) by definition.
Assume (k, rn) E ~Ay.t]p(Mx : = a) " then by definition m E t ]p(xM
a~ y
ek)"
Hence by IH, there is an n such that e n C_ a and m E [t]Mp(x
ert~ y
i.e. by definition (k, m) E ~Ay.t] M
This
concludes
the
verification
of
p(z := %)"
continuity, since the other direction follows by monotonicity. I-I
Once M -- RE, Pw is fixed or clear from the context, we simply write
[tip instead of It] M. Let p be an assignment in M and let ~ be the
structure for the language L(A) whose domain is M: then the initial
conditions
~l-(t
- s) [p] iff [tip - [sip and
~1-- (Nt) [p] iff [tip E IN] iff for some k E w, [tip -- {k},
fully determine the standard satisfaction relation for L(A)
AI~[= A[p] r
p satisfies A in .Ate,
once we interpret T as the empty set. Then we can state:
5.8. T H E O R E M . . ~ (with M = Pw, R E ) is a model of A (and hence OPA)
plus the surjective pairing axiom V x ( P A I R ( L E F T x ) ( R I G H T x ) =
Moreover .~]=--1 - I and hence Extop fails in MI~ (cf. 2.5).
x).
PROOF. We have to verify the schemata of ~-, a- and /?-conversion (see
2.2). First observe that = is preserved by application (5.2 (iii)). As to (~),
assume [[t]p(x := a) C [S]p(x := a) , for all a E .,~. Then, by definition of
inclusion for operators, Aa.[t]p(x := a) C Aa.[s]p(x := a); hence:
[Ax.t]p C_ lAx.sip (by 5.2 (vi) and 5.6 (iv)).
(c~)-conversion is essentially the first part of 5.7 (ii). As to (fl)-conversion, if
Y = Yi, p(i) -- a,
-
-
F
U
N
(
C
R
A
P
H
._
-
Introducing Operations
32
[Ch. 1
= it]p(x := a) = [[t[x "-- y]]p (apply 5.3, 5.7).
Hence Jig is a model of )~-calculus. By definition of IN], it is easy to check
that I X ] is the least subset of R E , which contains {0} and is closed under
[SUC]. Hence the N-induction schema is satisfied; all the remaining axioms
are verified with 5.4 (as to the surjective pairing axiom, observe that if
nEa,
either n - 2 m
and hence m E [ L E F T a ]
or m - 2 m + l
and
m E ~ R I G H T a ] ) . A simple computation also shows that 1 - I ( - MS.4 of
w fails in R E ; hence Extop is false in R E (and Pw). D
Pw and R E greatly differ as to the interpretation of the space of
number-theoretic operations.
5.9. T H E O R E M . R E is a model of Church's thesis and of the enumeration
axiom (see EA, after 4.14).
P R O O F . EA: this is a by-product of the enumeration theorem for closed
s
(cf. Scott 1976, Barendregt 1984, p.166 or Appendix), plus the
following fact: to each r.e. set a, we can effectively associate a closed
s
re, such that [ta] = a (Scott, cit. or Beeson 1985, pp.133-134).
CT: assume b E R E and for every a E IN], b.a E IN]. Define F ( n ) = m iff
m E b. {n). Then F is a function (as b. {n} is a singleton) and is total ; but
F has an r.e. graph: hence F is recursive . I"1
5.9.1. REMARK. The cited result for R E - {[t]" t closed term of s
should be contrasted with the fact that P w ~ R E ( C strict inclusion),
Pw ~ being the set {~t]: t closed term of pure lambda calculus}. This follows
from 5.7 (iii), 5.8 and the fact that that Pw ~ is not a model of ~-conversion
2.2 by Barendregt (1984, p.514).
5.10. T H E O R E M (Extension). Let X be a subspace of the topological space
Y. If F" X---,Pw is continuous, then there exists a continuous function
F" Y---,Pw, which extends F.
PROOF: one can define, if y E Y,
F(y) - U { M {F(x)" x E X M ~}" y E ~ , Rt open in Y}.
If we endow w with the discrete topology and we identify I N ] with w, w
can be regarded as a subspace of Pw, and hence by 5.10 and choice, we
have:
5.11. C O R O L L A R Y
(i)
If F" w--. Pw, then there is a continuous function F" Pw ~ Pw,
which extends (i.e. F({n}) - F(n), for n E w).
1.5]
The Graph Model
(ii)
33
If R C w x Pw and the domain of R is w, then R has a continuous
choice function, i.e. there exists a continuous operator F such that
for every n E ~, R ( n , F ( n ) ) .
5.12. D E F I N I T I O N
(i) AC N is the schema
Vn3yA(n,y) ~ 3fVnA(n, f n ) (A arbitrary formula of s or s
(ii) ACN! is the schema obtained from AC N by replacing 3y with
3!y( = there exists a unique y).
As an immediate application of 5.11, we obtain:
5.13. T H E O R E M . Pw is a model of OPA+AC N.
It is worth noting that OP)~+ACN! yields a model of full second-order
arithmetic (cf.40.2), if we interpret variables ranging over sets of numbers
simply as operations from w to {0, 1}. We shall see later that the restriction
to numbers in the choice schemata cannot be neglected (because of an
inconsistency); however, there are models of )~-calculus where the choice
schema holds, if the opening universal quantifier is bounded by a type
generated from w (one applies the models of Flagg 1989).
5.14. Generalization: Engeler models
In order to study computability over an arbitrary structure MI, with (nonempty) domain M, one would like to expand Mr, with a set of additional
objects, which represent programs or rules of constructions. Engeler (1981)
produces simple generalizations of the model Pw, the so-called D M-models ,
which meet the above desiderata.
Below, lower case Greek letters c~, /3, -/ stand for finite subsets of M. If
a c_ M is a (finite) subset of M, we write (a~b) for the ordered pair (a,b).
The notation is suggestive of the situation, where the elements of M are
atoms of a given language and (a~b) is a program clause with b as head
and a as body (see Engeler 1988).
5.14.1. D E F I N I T I O N
(i) Go(M ) - M;
Gk+I(M ) - Gk(M ) U {(a~b)" a C Gk(M), a finite and b E Gk(M)};
G(M) - U {Gn(M ) 9n C w}.
(ii) If X, Y C_ G(M) and ~P(G(M))is the power set of G(M), we define
application by
X , Y "- {b e G(M)" 3o~ C_ Y.(o~--,b) e X}.
34
Introducing Operations
[Ch.1
(iii) If F ' ~ ( G ( M ) ) - ~ ~P(G(M))is a continuous function (i.e. it is
C-monotone and preserves U ), )~-abstraction is interpreted with
the natural generalization of G R A P H :
/~M(F) "-- {(fl, a)" fl C_ G ( M ) and a E F(fl)}.
(iv) Finally, we put
D M "--(~(G(M)), *, )~M)"
It is then clear how to adapt the previous results in order to make sense of
the following
5.15. THEOREM. D M is a )~-model of OP)~.
5.16. REMARK. Graph models form a significant class of )~-models and
they have been studied in general. We mention that D M and Pw are not
isomorphic; Pw has no non-trivial automorphism (Schellinx 1991). For
extensive information on models of )~-calculus and combinatory logic, the
interested reader might consult Scott (1982), Longo(1983), Koymans (1984),
narendregt(1984), Lambek and Scott(1986), Hindley and Seldin(1986),
Asperti and Longo (1990).
w
A n effective version of the extensional m o d e l D oo
Henceforth we work within the model R E of the previous section; we use
capital letters X, Y, Z, P, Q for r.e. sets and we freely write X ( Y ) for
X - Y , and • X . P ( X ) for the graph of the operator X ~ P ( X ) .
By 5.3 (ii), R E satisfies the laws:
(7/-):
Q c_ ~ X . Q ( X )
(C):
Z C_ Y implies )~X.Z(X) C )~X.Y(X).
6.1. DEFINITION (after Scott 1976, 1980)
(i) P o Q := )~X.P(Q(X));
(ii) D o : = A X . X = I = { ( n , m ) : m E e n } ;
Dn+ 1 := ,~P.D n o P o D n.
Of course the operation o is associative.
6.2. LEMMA
1. We can find a primitive recursive function a such that for every n E w,
1.6]
The Extensional Model Doo
35
W~(n)-D n.
2. F o r every n,
(i)
(ii)
D n C Dn+i;
D n o D n - D n.
P R O O F . 1. The definition of A X . W a o X o W a is uniform in a and hence
there exists a primitive recursive function r such that
Wr(a)
--
) ~ X . W a O X O W a.
If we let (r(0) - i n d e x for the r.e. set I, ~r(n+l) - r(er(n)), we are done.
2. If n - 0 ,
(ii)is
trivial. By ( ~ - ) , Y C A X . Y ( X ) ;
hence by (~*),
D o - A Y . Y C_ A Y A X . Y ( X ) 0 1 , which verifies (i).
Induction step: we assume D n C_ D n + 1 and D n - D n o D n. Then we have
by monotonicity on the right and on the left (5.2 (iii)) plus IH,
Dn(P(Dn(X))
) C Dn+l(P(Dn+l(X))
).
(+)
D ~ + 1 c_ D n + 2 is a consequence of (+) and ((*). As to (ii), we have by
associativity of o and IH,
(D,~+I o D ~ + I ) ( P ) - D . + I ( D . + I ( P ) )
- Dn+I(D,~ o P o Dn) -
= D n o (D n o P o Dn)o D n - (D n o Dn)o P o (D n o Dn) = D n o P o D n -- D n + I ( P ).
Hence
D n + 1 o D n + 1 -- D n + 1
6.3. D E F I N I T I O N
6.3.1. F A C T .
by (~*). I-!
D o o - - tO { D n 9n C w}.
Dc~ is r.e.
P R O O F : by 6.2.1 Doo is the union of a primitive recursive family of r.e.
sets. ['1
N.B. By the proof of 5.9 and 5.9.1, there is a closed term t of L(A,) with
It] P w - Doo; but no such term exists in pure A-calculus by Longo (1983,
p.170).
We now verify that the collection of r.e. fixed points of F U N ( D o o ) is the
required extensional A-model.
6.4. MAIN L E M M A
(i)
D oo - D oo o Doo
(ii)
Doo - A P . D o o o P o Doo.
Introducing Operations
36
[Ch.1
PROOF. (i) D ~ C_ D ~ o D ~ : by monotonicity of application, 6.2.2 (ii).
0o0o D ~ C_ D ~ : if m E D ~ ( D ~ ( e k ) ), by continuity, there exists a j such
that ej C Dc~(ek) and m E Dc~(ej). Since ej is finite, we find a q such that
ejC_Dq(ek) and by monotonicity of application m E D~(Dq(ek) ). By
continuity of application in the first coordinate, and since )~x.D x is
increasing, we finally get, for p big enough, by 6.2 and monotonicity of
application,
m E Dp(Dp(ek) ) - Dp(ek) C_ D~(ek).
(ii) 9similar. D
6.4.1. REMARK. Since I - A X . X C_ D ~ (by 6.1 and 6.2), 6.4 (i) states that
Dc~ is a closure operator on R E (see 5.3 (ii)). By 6.4 (i) we also have
F I X ( D ~ ) - {X" X E R E A D ~ ( X ) - X } - { D ~ ( X ) 9X E R E ) .
Moreover, I C_ D ~ and 6.4 (i)-(ii) imply that D ~ ( D ~ ) -
D~.
6.5. DEFINITION
(i) D~ "- F I X ( D ~ )
(ii) Given any assignment p'w--+Dc~ , we inductively define the value
it].
Doo:
~xi]]pD oo -- p(i);
D - aX.O
;
- t-Dc~
(ll o(u := D
(X)))"
It]if ~ is well-defined by the theorem below; we simply write [tip instead of
gt]l_
L'~ and we neglect the interpretation of the additional constants
.11~.)
i1
of A.
6.6. THEOREM. Dc~ is a non trivial applicative substructure of R E , which
can be expanded to a model of OP)~ and extensionality for operations Extop.
PROOF. D ~ has at least two distinct elements T "- w and J_ "- 0.
Indeed, we have that Dc~(T ) - T holds by remark 6.4.1.
Trivially _L C_ D c~( _L ) and Do( _1_)C_ J_. Assume by IH Dn( J_ )C_ J_.
Then Dn( _L ( D n ( X ) ) ) C_ J_ (IH and since J_ ( X ) - J_ ); by ~* we get
Dn+l( _L ) Now let D ~ ( X ) hypothesis,
)~X.Dn( • ( n n ( X ) ) ) C_ )~X. _L -
_L .
X, D c~(Y ) - Y" we then obtain, by the main lemma and
X(Y) - D~(X)(Y) - (D~ o X o D~)(Y) - D~(X(D~(Y)))
- D~(X(Y)).
1.6]
37
The Extensional Model Doo
This shows that Doo is an applicative substructure of RE; hence it satisfies
left and right monotonicity of application. Moreover, for every term t, Doo
is closed under the operation p ~ [t]p, provided p" w ~ Doo. This is verified
by induction on t. In particular, by lemma 6.4, IH and fl-conversion in RE:
Doo([)~x.t]p ) - Doo o [~x.tlp o Doo = ~Z.Doo((~X.Doo([t]p(x := Doo(X))))(Doo(Z))) -- )~Z.Dco([t]p(x := Doo(Doo(Z)))"
But Doo o Doo - Doo; hence
Doo([)~x.t]p) -- AZ.Doo([t]p(x := Doo(Z))) -- lAx.tip, i.e. [Ax.t]]p E Doo.
~-conversion in Doo: since R E is a model of fl-conversion, we have for
[ x ] p - P,
[()~y.t)x]]p - ~Y.Doo([t]p(x := Doo(y))). P - Doo([[t]p(x := Doo(P))).
But [t]p(x := Doo(P) ) and P are fixed points of Dco , whence the conclusion.
~-conversion in Doo (cf. 2.2)" let p" w---~Doo and assume
[[t]p(x := p) = [s]p(x := p), for every P E Doo.
Then
[t]p(x
:=
Doo(p) ) = [S]p(x
:= D o o ( P ) ) ,
for every P E R E
and hence
by closure of D oo under p ~ I-Iv, we have for every P E RE,
Doo(~t]p(x := Doo(P))) -- Doo([S]p(x := Doo(P)))"
By ~-conversion in R E (5.8),
AX.Doo([[t]]p(x := Doo(X))) -- AX.Doo([s]p(x := Doo(X))),
whence by 6.5 (ii) l A x . t i p - Ax.[S]p.
~/-conversion (see 2.2)" it corresponds to the equation
[)~x.fxlp-[f]p
which holds by 6.5 and 6.4(ii). On the other hand, ~- and y-conversion
imply Extop. (a)-conversion is left to the reader. Finally, Doo can be
expanded to a full model of OP by choosing the denotations of the terms
given by 3.8. [:]
6.7. REMARK (Park's theorem). Doo and R E share an important feature:
in these models the paradoxical combinator F P (see w
coincides with
Tarski's fixed point operator, namely, if X E I}oo (or RE), F P ( X ) satisfies
the condition: X ( Z ) C_ Z implies F P ( X ) C_ Z, i.e. F P ( X ) is the C_-least
fixed point of the operator F U N ( X ) (for a proof, see Scott 1980).
Introducing Operations
38
By straightforward
immediately obtain:
arithmetization
of
the
[Ch.1
preceding
model,
we
6.8. THEOREM. OP+Extop is interpretable in PA.
Howev&, if we wish to refine 6.8 with OP 1 and PRA in place of OP
and PA respectively, it is not clear how to deal with D c~, in presence of
restricted inductions. The difficulty can be overcome by considering term
models of O P + E x t op"
According to the equivalence theorem of 2.9, it is enough to produce
models for the system A+Extop , i. e. the variant of OP based on )iabstraction as primitive; of course, Al+EXtop is obtained from A+Extop by
assuming the restricted induction schema 3(+)-NIND of 4.13 in place of the
full N-induction schema.
Let >-~n be the least reflexive transitive relation which preserves
application and is closed under the clauses (A), (P), (SUC), (D.1), (D.2) of
4.1 and
_>
(Z)
(,)
. - ,];
)~x.tx >__~, t (x not free in t);
from t _ f3u s infer Ax.t >_ ~,)ix.s.
By adapting the standard argument of Barendregt (1984), it turns out that
>__fl, satisfies the Church-Rosser property (cf. 4.3); hence the associated
conversion relation is a non-trivial congruence relation on the set TM~u of
all terms in the language of A+Extop , provably in primitive recursive
arithmetic.
To be definite, let TMf~o be the structure (TMf3o, 9 , = f3o, C, N/3o),
where TM~o is the set of all terms in the language of A and
1.
r
P A I R , L E F T , R I G H T , SUC, P R E D , 0};
2. 9 :TM~o-~ TMf3 o is the operation of juxtaposition of terms (i.e.
application);
3.
= f~o C_TM~o and t - zos holds iff t >_ zor and s _ f3o r, for
some r E TM~o;
4.
N/3,7 C_TM~o and t E N~o iff t _ f3o~, for some n.
Now TMzn is a non-trivial model of A+Extop , and satisfies CT, as in 4.11.
Thus by standard arithmetization, we get:
Appendix
I.A]
39
6.9. THEOREM. OP+Extop+CT (respectively OPl+Extop ) is interpretable
in PA (PRA respectively).
A final question involves the consistency of OP+Extop+EA+CT. At
present, we can only state the following partial result:
6.10. THEOREM. OP+Extop+EA is consistent.
For the proof, consider the theory OP(w), which includes OP plus an
infinitary w-rule for terms :
tr = sr, for each closed term r
t=s
Then define C T M ( w ) " - ( C T M ,
Lop-closed terms and
9 ,-w,
e, Nw) , where CTM is the set of
1. r "- {K, S, D, P A I R , L E F T , R I G H T , SUC, P R E D , 0};
2. 9 9CTM 2 ~ CTM is the operation of juxtaposition of terms (i.e.
application);
3. - w C _ C T M 2 a n d t - w s h o l d s i f f O P ( w )
Ft-s;
4. N w C_ CTM and t E N w iff OP(w) F Nt.
By Barendregt (1984, Ch.XVII and Ch.XIX, p. 508), de(w) is consistent;
hence CTM(w) is non-trivial and makes Ext op and EA obviously true (see
also Flagg-Myhill 1987).
Appendix
This appendix contains a few details about results, which were quickly
summarized in w First of all, we deal with the provability of the Church
Rosser theorem in PRA. It is fairly obvious to see that the proof of w works
in PA; however, it is not entirely obvious that only suitably restricted
instances of number-theoretic induction are needed.
A reminder on the chosen metatheory P RA is in order: the basic
language is a standard first-order language containing 0 (zero), successor
and function symbols for primitive recursive functions; terms and formulas
are defined as usual. (An occurrence of) A quantifier 3 (V)in a formula is
bounded, if 3 occurs in the context 3 x ( x < t A . . . )
(respectively
Vx(x < t ~ . . . ) ) . Formulas, which only contain bounded number quantifiers,
are called bounded (or A0); formulas of the form 3xB (Vy3xB) are called
E 1 (H2) , provided B is bounded. PRA is the formal system, based on
classical predicate calculus, which includes Peano axioms for zero and
successor, defining equations for primitive recursive functions and numbertheoretic induction for bounded formulas. By E l - i n d u c t i o n - EI-IND , we
40
Introducing Operations
[Ch.1
mean the induction schema extended to El-formulas; the rule of H 2induction, II2-INDR , is the rule: if A is a II2-formula , infer VxA from
A(O),Vx(A(x)~ A(x+I)). PA can be identified with the extension of PRA
which contains the number-theoretic induction schema for arbitrary
formulas.
A.1. THEOREM (Parsons 1972). PRA, PA 1 - P R A + ~ I - I N D and
PRA+II2-INDR have the same II2-lheorems.
A.1 grants that the provably recursive functions of the three systems are
exactly the primitive recursive functions.
We know from w that there exists a formal calculus ~', for deriving
expressions of the form tREDs (t,s terms of OP). ~" has the axioms (R),
(g), (S), (P), (SUC), (FRED), (D.1)-(D.2)of 4.1, while the inference rule
is (A); a ~-derivation is a finite sequence, whose elements are either axioms
or else expressions, obtained by previous ones by application of (A). If d is a
derivation, ending with tREDs, we put H e a d ( d ) - t and T a i l ( d ) - s. It is
folklore to find: 1) a bounded formula Dim(d), which formally represents in
PRA the metamathematical predicate "d is a derivation in the ~F-calculus";
2) terms representing the functions Head and Tail. Moreover, tREDs is
~'-derivable iff tREDs holds according to 4.4 (ii).
It is easy to see that (modulo encoding of tuples) the statement of the
crucial inversion lemma 4.7 has the form VxR(x, t(x)), where t(x)is a term
of PRA, actually built-up by inspecting the proof, and R is a bounded
formula. As a consequence of bounded induction, we get:
A.2. LEMMA. PRA proves the formalizalion of 4.7.
The Church-Rosser theorem CR(RED) for RED has the form of a
bounded condition
(VH < x)(Vd' < x)(x = d+d' A Dim(d) A Dim(d') A Head(d) = H e a d ( d ' ) ~
--. (Dim(CRl(d, d')) A Dim(CR2(d, d')) A Head(CRl(d, d')) = Tail(d) A
A nead(CR2(d, d')) = Tail(d')A Tail(CRl(d, d')) = Tail(CR2(d, d')) ),
where CRI(x,y), CR2(x,y ) are primitive recursive terms which can be
explicitly extracted from the proof of 4.8. If we apply bounded induction on
x, A.2 and we mimic the content of 4.8, we get
A.3. LEMMA. PRA F CR(RED).
Now we need CR for the transitive closure TC(RED) of RED,
provably in PRA; hence we extend ~" to the system ~* with the transitivity
rule (T) and we have an obvious notion of ~*-derivation and a
Appendix
I.A]
41
corresponding bounded formula Dim*, which represents it in PRA. Let"
Ro(d , a, b)"- Dim(d) A Head(d) - a A Tail(d) - b;
n*(d,a,b) . -
Dim*(d) A Head(d) - a A Tail(d) - b.
As Ro,R* are bounded, the following formula is 112:
C(n) "- VdVd'VaVbVc(Ro(d, a, b) A R*(d', a, c) A lh(d') - n
3r3r'3x(R*(r, b, x) A R*(r', c, x));
(here lh(x) is the primitive recursive term which computes the length of a
finite sequence).
By A.3, we can show PRA F C(0) and P R A + C ( n ) F C(n+I); hence a
first application of II2-INDR together with Parsons's theorem yields:
A.4. LEMMA. PRA F VnC(n).
Again II2-INDR applied to the condition
B(n) "- VdVd'VaVbVc(lh(d') - n A R*(d', a, b) A R*(d, a, c) ---,
b,
^
together with A.1, implies that PRA proves VnB(n), whence:
A.5. THEOREM. PRA proves the Church-Rosser property for the transitive
closure of RED.
Now let:
CT(x) "- "x is the code of a closed term of s
NUM(x) "- "x is the code of a numeral of s
CONV(x, y) "- 3z3d3d'(R*(d, x, z) A R*(d ', y, z));
NAT(x) "- 3z3d(R*(d,x,z) A NUM(z)).
Clearly CT(x), NUM(x) are bounded while CONV(x,y) and N A T ( x ) a r e
El" If A E L, Acre is the L0-formula which is obtained by replacing:
1) each atom of the form Nt, t - s respectively by NAT(t), CONV(t,s);
2) each quantifier Vx, 3y respectively by Vx(CT(x)--~...), 3 x ( C T ( x ) A . . . ) .
Now we give a more explicit statement of 4.15:
A.6. THEOREM
(i) If O P + E A F A(x), then PA F C T ( x ) ~ Acre(x);
(ii) O P I + E A F A(x) implies PA 1 F C T ( x ) - , Acm(X ).
Introducing Operations
42
[Ch.1
As to the proof of A.6(ii), we apply A.5 and we remark that each
instance of 3(+)-N-induction is sent into a suitable version of El-induction.
As to EA, it suffices to see that its verification in the closed term model
requires only El-induction (at most). The claim is made apparent by the
following informal argument. First, fix a primitive recursive bijection J*
from ~ x ~ onto ~ - { 0 , . . . , 8 } such that J*(n, m) > n, m (we can choose the
modified Cantor pairing function
J*(n,m)
"-
(n+m)2+3n+m-+-18
2
). Then primitive recursively define a
GSdel numbering GD of closed terms as follows:
G D ( 0 ) - 0;
GD(SUC)- 1; GD(PRED)- 2; G D ( D ) - 3;
GD(PAIR) - 4; GD(LEFT)- 5; GD(RIGHT)- 6; G D ( K ) - 7;
GD(S)-
8;
GD(Ap(t,s))- J*(GD(t), GD(s)).
The fixed point theorem for operations ensures the existence of a closed
term E such that
A.7. If 0 < n < 8, then:
E~E~
the unique constant C such that GD(C) - n; else:
-- (E(~)o)(E(~)I);
((~)0,(~)1 are the terms representing in O P - the projections of J*). A
straightforward induction on closed terms yields that E ~ - t ,
where ~ is
the numeral representing the value of GD(t). Clearly, Parsons's theorem (ii)
entails 4.14.
CHAPTER 2
EXTENDING OPERATIONS WITH REFLECTIVE TRUTH
w
w
w
w
w
w
w
Extending combinatory algebras with truth
The theory of operations and reflective truth: simple consequences
Type-free abstraction, predicates and classes
Operations on predicates and classes
The fixed point theorem for predicates
Applications to semantics and recursion theory
N on- extension ali ty
Appendix I
Appendix II
Appendix III
We introduce an axiomatic framework MF- (=Minimal Framework without
number-theoretic induction) and we derive a set of simple, but significant
consequences of MF-. The minimal fixed points of a natural monotone
operator over arbitrary combinatory algebras yield set-theoretic models of
MF-. This kind of models (in short, inductive models) are generated by
means of natural elementary semantic clauses.
The informal intuition is probably due to Curry and Fitch, and it freely
takes inspiration from the ideas of illative combinatory logic, later
reinterpreted by Aczel with the notion of Frege structure. The main
intuition can be summarized as follows: MF- describes an abslract logical
system, i.e. a pair given by an abstract syntax and a semantics. More
explicitly, we can imagine a non-empty set U of objects (if you like, terms),
which is endowed with a two-fold structure. The syntax establishes the rules
of combining elements of U; application is the basic combination mode and
U is an applicative structure with strong closure conditions and selfreferential abilities (indeed a combinatory algebra). Furthermore, the syntax
identifies two objects of U, whenever they are computationally equivalent in
a precise sense, specified by a conversion relation.
It must be stressed that these features are quite general and that they
can be reasonably specialized, as soon as we specify U with additional
constraints (for instance, we can always assume a set of primitive numbertheoretic operations, if we are interested in foundational applications).
44
Extending Operations with Reflective Truth
[Ch.2
On the other hand, the semantic structure comes on the scene, as soon
as we assert equalities and classify elements of U, e.g. we state that an
object truly enjoys a certain property, or that an algorithm yields welldefined values for arguments of a certain type. At this stage, we content
ourselves by choosing the simplest alternative, i.e. a truth predicate T,
naturally extending the standard Tarskian truth conditions. The basic
T-clauses reflect the idea of a reduclionist semantics: truth is assigned to
certain basic syntactic objects, (representing) atomic propositions, and it
propagates to more complex entities by means of appropriate reductive
clauses for logical operations. In agreement with the reductive spirit, atomic
propositions do not refer to the truth predicate T and their semantic value
only depends on the combinatory structure. Nevertheless, T strictly extends
the limits of Tarskian semantics. T itself becomes a propositional
constructor and it directly applies to expressions explicitly using T, like
T[~Tt] ([a] stands for the term representing the sentence a). Of course,
such "higher order" expressions cannot be regarded as atomic, and they
have a definite truth value, only if they can ultimately be reduced to
well-defined atoms (eft Kripke 1975).
There is, however, a price to pay, as one might expect from Tarski's
theorem: T cannot be consistent and complete at the same time. In spite of
this limitation, a logical system in the previous informal sense yields a
reasonable environment for an extended logic" predication, abstraction and
the notions of proposition and predicate (total or partial) can be easily
introduced by means of T and the combinatory structure, and the resulting
theory of abstraction has non-trivial aspects.
In details, w shows how to expand any given model of the theory of
operations with a reductive notion of self-referential truth, which satisfies
natural axioms. These axioms give rise to the basic axiomatic system MF-,
whose consequences are first discussed in w In w167
we define the
predicate abstraction operation {x: A} via ~-abstraction, and we introduce
a consistent reformulation of the type-free comprehension principle AP,
together with a few closure conditions on total properties ( = classes). It
turns out that AP can be finitely axiomatized by means of four primitive
predicates and eight generating operations. In w167
we exploit a kind of
second recursion theorem for predicates, which yields fixed point solutions
to a class of significant conditions (positive operators, definable in the
language of MF-). It follows that the system MF : - MF-plus the numbertheoretic induction schema, is proof-theoretically stronger than first-order
arithmetic PA, but still predicatively reducible in the sense of Feferman.
We then apply the fixed point technique to the formalization of semantics
and we obtain analogues of recursion-theoretic results, due to Rice and
Myhill. This last point naturally hints at possible connections with
II.7]
45
Combinatory Algebras with Truth
Generalized Recursion Theory, to be pursued in parts B and C. The final
section w shows that extensionality for classes and properties is violated in
MF-.
w7. Extending combinatory algebras with truth
We canonically associate to each OP--model an interpretation of the truth
predicate, which satisfies natural closure conditions. To this aim, we first
introduce canonical terms representing the "logical" functions, which are
defined by E-formulas.
7.1.
DEFINITION. (a) We choose:
I D "- ~x)~y.(1,{x,y));
NEe
"- Ax.{4, x);
T R : - )~x.(2, x);
A N D "- AxAy.(5, (z, y));
(b) We then define the map A H [ A ]
Z-formula:
(i)
(ii)
[t = s] =
(IDt)s;
[--A] = NEG[A];
N A T : - ~x.{3, x);
[Ns]- NATs;
A L L "- )~x.{6, x).
by induction on the notion of
[Tt]- TRt;
[A A B] - AND[A][B];
[VzA] - ALL(~x[A]).
7.1.1. FACT
(i)
I f L1, L 2 E L O G 1 - { N A T , N E G , T R , A L L } , then
O P - I- LlX - L2Y -+ L 1 - L 2 A x -- y;
(ii)
if G1, G 2 E L O G 2 - { I D , A N D } , then
O P - ~ GlXy - G2x'y'---+ G 1 - G 2 A x - x' A y - y';
(iii)
if L 1 E L O G 1, L 2 E L O G 2, then OP-I---1 L l X -
L2Yz;
if L1, L 2 are distinct elements of L O G 1 U LOG2, then
O P - I-- --1 L 1 - L 2.
(Verification by pairing axioms and #-conversion).
(iv)
We stress that [A] and A have the same free variables in common. It
would be possible to trivialize T R to ~x.x; but we stick to the present
choice, since it better suits to the generalizations of chapter VIII.
7.2.
Further notations and terminology
(i) We henceforth write T A as a shorthand for T[A].
(ii) To increase readability, we keep using --, A, V, etc., and infix
notation, instead of the terms N E G , A N D , A L L , etc.
Thus t A s, V f , Vx.t , ~t stand for the terms ( A g D t ) s , A L L f , A L L ( ~ x . t ) ,
Extending Operations with Reflective Truth
46
[Ch.2
N E G t (in the given order); we also adopt the obvious shorthands ~ t ,
t V s, t ~ s , in place of -~(-~t), -~(~t A-~s), (-~t V s) (respectively). As to the
existential operator, we define:
3(f) "-~(V()~x.-~(fx)))and 3x.t "-3()~x.t).
(iii) s
is the (operational) fragment of s which omits the predicate
T. The atoms of Lop, i.e. Nt, t - s are called elementary atoms, e-atoms for
short. Atoms of the form Tt of the full language are called T-atoms. s and
s clearly have the same terms.
We now fix a model ~1~ of O P - (i.e. OP without number-theoretic
induction) with domain M.
7.3.
DEFINITION
(i) L op(~t~), L ( ~ ) are the languages Lop , L (respectively), expanded
with distinct individual constants, for each element of M. If t is a closed
term of the expanded languages, Jtt~(t) denotes the (unique) value of t in M.
For the sake of simplicity, we shall use a, b, c, d, e,..., both for the
elements of M and the corresponding constants (we identify ~ ( a ) with a).
(ii) If P is a unary predicate (possibly T itself), L op(P ) is the language
Lop expanded with P; so, Pt is a new atomic formula ( = atom) of Lop(P).
(iii) Let S be any subset of M: ( ~ , S ) i s
the realization of Lop(P),
which interprets P by the set S: if t is an arbitrary closed term of s
( ~ , S ) I= Pt iff ~t~(t)E S. If P - T, then L - Lop(T) and (Jtl~,S) is the
realization of s which interprets T by the set S.
P F O R ( x ) i s the Lop-formula:
(iv)
~y3z(~
[~
-
z]
-
v
~
-
[g~]
v
9 -
[Ty]
v
9 -
(-~y)
v
x
-
(y ^ z)
v
~
-
vy);
.
M-PFOR-
{a E M" Jft~l=PFOR(a)}.
If a C M - P F O R , we say that a is (the code of) a pseudo-formula (p-form,
for short).
(v) We define:
P(x, P ) ' - 3u3v((x - (~u) A-~PFOR(u)) V (x - (~-~u) A Pu) V
v (~ - [~ - v]A ~ - v) v (~ - [~(u - v)] ^ ~(~ - v)) v
V (x - [gu] A g u ) V (x - [-~Nu] A ~ g u ) V
V ( x - [-~Tu] A P(~u)) V (x - [Tu] A Pu) V
V
(x
-
(u A v) A Pu A By)
V
(x
-
[-~(u A v)] A (P(-~u) V P(-~v))) V
V (x - (Vu) A VyP(uy)) V (x - -~(Vu) A 3yP(~(uy))));
II.7]
Combinatory Algebras with Truth
47
If S C_ M, we put:
r(S) "- {a E M" ( ~ , S ) l = r ( a , P ) } .
(vi) A subset S of M is consistent (complete) iff for every a E M, either
a ~ S or (--,a) ~ S (a E S or (--,a)E S).
(vii) Put atl~(g)"- {dig(t)" t numeral}; then att,(SUC) ^ (ati,(PRED) ^) is
the unary function atl,(N)~,&(N), represented by SUC ( F R E D ) in art,.
Jig is an w-model iff the structure (.3g(g), Jlg(0), JIg(FRED)', JIg(SUC) ^) is
isomorphic with (w,O, pred, suc) ( - s e t of natural numbers with zero,
predecessor and successor).
(viii) S C_ M is F-dense (F-closed) iff S C_ F(S) (F(S) C S).
Once ~ is a fixed combinatory algebra of domain M and b, c E M,
we shall write be, ~b, Vb, b Ac, instead of the proper ~t,(Ap(b,c)),
JlI,(NEGb), JtI,(ALLb), Jlg(ANDbc)(in the given order); we also let id(b,c)
(tr(b)) stand for JIg([b- c]) (respectively Jlg([Tb])).
7.3.1. REMARK. I'(x,S) formalizes the clauses of the intended semantic
schema, to be used for interpreting the truth predicate T. As to PFOR(x),
it defines the range of application of T: we stress that p-forms are not
inductively defined entities (like sentences), but only objects of the ground
algebra, possibly representing semantical information.
7.4. LEMMA
(i) /f S C_ M, F(S) C_ M - P F O R .
(ii) r " ~ 2 ( M ) ~ ( M )
is monotone: S C_S' implies F(S) C_ F(S').
(iii) Assume a, b, f E M:
if a ~ M - P F O R , (~a) E F(S);
.Ag([A]) E F(S) iff A holds in att~ (A closed e-atom or a negated
e-atom);
(aAb) EF(S) iff a E S and b E S;
(~(a /k b)) E F(S) iff (-~a) E S or (-~b) E S;
(Vf) E F(S) iff (fa) E S, for all a E M;
(--,(Vf)) E F(S) iff (-~(fc)) E S, for some c E M;
(-~tr(a)) E F(S) iff (-~a) E S;
(tr(a)) E F(S) if]" a E S;
(-,--,a) E F(S) iff a E S.
(iv) If S is consistent and F-dense (complete and F-closed), then r ( s )
is consistent and F-dense (complete and F-closed).
PROOF. (i): trivial.
48
Extending Operations with Reflective Truth
[Ch.2
(ii): r ( z , P ) i s positive in P, i.e no negated atom Pt occurs in r(z, P).
(iii): by 7.1, 7.3, 7.1.1 and the pairing axioms.
(iv)" we repeatedly apply (iii). Let S be consistent and F-dense; we claim"
a ~ F(S) or (--a) ~ F(S).
(,)
Case 1. a ~ M-PFOR: then a ~ F(S) follows by (i) above.
Case 2. a = JII,([A])with A = gb, (b = c ) o r a = (Vb), tr(b), (b A c).
Then (,) is a consequence of the consistency of S and (iii). For instance, if
a = (b Ac) E F(S), then b E S and c E S. Were (-,a) E F(S), we ought to
have (-,b)E S or (--,c)E S: either alternative contradicts the consistency of
S.
Case 3. a = (-,b). Assume by contradiction (-,b) E F(S), (-~-~b) E F(S).
Then b E S and by F-density b E F(S), which also implies b E M - P F O R .
3.1. b=-,c: then we have ( - - c ) e r ( s ) a n d
(-,c) E S , whence
c E S and (-~c) E S, against the consistency of S.
3.2. b = (c A d), (Yc), tr(c), NI,([A])with A e-atom or negated e-atom.
By F-density, we are led to case 2.
Note that if S is F-dense, so is F(S) by F-monotonicity. If S is complete
and F-closed, the argument is similar. I-1
7.5. DEFINITION. FIX(r,~).- {S C_M. r ( s ) - s}.
F I X ( F , MI,) is the set of fixed points of F over Mr,. In chapter VII we shall
investigate the global structure of the fixed points of F; however, in the
following we concentrate upon O(M1,) "- the C-least fixed point, which is
generated from below by transfinite iteration of r.
7.6. DEFINITION (by recursion on ordinals).
(i)
-
0;
(ii) O(Jtt~, c~ + 1) - F(O(~I,, or));
(iii) O(.Ab, A) - U O(~l,, c~) (A limit).
a<,k
7.7. LEMMA
(i)
c~ < ~ implies O(Jtt,, c~) C_ O(Jtt,,/~);
(ii)
O(.At~,c~) C_ M - P F O R and O(.Ai,,c~) is consistent and F-dense,
for each ~.
P R O O F . (i). We verify by induction on ~:
for each a < fl, O(~,c~) C_ O(.tl,,~).
(,)
If fl = 0 or ~ is a limit, the claim ( , ) i s trivial. Assume that (,) holds and
let a < fl + 1. It suffices to check O(dtt,, fl) C_ O(.At~, fl + 1). If fl = 0, we are
done; if fl = 5 + 1, we have O(Nl,,5) C_ O(.&,5 + 1) by In, which implies
O ( ~ , fl) C_ O ( ~ , fl + 1) by F-monotonicity. If fl is a limit and 7 < fl, then
Combinatory Algebras with Truth
II.7]
49
by IH and F-monotonicity O(all,, 3') C_ O(all,, 7 + 1) C_ O(,&,/3 + 1). Hence
O(ag, fl) _C O(all,, fl + 1).
(ii) O(atl,,c~) is r-dense by (i) above and hence O(aM,,a)C_ M - P F O R by
7.4 (i). On the other hand, O(.Al,,c~) is consistent by induction on a, using
(i) and lemma 7.4 (iv). E]
The _C-chain {O(alg, c~)" a E ON} cannot be strictly C_-increasing by the
well-known Cantor's theorem: hence, there exists an ordinal 6 < card(All,) +
(+ "- successor operation on cardinals), such that O(all,, 5) - O(all,, 5 + 1).
7.8. DEFINITION. We set"
.-
6)
where 5 - the least a such that O(all,, a) - O(all,, a + 1).
Then O(./11,) - U {O(Jtt~, a)" a E ON}.
7.9. PROPOSITION. Let ag be a model of O P - (OP). O(all,) is consistent
and is the C_-least fixed point of F:
r ( o ( . ~ ) ) c O ( ~ ) (r-closure);
(,)
if F(S) _C S, then O(atl,) C_S (F-induction).
(**)
PROOF. The consistency follows by lemma 7.7, while (,) holds by choice of
5 in 7.8. As to (**), simply prove O(Ml~,a)C_ S by transfinite induction on
a, applying F-closure of S and F-monotonicity. V1
7.10.THEOREM. (i) If JM,I=OP- , the structure (~,O(3t~))
universal closures of the following s
T.1
TARA,
if A - ( x - y ) ,
T.2.1
TTx~Tx;
T.3
T',-,x ~ Tx;
T.4.1
satisfies the
Nx, ( - x - y), ~Nx;
T.2.2
T-~Tx ~ T-~x;
T(x A y)+-+Tx A Ty;
T.4.2
T~(x A y ) ~ T ~ x V T~y;
T.5.1
T(Vf)~VxT(fx);
T.5.2
T--(Vf) ~ 3xT~(fx);
T.6
~(Tx A T~x) ( - C O N S ) ;
RES
Tx---, PFOR(x);
~PFOR(x)---~T~x.
(ii) If .]g is an co-model (i.e the denotation of N is isomorphic with the
standard set of natural numbers), (all,,O(alt,)) satisfies the N-induction
schema for arbitrary formulas of 2..
PROOF. Part (ii) is trivial. As to part (i), CONS is true in O(31,) by 7.9,
50
Extending Operations with Reflective Truth
[Ch.2
while T x ~ P F O R ( x ) holds by 7.7(ii). The other axioms are immediate
consequences of 7.4 (iii) and the fact that O(Jtl~)is F-closed and r-dense. [3
7.10.1. REMARK. Let O(~t~) d : : {a: a E M and (-~a)~ O(.~1~)]. The reader
can easily check that O(~l~)dl: T.1-T.5 + RES + COMB : : Vx(Tx V T~x);
O(Jtt~) d is the C_-largest fixed point of F over ~ (see Ch.VII).
7.11. DEFINITION. (i) The theory M F - ( - m i n i m a l framework for selfreferential truth and abstraction) is the finite extension of O P - by means of
axioms T.1-T.6 of 7.10.
(ii) MF is M F - plus the schema of N-induction for arbitrary s
NIND "- A(0) A Vx(A(x)-~ A(x + 1))-~ V x ( g x ~ A(x)).
NB. RES is omitted (unless we explicitly mention it).
In the sequel, we mostly deal with MF- or with subsystems of MF, which
contain restricted forms of number-theoretic induction. The restriction
axioms RES will play a marginal role in our investigation; also, the second
restriction axiom is certainly a matter of convention (for alternatives, see
Ch.VII). However, RES is needed for a full characterization of models of
theories of reflective truth in Ch.VII, as it can be guessed from the
following"
7.12. PROPOSITION (Alternative axiomatization of M F - + R E S )
Let the fixed point axiom for truth FPT be the sentence
Vx(Tx ~ F(x, T)),
where r(x,T) is obtained from the formula r(x,p) of 7.3 (v) by replacing
every subformula of lhe form Pt with Tt . Then we have:
M F - + RES C_ O P - + FPT + CONS and M F - + RES ~ FPT.
The verification makes use of the independence properties of 7.1.1; we
underline that by 7.12 M F - + R E S is a genuine fixed point theory in the
sense of Feferman (1982), and it axiomatizes the property of being an
arbitrary fixed point of F (see Ch.VII).
II.8]
Operations
and Reflective Truth: Simple Consequences
51
w8. The theory of operations and reflective truth: simple consequences
In this section we start working axiomatically within the system M F without number-theoretic induction; since we are interested in general
properties of truth and propositions, the number-theoretic axioms are not
needed. Towards the end of the section, we sketch a version of M F - , where
the consistency axiom CONS is replaced by its dual, i.e. completeness.
First of all, we must distinguish between T-~t, which can be read as "t
is internally false", from -~Tt; so we define a notion of internal falsehood F:
8.1.
F x := T ~ z .
8.2.
PROPOSITION.
The following
formulas are provable in M F -
without consistency:
(i)
(ii)
(iii)
T z ~ FFx;
Fz ~ T F z ~ FTz;
T(x V y ) ~ Tx V Ty;
F ( x V y ) ~ Fx A Fy;
T3(f)~
3zT(fx);
F 3 ( f ) ~ VxF(fx).
Closure under cut: M F - p r o v e s
(iv)
T(x ~ y ) ~ (Tx ~ Ty).
P R O O F . (i)-(ii): apply T.3 and T.2, T.4. As to (iii), recall the definition of
3 ( f ) and apply ~-conversion 2.2(ii) and T.5. The statement ( i v ) i s a
consequence of (ii) and consistency. O
8.3.
DEFINITION
(i)
(ii)
Prop(x):= Tx V T-~x = "x is a proposition".
Propfunn(f):= VXl... VxnProp(fxl...Xn) = " / i s a n-ary
propositional function" (n > 1; if n -- 1, we simply omit the index).
Clearly we have in pure logic :
Propfunk+l(f) ~ VxPropfunk(fz ) (k > 1);
8.3.1.
hence we can restrict our attention to unary propositional functions.
We now investigate the closure properties of Prop under standard
logical operations and the behaviour of T, whenever T is restricted to Prop.
Abbreviation: Prop(A):= Prop([A]).
8.4.
LEMMA.
(i)
M F - proves:
Prop(A), whenever A = (-~)Nx, (-~)x = y;
Tx--. Prop(x);
52
Extending Operations with Reflective Truth
[Ch.2
(ii) Prop(z)~ Prop(Tz)~ T(Prop(z));
(iii) Prop(z)~ Prop(Prop(z));
(iv) Prop(z)~ Prop(--,z);
(v) Prop(z) A Prop(y)---, Prop(x A y) A Prop(z V y);
(vi) Prop(z) A (Tz --, Prop(y))---, Prop(z ---,y);
(vii) Prop fun(f)---, Prop(V(f)) A Prop(3(f));
(viii) Prop(x,y)~ Prop(z) V Prop(y) (, = V, A,---,);
(ix) Prop(Q(f))~3zProp(fz) (Q = 3, V);
(x) ,F(Prop(z)).
PROOF: straightforward application of 8.2 and T-axioms. As to the final
point, if FProp(z)is assumed, we have F T z A F F z (8.2(ii)), whence
Fz A Tz (by T.2.1), against consistency T . 6 . 0
8.4.1. REMARK. 8.4 (x)implies T(Vz(Prop(z)---, f z ) ) ~ T(Vf); this means
that internal truth (i.e. truth with respect to T) disregards quantification on
propositions; thus, there is no hope to produce propositions by means of
Vx(Prop(x)---,... ), except for trivial cases.
We stress that the internal truth predicate is partial and that 8.4 (viii)(ix) cannot be improved by replacing V, 3 in the right member of the
implications with A, V respectively; for instance, there are disjunctive
propositions with a member which is not itself a proposition. Therefore the
behaviour of logical operators on Prop is non-strict. By 8.4 (x), the notion
of proposition is essentially external and positive; we cannot come to know
that p is not a proposition by adopting the semantical schema embodied by
T.
8.5. PROPOSITION. MF-proves:
(i) 3x(-,Prop(x)) A -,Propf un(Iz.[Prop(z)]);
(ii) 3z3y(Prop(x V y) A-,(Prop(x) A Prop(y)));
(iii) 3f(Prop(3f) A--,VxProp(fz)).
PROOF. (i) We consider the fixed point L of )~x.[Fz], i.e. L - [FL] (apply
2.2). Then Prop(L)implies both TL and FL (by 8.2 (i)), against
consistency. If .kz.[Prop(x)] were a propositional function, we could
conclude by 8.4 (x) Vx.TProp(x), which contradicts the previous result.
(ii): choose y - [ 0 - 0] and x - [L].
II.8]
Operationsand Reflective Truth: Simple Consequences
(iii): choose f -
53
~y.[Fy]. Vl
On the other hand, T is consistent and complete on propositions and
satisfies the standard Tarski conditions; indeed, the essential content of 8.2
and 8.4 can be summarized as follows:
8.6.THEOREM. MF- proves:
(i) Prop(A) A (TA ~ A), whenever A = (-,) x = y, (--)Nx;
(ii) P r o p ( x ) ~ Prop(-,x) A (T-,x ~ - , T x ) ;
(iii)
Prop(x) A (Tx~Prop(y))---,Prop(x ~ y) A (T(x ~ y ) ~ ( T x ~ Ty));
(iv) Prop fun(f)---, Prop(V f) A (T(Y f ) ~ Y x T ( f x));
(v) Prop(x)--, Prop(Tx) A ( T ( T x ) ~ Tx).
8.6.1. REMARK. (i) 8.6 shows that MF-essentially contains the (classical)
theory of Frege structures (see Aczel 1977, 1980).
(ii) The Curry paradox (Curry 1942). We cannot consistently add to M F - a
strengthened introduction axiom for implication, which omits the
hypothesis Prop(x)in 8.6 (iii): ME- + ( , ) i s inconsistent, where ( , ) i s the
statement
(Tx ~ Prop(y))--, ((Tx -~ T y ) ~ T(x ---,y)).
(,)
Indeed, we can find c such that c-c---, y ( c - FP(,~x.[x---, y]), see 2.3)and
clearly Tc---, Ty (by 8.2 (ii), consistency and ---logic). If we assume (,), we
can infer T(c---,y), i.e. Tc, whence Ty by 8.2(iv): contradiction (choose
[0-1]).
m
As to the general Tarski schema T A ~ A, it can be justified "from left
to right" and also for positive conditions.
8.7. DEFINITION. (i) A formula B is T-free if T does not occur in B.
(ii) The collection T-Pos (T-Neg) of T-positive (T-negative) formulas
is inductively generated by the following clauses:
1. each e-atom is both T-positive and T-negative; each atom of the
form Ts (-~Ts)is T-positive (T-negative);
2. if B is T-positive (T-negative), ~B is T-negative (T-positive);
3. if B, C are T-positive (T-negative), then so is B A C;
4. if B is T-positive (T-negative), then so is VxB.
Extending Operations with Reflective Truth
54
[Ch.2
8.8. T H E O R E M
(i)
(ii)
The soundness schema: M F - proves ( T A ~ A), for arbitrary A;
if A is T-positive (T-negative), M F - minus consistency proves:
A ~ T A (-~A---, FA, respectively);
(iii)
if A is T-free, M F - minus consistency proves T A Y T-~A.
P R O O F . (i): by induction on A and by considering the form of B whenever
A - - - l B . If A is an atom, we apply T.1 and T.2.1, while, if A is a
conjunctive or universally quantified formula, we use IH~ T.4.1, T.5.1 plus
/?-conversion. If A = - ~ T t , we apply T.2.2 and consistency; in the remaining
cases, we make use of T.4.2, T.5.2 coupled with IH.
(ii): by simultaneous induction on the definition of T-Pos and T-Neg.
(iii): by ( i i ) a n d tertium non datur. F!
We conclude with a simple, but useful duality property, whose semantic
content will be made clear in Ch.VII (w
8.9. DEFINITION
1. ^ is the (unique) map of the basic language s into itself such that
(i) ^ is the identity map on e-atoms and (Tt)^ =--,Ft;
(ii) ^ commutes with the logical operations:
( A ^ B ) ^ = A ^ ^ B ^, ( W A ) ^ =
W(A^),
(-,A) ^ =
2. Put COMP := Vx(Tx V T-~x)(Completeness);
NMF ( = the neutral MF) is MF minus CONS, where CONS = T.6;
MF ^ := NMF + COMP.
As usual, N M F - is NMF without N-induction.
3. x=~y := (Tx ---, Ty) A (Fx ~ Fy) and xc~y := ( x ~ y ) A (y=~x).
Then CONS ~ (COMP) ^ and COMP ~ (CONS) ^, provably in NMF (use
axiom T.3); more generally, we can easily check 9
8.10. LEMMA. NMF proves:
(i) A ~ A
;
(ii) ( ~ y ) ^ ~ ( y ~ ) ; ( ~ r
(~r
8.11. T H E O R E M (Self-duality of NMF)
For every A, NMF F A iff NMF F A ^. The same holds if the restriction
axioms RES are added to NMF.
P R O O F . By the previous lemma, it is enough to check the theorem from
II.9A]
Type-free Abstraction, Predicates and Classes
55
left to right. The verification runs by induction on the length of the formal
proof of A in NMF; the induction step is immediate by definition of ^ and
IH. If A is an axiom, either it is self-dual (i.e. equivalent to its ~-transform,
like NIND, T.3) or it can be proved by the axiom lying on the same line in
the statement 7.10 (e.g. (T.2.1) ^ requires T.2.2). F!
8.12. COROLLARY. For every A, MF F A iff MF^F A ^. MF and MF ^ have
the same T-free theorems and hence they are equiconsistenr
NMF is a possible axiomatic counterpart of the four-valued approach to
semantics (Belnap 1977, Woodruff 1984, Visser 1984), according to which
self-reference leads to underdefined (neither true nor false), as well as to
overdefined (both true and false) sentences. For a general account of the
NMF-models, we send the reader to Ch.VII.
w9A. Type-free abstraction, predicates and classes
We will show that M F - s u p p o r t s
a reasonable theory of type-free
abstraction. To this aim, we observe that internal truth yields a wellbehaved notion of general predicate application (in short predication), and
that the underlying combinatory structure grants a systematic notation for
partial predicates defined by abstraction. Furthermore, if we identify total
predicates withpropositional functions, we obtain a rich domain, satisfying
natural closure conditions for abstraction. Henceforth, we shall adopt
Feferman's terminology by using the shorter term class instead of
propositional function. Of course, as we already know from the previous
section, there exists a stumbling block in any theory of abstraction, based
on such an identification" the notion of propositional function (or class) is
itself non-total and this is an essential limitation for deriving impredicative
fragments of second-order logic. On the other hand, the limitation is not
surprising, in view of the reductive, predicativistic interpretation, which is
suggested by the C - m i n i m a l model of w7.
9.1. DEFINITION
(i) (Xl...Xn)~Ty :- T(YXl...Xn);
(ii)
(Xl...xn)-~y - F(YXl...Xn);
{Xl...xn: A} := ~Xl...)~xn[A]("the n-ary predicate defined by A");
(iii) Cl(y):= Yx(xrly V xfiy) ("y is a class");
CL
:=
Note that e l ( y ) = Prop fun(y). We also recall (see 8.10):
xVVy := (Tx ~ Ty) A (Fx ~ Fy) and AC~B := [A]c:~[B].
Extending Operations with Reflective Truth
56
[Ch.2
9.1.1. REMARK. (i) The definition does not ensure the injectivity condition
[~] =
[u~v]--,
9 = u ^ y = v
(,)
If (,) is needed, choose P D := ~xy. Ix = x A y = y A yx] and define
xrly := T ( P D x y ) , x-~y := T-~(PDxy). Then (,) is met and we can prove in
N M F - the formula:
T(PDxy) ~ T(yx) A T~(PDxy)~
T~(yx).
(ii) Of course, 77, ~, { } might be accepted as primitive symbols of s
and the definitions of 9.1 (i) would become axioms. A similar choice might
be advisable in applications, or if one wishes to avoid combinatory logic (see
appendix I and Ch.XIV).
9.2. PROPOSITION. (i) The Abstraction principle AP: for every formula
A, N M F - (i.e MF- minus consistency) proves:
VUl... Vun((Ul... Un)r]{Xl... Xn 9A} ~ A[x 1 "- Ul,... , Xn "-- an]);
(ii) NMF- proves:
((=1... ~ , ) , y ~ T[(~I... ~,),y]) ^ ((~1... ~,)~Y ~ F[(~l... ~,),y]);
(iii)
NMF- F- (uT]{x" A} ~ TA[x : - u]) A (u~{x" A} ~ FA[x "- u]);
(iv)
NMF- F- T[Cl(x)] ~-, Cl(x) ~-, x~CL;
(v)
M F - }- Cl(x) ~ (-(y~x)~y-~x).
PROOF. (i) Assume n - 1 " then T(ur]{x" A } ) ~ T ( A x . [ A ] ) u ~ T A [ x
"-u]
(by T.2.1, fl-conversion and [A][x " - u ] - [A[x "-u]]). A similar argument
works for F (we need T.2.2.). (ii)-(v)" left to the reader. [3
By the Russell paradox, there exist predicates, which are not classes,
and the notion of class does not determine a class. More generally:
9.3. PROPOSITION. Let r -
{x" ~xrlx }. Then:
MF- ~ ~3~(Ct(~) ^ W(~,~ ~ ~,~)) ^-~3y(Cl(y) ^ W(u,~ ~ Cl(~))).
PROOF. Let x be a class such that Vu(urlx ~ urir). Then by AP we have:
xrlx ~ x~r ~ x-~x, whence by consistency ~(T(xrlx ) V F(xrlx)), i.e. ~Cl(x). If
y is a class, which exactly contains all classes, b - {x" x~y A-~x~x} is a
class and br]y: hence br]b~(bqyAb-~b)~b-~b, i.e. b is not a class"
contradiction !El
n-ary predication can be reduced to unary one by adding parameters:
9.4. LEMMA (Parametrization). NMF-proves:
II.9A]
57
Type-free Abstraction, Predicates and Classes
VXl... VXn+m((Xl...
g::}(xn_l_l... X n + r n ) r l { U n + l .
Xn+rn)rl{Ul...
Un_kr n 9 A } r
. . un+rn:A[Ul " - X l , . . . , ttn . -
xn] }.
We now consider some useful approximations to the naive abstraction
principle, i.e. to versions of AP where r is replaced by the standard
biconditional.
9.5. DEFINITION
(i) A formula B is elementary in the list X l , . . . , x n iff B is built up
from e-atoms, negated e-atoms, T-atoms of the form trlx i and their
negations -~trlxi (1 < i < n ), by means of A, V ,Vy, 3y (y ~ {Xl,...,Xn});
(ii) a formula B is quasi-elementary in x l , . . . , x n iff B is built up from
e-atoms and negated e-atoms, arbitrary T-atoms, and negated T-atoms of
the form -~t~x i (1 < i _< n), by means of A, V and Vy, 3y (y ~ {Xl,...,Xn} ).
9.5.1. REMARK. (i) B is T-positive iff B is (up to logical equivalence)
quasi-elementary in the empty list of variables.
(ii) If B is elementary, then B is trivially quasi-elementary; moreover, the
negation of an elementary formula is always elementary (up to logical
equivalence).
The notion of elementary condition for type-free languages is adapted
from Feferman (1975). We say that B is (quasi-) elementary tout court iff
B is (quasi-) elementary in some list X l , . . . , x n. With the notions of 9.5, we
obtain a useful generalization of 8.8 (i)-(ii):
9.6. LEMMA
(i)
Let A be quasi-elementary in Xl,... , x n. Then:
MF(ii)
F CI(Xl) A . . .
A Cl(xn)---+
(A ~ TA).
If A is elementary in Xl,... , xn,
MF- F C l ( x l )
A...
A Cl(xn)--+
Prop(A).
PROOF. (i) By 8.8 and induction on A, using the hypothesis on Xl,...,Xn,
whenever A - - , t q x i. (ii): by (i), classical logic and 9.5.1 (ii). F!
Lemma 9.6 and the abstraction principle 9.2 immediately imply 9
9.7. COROLLARY. (i) If A(V, X l , . . . , x n ) is quasi-elementary in
Xl,...,Xn,
M F - ~ Cl(xl) A ... A Cl(xn)--+ Vy(y~{v: A(V, Xl,...,Xn) } ~-+A(y, x l , . . . , X n ) ).
(ii)
If A(v, Xl, . . . , Xn) is elementary in Xl,... , Xn,
M F - F Cl(xl) A . . . A Cl(xn)--+Cl({v: A(V, Xl,...,Xn)}).
58
Extending Operations with Reflective Truth
[Ch.2
Corollary 9.7 yields the so-called elementary comprehension schema, in
short EC (Feferman 1975). It may be asked whether classes are closed under
a strengthened schema, where "elementary" is replaced by some reasonable
notion of "second-order condition", e.g. the formula A admits a standard
interpretation in second-order logic. In Ch.VII, we shall prove that a
second-order impredicative comprehension schema is consistent with MF,
but there are models (e. g. the inductive model of w which falsify it.
9.S. DEFINITION
(i) Vxrlt.A "- Vx(xrlt --. A);
3xrlt.A "- 3x(xrlt A A);
(ii)
"f is a family of classes indexed by a " : - Vxrla.Cl(fx);
(iii)
{{Xl,...,xn)" A(Xl,...,xn) } ": - { x ' x - ( ( X ) l , . . . , (X)n) A A((x)I , . .., (x)n)} (for (x)i, see w
(iv)
E(a, f) "- {(x, y)" xTla A yrl(fx)} ( - generalized direct sum or join);
(vi)
I I ( a , f ) : - {g" Vxrla. (gx)rl(fx)} (-generalized product).
(iii) is justified in N M F - w i t h pairing axioms:
9.8.1.
(al,...,an)rl{(Xl,...,xn): A(Xl,...,xn) } r A[x 1 := a l , . . . , x n := an].
9.9. PROPOSITION (The Join Principle J). CL is closed under generalized
sums over families of classes, indexed by classes. Formally, MF-proves:
(i) w(~,r,(b, f ) ~ 3~3y(~ = (~, ~)^ ~,b ^ y,(f~)));
(ii) Yx~lb.Cl(fx) A Cl(b) --, Cl(E(b, f)).
PROOF. (i) is an immediate application of 9.7 (i) and 9.5.1.
(ii): let f be a family of classes indexed by the class b and let:
A(u) := 3x3y(u = {x, y) A xrlb A yrl(fx)).
By T.1, Cl(b), 8.2 (ii)-(iii), T.4.2, 9.2 (v)we get:
FA(u) ~ VxVyF(u = {x, y) A xrlb A yrl(f x))
VxVy(u r (x, y)V F(xrlb) V F(yrl(fx)))
wvy(,., 7: (~, y) v-~(~,Tb) v y~(f~))
wvy(,_, = (~, y) ^ ~,Tb ~ ~,~(f~))
w v y ( u = (~, y) ^ =,Tb ~-~(~,,7(.f~)))
~A(u).
Together with (i) and 9.2 (i), this yields:
Operations on Predicates and Classes
II.9B]
59
-~(u~)E(b, f)) ~-~A(u) ~ FA(u) ~ u~E(b, f). [3
9.9.1. REMARK. 9.9 proves the so-called join principle J; 9.9 and 9.7 show
that MF contains Feferman's system EM+J for explicit mathematics
(Feferman 1975, 1979; Beeson 1985; cf. appendix II).
As an exercise, the reader may verify the dual principle for II:
9.10. PROPOSITION (Closure under generalized products). MF-proves:
(i) C l ( b ) ~ Vg(g~lII(b, f ) ~
Vx~lb. (gx)~l(fz));
(ii) el(b) A VxTib.Cl(fx) ---, Cl(H(b, f)).
Similar arguments prove that Prop, the notion of (internal) proposition, is
closed under infinitary conjunctions and disjunctions in the following sense:
if f is a family of propositions indexed by any class c (for instance
c = {x: Nx}), there exist propositions A { f x : xTIc}, V { f x : x~lc}, satisfying:
T( A { f x : xrlc}) ~-, Vxzlc.T(f x);
T( V { f x : x~lc)).-. 3xrlc.T(f x).
w9B. Operations on predicates and classes
We extend the standard operations of the algebra of (extensional) classes
and relations to the general domain of partial properties. In particular, each
definable predicate can be generated starting from four primitive predicates
by means of eight predicate operations.
9.11. DEFINITION
1. Initial Predicates:
~Pe := {(~,y,z):
OD := { ( ~ , y ) :
9 = yz);
~ = y);
N :- {x:Nx};
~-:= {x:
Tx}.
2. Basic Operations:
Singleton
{a} := {x : x = a} ;
Complement - a
:= {x: -~xTla};
Intersection
a f-1 b "- {x" xqa A xrlb};
Domain
dom(a) := {x: 3y.(x,y)~?a}.
Extending Operations with Reflective Truth
60
[Ch.2
3. Combinatorial operations:
Expansion
Exp(a) :-- {(x,y): y~a};
Converse
Cony(a) :- {(x,y) : (y,x)rla};
Cycle
Cyc(a) :-- {(x,y,z) : (z,x,y)ria};
Transpose
Tress(a) :- {(x,y,z): (x,z, ylria};
(x, y, z, a, b denote distinct variables; remind that ( x , y , z ) - ((x, y),z)).
4. We say that CL is closed under a given n-ary operation H, provably in a
theory ~l', if C l ( a l ) A . . . A C l ( a n ) ~ Cl(H(al,..., an) ) is provable in ~';
5.
a - b :- Vx(xria r xrib);
6.
a C_ b "- Vx(x~a - , x~b) and a - eb "- a C_ b A b C_ a.
{Clearly, if a and b are classes, then a - e b ~ a - b)}.
7. EXPL is the collection of s
which is inductively generated by
the clauses: ~PP, DD, N, Y E EXPL; if t is a variable or a constant of .5o_,
{t} E EXPL; if t E EXPL and s E EXPL, then dora(t), tM s , - t , Conv(tl,
Exp(t), Cyc(t), Trans(t) are elements of EXPL.
If t E EXPL, we say that t is an explicit predicate of .5. The subcollection
ELP of elementary predicates is inductively generated as EXPL, except that
we omit Y from the initial clauses of EXPL, and we add the condition" if t
is a variable, Exp(t) E ELP.
A trivial application of elementary comprehension 9.7 (ii) yields"
9.12. LEMMA. CL contains f~DD, DD, N and is closed under the operations
of 9.11.2-9.11.3, provably in MF-.
9.13.THEOREM (Explicit abstraction)
(i) For every formula A of s for every n, we can effectively define a
term vn(A) E E X P L with F V ( r n ( A ) ) - F V ( A ) - { x l , . . . , x n } ,
such that,
provably in MF-:
vn(A ) - {(xl,...,xn)" A}.
(*)
(ii) Assume that A is elementary in u (where u may be a finite list of
variables): then the term rn(A ) can be chosen in ELP.
P R O O F . (i) The argument
parallels the well-known class theorem for
Gbdel-Bernays set theory. We proceed by induction on the built-up of A.
The inductive step is easily handled by means of complement, intersection
and domain; therefore we only need to find, for each n, a predicate rn(A),
Operations on Predicates and Classes
II.9B]
61
satisfying (~) above, whenever A - Nt, Tt, t I - t 2. On the other hand,
t 1 - t 2 is equivalent (mod r
to 3 y ( y - t l A y - t 2 ) .
Now we can find,
uniformly in r, n, i with 1 _< i _< n, a term er'n(r ) such that:
crin(r)- {(Xl,...,Xn)" X i - - r } .
Hence:
{(Xl,. ..,
t2} -
Xn > 9tl _
~.n-I- 1
d o m ( ~ nn-t-1
+ l ( t l ) fq~n+](t2)).
The definition of ~r/n(r) can be reduced to the construction of elementary
predicates I~PPn(i , j , k ) and ODn(i, j), where 1 < i, j < n such that:
f~pPn(i,j,k)-
{<Xl,...,Xn). x i - X j X k }
ODn(i,j ) - { ( x i , . . . , X n ) "
x i -xj}.
(1)
In turn, (1) is verified by induction on n with the help of ~PP, 0I) and the
combinatorial functions (details are in appendix II). A similar argument
works if A - N t or T t where, of course, we must use N and 3-.
(ii) It is enough to deal with the atomic case A(Xl, u) - tou , where Xl, u are
distinct variables, possibly free in t. But there is an elementary predicate
(r2(t) -- {(Xi,X2>" Z 2 -- t}; if we choose ezt(u,t) - dom(a2(t) f3 E x p ( u ) ) ,
ext(u, t ) i s elementary and ext(u, t ) - {xl" trlu }. [3
By application of elementary comprehension
generalized sums and products, we easily have:
and
closure
under
9.14. PROPOSITION. C L is closed, provably in MF-, under the operations
defined in the following list.
1.
Pair"
{a,b} : - {x" x - a V x - b};
2.
PirectSum:
a@b:--{(x,y)'(xT?aAy--O)V(x~bAy--1)};
3.
Cartesian Product:
4.
Exponenlialion: [a--,b] "- { f " Vxqa.(fx)qb};
5.
Universe and empty class:
m
v.-
6.
m
a | b "- {(x, y)" xrla A yrlb};
-
0
.
-
Generalized union and intersection:
n fz .- {u. Vz,Tb.u,7(fz)},
U far
{u 93xrlb. u,l(fx)} , xob
xrlb
provided b is a class and f : b ~ CL.
9-
9.14.1. REMARK. In analogy with the generalized closure condition of
P r o p under implication, it holds, provably in MF-:
Extending Operations with Reflective Truth
62
[Ch.2
Cl(a) A (3x(xrla) ~ Cl(b))----, Cl([a---, b]) A V f(frl[a ~ b] ~ Vxqa. (fx)rlb).
We end up the section by considering a natural question: is there any
reasonable counterpart at the level of predication of the set-theoretic power
set operation? Let us try two obvious alternatives:
P(a) "- {x" x C_ a}; P+(a) "- {x" xrlCL A x C_ a}.
Unfortunately, the former is unmanageable, while CL is not closed under
the operation P+:
9.15. PROPOSITION
(i)
M F - F ~VyVx(x C_ y ~ x~?P(y));
(ii)
M F - F Vz(z~?P+(y)~ Cl(x) A z C_ y);
(iii)
MF-F~Vx(CI(x)~CI(P+(x)).
PROOF.(i): choose x - CL and apply 9.3.
(ii) is obvious. As to (iii), choose y - V ( - universal class) and verify that
Cl(P+(V)) implies Cl(CL): contradiction !O
Clearly, both operations ~x.P(x) and ~x.P+(x) are C_-monotone (e.g.
a C_ b implies P(a)C_ P(b), etc.); however, in the present framework we can
also define an intensional power set operation"
9.16. DEFINITION (Weak power set). P w ( b ) " - {x " 3 y ( x - y
gl b)}.
Then by elementary comprehension 9.7 we have, provably in MF-:
9.17.
Cl(Pw(b)) A Vu(u C_ b---, 3v(wlPw(b) A v -- e u)) A
where - e is the extensional equality of 9.11.6.
The interest of Pw(b) is limited by the fact that, even if b is a class, we
cannot predict whether any given subproperty of b is itself total. A
weakening of the power set operation along constructivistic patterns, looks
more promising and it leads to consider the decidable subclasses of a given
class x. If c is a class, a decidable subclass of c has the form
{x" x~c A f x -- 0}, for some f" c ~ 2 (here 2 - {0,1}).
9.18. DEFINITION. P d ( a ) " - {y" 3f3y(y - {z" xrla A f x -- 0} A f~[a --, 2]}.
Then we have:
9.19. PROPOSITION. MF-proves:
Cl(c) ~ (Cl(Pd(c)) A Vb(brlPd(c ) ---, Cl(b) A b C_ c)).
Fixed Point Theorem for Predicates
II.10A]
63
We underline that Pd(c), for certain c's, can be very large: it is consistent
to assume that P d ( N ) i s a model of second-order comprehension (cf. 5.13).
w10A. The fixed point theorem for predicates
We know from the previous sections that the logical universe described by
MF-, though not well behaved under impredicative quantification over
classes and propositions, is closed under natural infinitary constructive
operations, which go beyond the limits of elementary logic. We now
consider the problem of inductive definitions; we shall see that M F - can
prove the existence of solutions to a number of recursive conditions, but in
general such solutions cannot be shown to define total predicates, or to be
extremal (i.e. minimal or maximal). The main tool is the property-theoretic
analogue of the second recursion theorem. Indeed, if we combine the fixed
point theorem for operations and the abstraction schema, we get the simple
and fundamental:
10.1. T H E O R E M (Fixed point for predicates)
(i) Let A(x,y,v) be a formula with the free variables shown only. We
can f i n d - uniformly in A - a
is v, such that M F - p r o v e s :
term IxyA(x,y,v),
whose only free variable
IxyA(x, y, v) = {x: A(x, IxyA(x, y, v), v)} A
A Vu(u~IxyA(x, y, v) r A(u, IxyA(x, y, v), v)).
(ii) /f A(x,y,v) is quasi-elementary in v (a fortiori if A is T-positive,
see 8.7), and we assume that v is a class, r can be replaced by ~-~.
PROOF. Choose:
IxyA(x,y,v) := FP(Ay.{x: A(x,y,v)}).
Hence u q l x y A ( x , y , v ) c ~ A ( u , IxyA(x, y, v), v) by 2.3 and 9.2. The second
part follows with 9.7. D
10.1.1. NOTATION. 10.1 introduces a new variable binding operator I. If
x, y are clear from the context, we shorten IxyA(x,y, v) to I(A, v); we also
let I ( A , v ) : = I(A) if the dependence on the variable v is not explicitly
needed. Of course, the definition of I makes sense in the general case where
v is a finite (possibly empty) list of variables, apart from the bound x and
y.
10.2. Examples
(i)
The notion I S of "ileralive set" (or "hereditary class").
64
Extending Operations with Reflective Truth
Let I T S ( x , y ) := Cl(x) A x C_ y and choose I S ( x ) : = xrII(ITS).
have, by 10.1(ii) and exploiting the classhood of x:
10.2.1
MF- ~ I S ( x ) ~
[Ch.2
Then we
Cl(x) A Vz(z,lx ---, IS(z)).
In chapter V we shall prove that I S (actually, a generalization) yields a
well-behaved set-theoretic universe.
(it) The intensional notion of (finite) type over N.
Let T P ( x , y ) be the formula:
[(= = N) v 3c3b(~ = [b-~ ~] ^ b~y ^ c ~ ) v ~b3c(~ = b | c ^ b ~ ^ c~y)].
If T Y P E ] " - I ( T P ) , we get, provably in MF-, that T Y P E / c o n t a i n s N and
is closed under Cartesian product and exponentiation.
(iii)
Let:
The construclive second number class O.
O~d(., y ) . -
=~N ^ [(. - T) v 3 k ( . - 2 k ^ k~y) v
V 3kVn3m({k}(n) ~_ m A mTly A x -- 3k)]
(for the notations 2 k, 3 k, recall the convention 3.6.1). If O "- I(Ord), M F proves the fixed point axiom
10.2.2.
Vm(m~O ~ Ord(m, 0)).
By inspecting the examples above, we are naturally led to lift to the
present framework the notion of (positive) operator, which is familiar from
the standard theory of inductive definitions (Moschovakis 1974).
10.3. DEFINITION
(i) A formula A(v) is operative in v iff either v does not occur in A or
A belongs to the least class of formulas, which is inductively generated by
means of A, V, Vy, 3y (y distinct from v) from formulas of the form Nt,
-~Nt, t - s, -~t - s, tTlv, Tr, -~Tr, provided v does not occur in t, r, s.
(it) If A is a formula, which is operative in v and T-positive, the term
~ v . { x l . . . x n : A ( x l , . . . , X n , V)} is called an operator, n being the arity of the
operator; A may contain free variables ~ {Xl,... ,Xn, v}.
(iii) An operator Av.{xl...Xn: A(Xl,...,Xn, V)} is elementary if every
T-atom, which occurs in A, has the form t,lv.
(iv) An operator ~ v . { x l , . . . , Z n : A(Xl,...,Xn, V)} is existential iff no
universal quantifier occurs in A, except possibly for universal bounded
number quantifiers of the form Vm < n (see 3.1).
II.10A]
Fixed Point Theorem for Predicates
65
10.3.1. CONVENTION. We henceforth identify any given operator with the
formula defining it; thus we simply call A(Xl,...,Xn, V) an operator in v tout
court, and X l , . . . , x n show its arity. In the following we only consider
operators of arity 1 (or 2 at most; this is not restrictive by pairing).
The idea is that any given operator defines a monotone operation (with
respect to C_ of 9.11), transforming relations (represented by) v into
relations (represented by) { X l . . . x n ' A ( X l , . . . , X n , V)}. The notion of
existential operator is suggested by the formula involved in the definition of
T Y P E I , while Ord(x, y) is elementary and I T S ( x , y) is not.
In set theory (or in a suitable fragment of second-order logic), the
Knaster-Tarski theorem grants the existence of fixed points; here we have,
by induction on the build-up of the given operator and by 10.1 (ii)"
10.4. PROPOSITION. ( i ) / f A(v) is operative in v, then in pure logic
A(v) A v C_ u ~ A(u).
Hence if A(x, v) is an operator, )~v.{x" A(x, v)} is C_-monotone, provably in
MF-:
a C b ~ {x" A(x,a)} C_ {x" A(x,b)}.
(ii) For every operator A(x, v), I ( A ) : - IxvA(x, v) is a fixed point, provably
in MF-:
Vu(urlI(A ) ~ A(u, I(A)).
10.4.1. REMARK. We warn the reader that the class of T-positive
formulas, which are operative in the variable v, is not closed under
substitution" e.g. A(x, v) - XrlV is an operator, but A(v, v) - VrlV is not.
Proposition 10.4 raises two natural problems. We first wonder whether
there exists a more mathematical characterization of operators, which does
not refer to the syntax: the question will be answdred in the positive in
Ch. IV, where we shall prove that operators coincide with extensional (or
C_-monotone) operations and have a natural topological interpretation. A
second question concerns the apparent weakness of 10.4(ii): it only offers
implicit solutions to the given condition and states nothing about the
minimality (or the maximality) of the solution I(A), which is essential to
argue by (generalized) induction on the given I(A). In the next chapter, we
shall see that such limitation is essential: MF- is consistent with different
hypotheses on I(A) and hence MF- is formally unable to distinguish among
them.
However, proposition 10.4(ii) already establishes a simple link with
standard subsystems of second-order arithmetic ( - analysis, in short).
Extending Operations with Reflective Truth
66
[Ch.2
Indeed, let us consider the standard language s of PA ( - first-order
arithmetic), expanded by a new unary predicate symbol P; an arithmetical
operator is simply a formula A(x,P) of s which is positive in P (i.e. A is
logically equivalent to a formula, built up from atoms of the form Pt, t = s,
--t = s by means of V, A, 3, V), and has the free variable shown only.
s
, the language of the elementary theory of inductive definitions, is
obtained from the language of PA, by adding a distinct unary predicate
symbol I A, for each arithmetical operator A(x,P).
A
10.5. DEFINITION. ID 1 is the first-order theory in the language Z(ID1) ,
which contains Peano arithmetic PA (with the induction schema, extended
to all formulas of s
) and the fixed point axiom:
FP(A)
VX(IAx~A(X, IA)),
for each arithmetical operator A (of course, A(X, IA)results from A(x,P)
by replacing each subformula of the form Pt with IAt ).
It is clear that every arithmetical operator becomes an elementary operator
in the sense of 10.3 above, once we replace "Px" with "xr/v" (we can
assume that v is a fresh variable); thus, if we choose N as the range of
individual variables and we interpret "IAX" by "xrlI(A)" , we readily
obtain:
A
10.6. PROPOSITION. ID 1 is interpretable in MF.
Since I131 has the same arithmetical theorems as the subsystem EI-AC or,
equivalently, Predicative Analysis of levels < e0 (cf. Aczel 1977, Feferman
1982 and Ch.VIII), we also have a lower bound on the arithmetical content
of MF (which turns out to be sharp by the proof theory of chapters IX-XI).
At this stage of the investigation, it becomes essential to calibrate the
strength of the number-theoretic induction available; so we explicitly
introduce two finitely axiomatized subsystems of MF, which restrict NIND
to arbitrary properties and to classes respectively.
10.7. DEFINITION
induction).
(Subsystems of MF with restricted number-theoretic
(i) Set Clos(y) "- (Orly) A Vx(xrly~(x+l)77y). Then"
Property N-induction P-NIND:
Class N-induction CL-NIND:
(for C see 9.11.6).
Vy(Clos(y)~N C_ y);
Vy(Clos(y) A C l ( y ) ~ N C_ y);
II.10A]
Fixed Point Theorem for Predicates
(ii) MFp "- MF-+P-NIND;
67
MFc := MF-+CL-NIND.
Later, we shall prove that the inclusions MF c C_ MFp C_ MF are proper;
while MF c is proof-theoretically equivalent to OP, MFp already proves the
consistency of OP. If we restrict our attention to existential operators, then
we can give a standard inductive definition of minimal fixed points.
10.8. LEMMA. Let A(x,u) be an existential operator. Then we can find a
term Ax.IXA such that, provably in MF"
(i) Vn(I~ -- { x : - , x -
x} A InA+1 -- {x" A(x,I~4)});
(ii) VnVp(n < p---, InA C I~);
(iii) Vx(A(X, I A ) ~ 3kA(x, IkA)).
PROOF. (i)-(ii): we apply recursion on N (see 3.2), N-induction and
proposition 10.4(i). As to (iii), proceed by outer induction on the build up
of A; here it is necessary that A is existential and we apply the true
arithmetical schema:
Vn(n < m--~ 3 k B ) ~ 3jVn(n < m ~ 3k(k < j A B)),
which is provable by induction on m. O
Now choose:
10.9. THEOREM. MF proves (for B arbitrary, A existential operator):
(i) gx(A(x, IA) ~ x~7IA);
(ii) Vx(A(x,B)---, B(x))---~ Vx(xOI A - , B(x)).
PROOF. (i): by 10.8 (iii), (i).
(ii)" if we assume the premise of (ii), VnVx(xrlI~4---, B(x)) is derivable by
means of N-induction and monotonicity. O
10.9.1. REMARK. If the existential operator defined by A(x,u) is
elementary in u, then by property N-induction, elementary comprehension
9.6 and 9.14, we have:
MFp F VnCl(InA)A CI(IA).
(1)
Under the same hypothesis on A, (1)implies that 10.8 (ii)-(iii) are provable
in MFp. Hence, 10.9 (i) together with
Extending Operations with Reflective Truth
68
[Ch.2
is already derivable in MFp. (1) also holds if every T - a t o m occurring in
A ( x , u ) - e x c e p t those of the form trlu-is of the form sr/r where r can be
proven to be a class. Hence 10.9(i) and the special case of 10.9(ii) with
B(x) = xTlu are derivable in MF p "
wlOB. Applications to semantics and recursion theory
The fixed point theorem 10.1 turns out to be a significant tool in general: in
most applications, one only needs the existence of a solution and not its
m i n i m a l i t y / m a x i m a l i t y . We illustrate the theme by showing the existence
of a partial satisfaction predicate for 2, and then by proving two abstract
versions of well-known results, due to Rice and Myhill.
First of all, OP is obviously sufficient (see w to carry out a primitive
recursive arithmetization of the L-syntax; so, we can fix an effective GSdel
numbering [ ] and we let [E] stand for the (canonical term of L
representing the) GSdel number (in short gn) of the expression E. For later
applications, it is also convenient to define the satisfaction predicate over
GSdel numbers of arbitrary terms, possibly encoding formulas via [... ]. To
this aim, we say that t is a formula-term if t = [A], for some formula of s
10.10. D E F I N I T I O N
(i)
If f is any term, f( +
) is the operation defined by
x
f ( xi ) i - x
,
f ( xi ) n - fn, if - - , n - i .
Here we suppose that n, k, i range over N. Clearly f(/=)is well-defined,
uniformly in i, x (apply definition by cases on N ; see 2.1, NAT.2).
(ii) By 3.6, we may assume that there are formulas and terms in the
T-free part Lop of the language s defining the following notions:
Ter(x) := x is the gn of a term;
F o r ( x ) "- x is the gn of a formula of s
Fort(x): = x is the gn of a formula-term [A];
f t r ( z ) := the gn of [A] if z is the gn of A (i.e. ftr([A]) = [[A]]);
vr(x) := the gn of the z-th variable;
tr(z) := the gn of [Tt], if z is the gn of t;
id(x, y):= the gn of [t = s] if z = It], y = Is];
nat(z) := the gn of [Nt] if z = [t];
neg(z) := the gn of [-~A], if z is the gn of [A];
and(x, y):= the gn of [A A B] (x gn of [A], y gn of [B]);
all(x, y):= the gn of [VziA ] (x = [zi] , y gn of [A]);
Applications to Semantics and Recursion Theory
II.IOB]
69
app(x, y):-- the gn of (ts) (x gn of t, y gn of s).
10.11.LEMMA. There exists a term Val such that, provably in OP-,
Yal([tl, f ) -
t[x o "-- f O , . . . ,x n "-- f-i],
for every term t with free variables in the list Z o , . . . , x n.
In particular, O P - proves
Val([[A]], y ) -
m
[A[~ o " - f 0 , . . . , ~ n " - f n ] ] ,
for every L-formula A with free variables in the list Zo,... , z n.
PROOF: we define an operation Val such that"
Yal ([vi], f) - f~;
Val ([c], f ) - c if c is a constant;
Val ([t~l, f ) -- Val(rtl,/)Va/(r~l, f).
Val is well-defined by means of the operation D and fixed point theorem
2.3 (it is essential that the operation F 1 is N-valued and that the language
has a finite number of constants). E!
In order to introduce the satisfaction predicate, let S ( z , v )
formula saying that z has the form (m, f) and
1.
2.
3.
4.
5.
6.
m -mmm m m -
be the
id(n, k) and Val(n, f ) - Val(k, f ), or
nat(n) and N ( Y a l ( n , f)), or
tr(n) and T ( Y a l ( n , f)), or
nag(n) and -~(<n, f)r/v), or
and(n, k) and ((n, f)rlv A. (k, f)rlv), or
all(vr(i), n) and Vx((n, f(~))rlv ).
Then, if we apply the fixed point for properties (10.1) and we define
Sat(x, y) "- <x, y)r/I(S),
we have:
10.12.THEOREM. (i) For every L-formula A,
MFc ~ Sat([[A]l, f ) <=~A[xo "- f O , . . . , x n "- f-i];
( X o , . . . , x n contain all the free variables of A).
~f v ( ~ ) . - Sat(~,go), w~ ha~ fo~ ~v~y Z-~nt~.c~ C:
MF c ~ V ( [ [ C ] l ) ~ C.
(ii)
Moreover Sat satisfies, provably in MFe, the adequacy conditions:
Sat(nag(n), f ) r -~Sat(n, f);
Extending Operations with Reflective Truth
70
3.
4.
5.
6.
Sat(and(n, m), f) ~-, Sat(n, f) A Sat(m, f);
Sat(all(vr(n),m),f)~Vu.Sat(m,f( ~ ));
Sat(tr(n), f) ~ T(Val(n,/));
Sat(nat(n), f) ~ g(Yal(n, f));
Sat(id(n,m),f) ~-, Yal(n,f) - gel(re, f).
(iii)
ME c F -~V([[L]])A--V([[~L]]), for some L-sentence L.
.
[Oh.2
PROOF" the definition of Sat is already available. Condition (i) follows by
metamathematical induction on A, while (ii) is a consequence of the
definition of Sat and (iii) is Tarski's theorem. 0
Of course, by the first clause of 10.12 (ii), V cannot be an adequate
truth definition; however, if A is a sentence of OP, MF c F V ( [ [ A ] ] ) ~ A
and we obtain:
10.13. COROLLARY. MFp proves the consistency of OP and hence is
proof-theoretically stronger than Peano Arithmetic.
PROOF. Let Provop(X ) be the formalization of "x is the gn of a formula
provable in OP". Then we apply P-NIND with X - {x" Nx A Y(ftr(x))}
and we verify the formalized soundness theorem:
Vn(Provop(n)---,n~X ).
The argument does not work in the subsystem with class N-induction only,
because the predicate involved is strictly partial.
After a glimpse into formal semantics, we illustrate the fixed point
technique by lifting two well-known propositions of Recursion Theory to the
present setting. Indeed, we can easily formalize a general form of Rice's
theorem (see Rogers 1967, Fitting 1981), which is a useful tool to test the
non-classhood of certain properties.
10.14. DEFINITION. We recall that x - e
equality modulo 7/);
Y "-Vu(urlx~u~Y) (extensional
(i) b is eztensional iff, for every z, y, if x - e Y and yrlb, then xrlb.
(ii) b c c "- (b C_ c) A~(cC_b);
b is proper if b C V - {z" x - x}.
For instance, {u: 3y(yqu)} is extensional; but C L = {x: Cl(z)} is not
extensional (provably in MF-). In fact, recall that N - {x: Nx} is a class
and clearly, if r is the Russell property of 9.3, b - {x: Nx V r r / r } - - a N .
Were b a class, we should have Vu((Nu V r~r) V (-,gu A r~r)). Since r~r,
rr/r are provably false, we should have Vu.Nu, against remark 3.9.2.
II.IOB]
Applications to Semantics and Recursion Theory
71
10.15. T H E O R E M ("Rice generalized";provable in MF-). If a is non-empty,
extensional and proper, then there exists no class c such that c - e a.
PROOF: by contradiction, assume that d is a class such that d - e a and
choose b, c such that br/a and not cr/a. Then by 10.1 we can find a property
I such that I - {x:(xr/b A - I r / d ) V (xr/c A/r/d)}; so, by classhood of d,
Vx(xr/I ~ (xr/b A -~Ir/d) V (xr/c A Ir/d)).
(1)
If Ir/a, then Ir/d and by ( 1 ) a n d consistency V x ( x r / I ~ x r / c A I r / d ) , i.e.
I - e c, whence by extensionality, cr/a: contradiction. If-~Ir/a, then -~Ir/d
and I - e b, whence Ir/a" contradiction! 0
10.15.1. APPLICATION
{x" 3y(yr/x)}, { x ' c C_ x} (c is a class), cannot be made equiextensional to
classes. The same holds for the property of being a finite property. Indeed,
let f" k H b (k in N) mean that f is a bijection between {x" N x A x < k}
and b, and assume that F I N - {b" 3 k 3 f ( f " k H b ) ) is a class. Then b is an
element of FIN iff, for some k and f, f" k ~ b. Clearly, if c - e b holds, also
f" k~--~c, i.e. cr/FIN. Hence, since FIN is non-empty, we must have
V C_ FIN, which is absurd (N is not in FIN provably in MFc).
We outline an extension of Myhill's theorem about recursive equivalence
of creative sets (Rogers 1967). Being a creative property, again implies nonclasshood. We closely follow the proof, given by Fitting (1981).
10.16. DEFINITION. Let a U b "- {x:xr/a V xr/b}. Then:
(i) b is creative iff 3fVa(a f3 b - e 0 ~ ~(fa)r/(b U a)).
(ii) b is m-reducible to c (in short b < m e) iff 3fVx(xr/b ~ (fx)r/c).
10.16.1. Example: b - {x" xr/x} is creative. Indeed, choose f if a fq b - e 0, we get -~ar/(b U a).
Ax.x. Then,
10.17. T H E O R E M (MF-). b is creative iff every c is m-reducible to b.
The proof is a consequence of two lemmata.
10.17.1. LEMMA. If b is creative and b <_ m c, then c is creative.
PROOF. Assume, for some f,
(1)
There exists an operation g, such that
a n b - e 0--,-~(ga),(a U b).
Choose h -
~x.f(g(tx)), where t x -
{u" (fu)~x}. The claim is that
(2)
72
Extending Operations with Reflective Truth
a I'1 c -- e 0--*-~((ha)rl(a U c)).
[Ch.2
(3)
First note that a N c - e 0 implies ( t a ) N b - e O (by (1)). Hence by (2), g(ta)
does not belong to (re)U b, which implies by definition ~(ha)~(a U c). D
10.17.2. LEMMA
We can define a term f such that, if y is in b, then f y = e ( g ( f y ) )
is no~ in b, then f y =cO"
and if y
PROOF. Define hey = {z: yrlb A x = gey}. By fixed point, choose f such
that h f y = f y: hence
w(~fy
~ v~b ^ 9 =
gfv).
n
P R O O F of 10.17. Assume that b is creative; for some g, we have
cnb
- ~ 0 - ~ ~ ( g c ) ~ ( c u b).
(1)
It follows:
c = ~ 0--,-,(gc)~b;
c = e {x} A -~xrlb --, ~gc = x.
(2.1)
(2.2)
By 10.17.2, we find an operation f such that:
uTlc---~ f u - e {g(fu)};
(3)
-~u~c ---*f u = e O.
It is immediate to check that c is m-reducible to b via h - A z . g ( f x) using
(2)-(3).
In the other direction, assume that every property is m-reducible to b" then
{z: zrlz} < m b . Therefore b is creative by 10.16.1 (example) and 10.17.1.17
10.18. COROLLARY (provable in MF-)
No creative property can be equieztensional to a class.
PROOF: pick any property c not equiextensional to a class (such c exists by
9.3 or 10.15.1). Were b creative and equiextensional to some class b', we
should have that c is extensionally equivalent to c ' = {x: (fx)~b'}, for some
f. But c' is a class: contradiction. 13
Moreover, by Rice's theorem, the property of being creative does not
define a class.
Non-Extensiona lity
II. 11]
73
w11. Non-extensionality
The informal intuition behind the theory M F - and its extensions obviously
suggests an interpretation of properties and classes, which is radically
opposed to a world of extensional entities. We show that extensionality
fails, already in a very weak fragment of MF-. It is impossible, in general,
to identify arbitrary empty properties.
11.1 THEOREM. (Gilmore 1974)
(i) We can produce terms t, s such that ( t intuitionistic logic with
where A is T - p o s i t i v e .
(ii)
e s A-~t -- s) is provable in
equality plus the schema V u ( u ~ { x " A } ~ A ( u ) ) ,
We can show in M F - that t and s are classes.
PROOF. Define:
9-
{z" x~x}
f y "-- {x" x -
~ A YrIX }
a "-- {u" f u -- f u n q)},
where, as usual, 0 "- { x ' - ~ x - x). We first verify
fanO
-- e f a .
(1)
Trivially, urlf a N O implies u y f a. Let u ~ f a; then u - tc A a~lu , i.e. aya,
whence aya, which yields f a N 0 -- f a , i.e. u ~ f a N 0, by - -logic.
We now prove:
- - , f a n 0 -- f a .
(2)
If f a n 0 - - f a , then a~la, i.e. a~/~, whence tc~lfa and by (1) ~7/0:
contradiction!
(1)-(2) complete the proof of (i).
Ad(ii). f a n 0 is a class, because we have Vu(u-fffa n 0). As to the classhood
of f a , if -~u - to, we trivially obtain:
(u - ~ A arlu) V --,u -- ~ V a -~u,
i.e. by T.1 u~lfa V u-~fa. Assume u - ~; by (2), we have a~a, i.e. @ u ,
whence @ f a V urlfa. It follows that (ur/fa V @ f a ) holds for arbitrary u, i.e.
C l ( f a). El
Gilmore's tricky empty classes are constructed via a detour through
properties, which are not classes (like ~; see 10.16.1, 10.18). However, no
essential use is made of the combinatory structure underlying the semantic
structure. Gordeev found a quicker refutation of extensionality for classes; in
his proof, the self-referential aspect is entirely absorbed by the fixed point
for operations, and the argument has the advantage of being intrinsic (no
74
Extending Operations with Reflective Truth
[Ch.2
detour through properties) and acceptable in a fully constructive context.
The informal idea is to consider the fixed point of the property "to be equal
to the empty property""
11.2. Gordeev's Paradox. MF-proves:
3 a ( C l ( a ) A a = {x: x = 0 A x = a} A a = eq} A - a = 0).
P R O O F . By 10.1 we can find a n f s u c h t h a t f = { u : u = 0 / ~ u = f } .
By axiom T.1, f and 0 are classes. Moreover, f = e0: trivially 0 C f and
u r l f implies u = 0 and u = f, i.e. ur/0 and f C_ 0. Now f = 0 yields fr/0,
which is a contradiction. [:]
11.3. R E M A R K
(i) 11.1-11.2 imply that there are very simple non-extensional
operations on classes ; here we say that g" C L ~ C L is extensional if a - e b
implies ga - e gb for every a, b C C L .
For instance, if we choose g x - { z } - { u ' u x} and f as in the proof of
11.2, we have f - e 0, but --, {f} - e {0}" hence g is non-extensional.
(ii)
11.2 can be generalized; indeed MF-proves
Va(Cl(a)---, 3b(Cl(b) A ~a - b A a - e b)).
Hint (by Minari)" choose b - {x "(a - b A ~arla ) V (~a -- b A x~Ta)}.
11.1-11.3 do not completely settle the situation: for instance,
remains to see whether it is consistent to assume:
a - b---, a - b,
it
(,)
where a - b " - a - e b A Vx (x~a ~-, x~b) and a, b are not classes.
A straightforward modification of Gordeev's argument shows that ( , ) leads
to inconsistencies. Choose ' E - - {z" --z - x V rr/r}, where r - {x" --xr/x}
and G - {x "(x - E/~ x - G) V rr/r}. It is easy to verify that, in ME-:
-
Cl(E) ^ E -
0^ -
Cl(a) ^ E - a ^
- a.
We don't know whether the non-uniform version of abstraction
3 y ( V u ( u q y C : ~ A ( u ) ) ) is consistent with extensionality for properties. Of
course, it is possible to maintain extensionality for properties, as soon as a
specific equality relation for properties is adopted. This suggests to extend
the language with a new sort of variables, ranging over properties in
extension, in order to distinguish properties as data types (or partial
functions) from properties as f o r m a l constructs (or names); the interested
reader can consult Js
(1987), Marzetta (1993).
II. 11]
Non-Extensionality
75
11.4. On formalizing mathematical arguments
Beginning with the work of Bishop (1967), it has been demonstrated that
non-extensionality of the basic notions does not affect the formalization of
mathematical practice. Since we always work within specific mathematical
structures, we are simply required to make explicit the equality relation,
which is appropriate to the structures in question. In other words,
mathematical practice has to cope with different kinds of "equalities",
which are simply congruence relations and have to be explicitly given: think
of the introduction of "equality" in the different number systems or when a
basic equality on a given domain U of urelements is lifted to the type
structure, built upon U. Moreover, equality on a given structure may
depend on the presentation of the structure itself; for instance, consider
equality on Cauchy reals versus equality on Dedekind reals.
The reader versed in foundational applications will find the philosophy
of constructive and explicit mathematics appropriate here (see Bishop 1967,
Feferman 1979, 1985, Beeson 1985). Accordingly, one has to consider not
classes (or properties), but objects of the form ~4r- (W, - w ) , where - w is
an equivalence relation on W. This also means that the notion of operation
itself has to be specialized. If ~ - ( W , - w )
and ~ - ( U , - u )
are
structures (of the same similarity type), a function f . ~r___,~ is an
operation f" W ~ U, which "preserves equality""
x, y are in W and x - w Y
::~fx - u f Y "
Incidentally, we recall that, according to the views of Poincar~ (1913,
p.133), a class is well-defined not only if it is predicatively defined, but also
if it is possible to produce a predicative definition of the equality on
elements of that class.
As usual in the constructive practice, the power set operation z2(~r) can
be replaced by the function space 2 ~r (see w Of course, the classical step
from a structure ~ to its quotient ~47"/E, modulo a given congruence
relation, cannot be carried out: if two elements x, y of W are E-equivalent,
it is not true in general that the corresponding equivalence classes [X]E, [Y]E
are equal (in the sense of ground equality). At any rate, one can use
equivalence classes as "contextually eliminable symbols"; one should also
not forget that, in axiomatic set theory, the definition of Ix] requires some
trickery.
To conclude, let us recall that the present framework and its extension
look promising for the treatment of categorial concepts; some hints may be
found in Feferman (1977a) (although the theory is somewhat different), and
interesting developments are due to Gilmore (1990).
76
Extending Operations with Reflective Truth
[Ch.2
Appendix I" a property theoretic definition of the f'Lxed point operator for
predicates
In w
we discussed a fixed point theorem for predicates which heavily
relies on the full definitional strength of combinatory logic. It may be of
interest to observe that the essence of the construction can be recovered
under much weaker conditions.
1. DEFINITION. Let A(z,y) be a formula with the free variables shown.
We define:
D(A) := {(x, f ) : A(x, {u: (u, f)rlf })};
V(A) := {z: (z,D(A))qD(A)}.
The definition of V(A) is essentially in Visser (1989, pp. 695-96), though in
semantical context (77 being replaced by a satisfaction predicate).
The verification of the following lemma is straightforward:
2. LEMMA ("Second diagonalizalion').
(i) /f A(x,y) is an arbitrary L-formula with the free variables shown
only, then we can find a term V(A), not containing the paradoxical
combinator FP, such that MF-proves:
Vz(xT} V(A) r A(z, V(A))).
(ii) If A is T-posilive, r
can be replaced by ,-,.
We underline that V(A) can be used in place of I ( A ) i n all the relevant
applications of this book. This is important when we wish 1o weaken lhe
operational basis of the theory and to avoid full combinatory logic. In fact,
the lemma suggests the consideration of a pure properly theory PT, based
on the following language:
(i) three binary predicates - ,
77, ~ and a unary predicate T;
(ii) three function symbols for pairing function (-,-) and left and right
projection ()1, ()2;
(iii) the operators [] and { }.
Terms and formulas of PT are defined by simultaneous induction: variables
are terms; if t,s are terms, t - s , trls, t~s, Tt are formulas; if A is a
formula, [A] and {z: A} are terms (the free variables of [A] are exactly the
free variables of A; the free variables of {z: A} are exactly the free variables
of A minus z); if A and B are formulas, -~A, A A B, VzA are formulas; if t,
s are terms, then (t,s) and (t)l , (t)2 are terms. We again use T A as an
Appendix !
II.A.1]
77
abbreviation for T[A].
3. DEFINITION. P T is the theory based on classical first-order logic with
identity, which contains the following axioms and schemata:
AP:
Vx(xrl{u" A} ~ TA[u "- x]) A Vx(x~{u" A} ~ T-~A[u "- x]);
T.1.
T A H A , if A - (x - y), Nx, ~(x - y), ~Nx, x~?y, x~y;
T.2.1.
TTA~TA;
T.3.
T-~-~A ~ TA;
T.4.1.
T.2.2.
T-~TA~T-~A;
T(A A B ) ~ T A A TB;
T.4.2.
T-~(A A B ) H T-~A V T-~B;
T.5.1.
TVxA H VxTA;
T.5.2.
T~VxAH3xT~A;
CONS
~ ( T A A T-~A);
SYM
(x~y ~ T-~(xOy)) A (x~y ~ T--,(xrly));
PAIR
VxVy(((x, Y))I
- x A ((x, Y))2 -- Y)"
Clearly P T can be interpreted into MF- and considered as a fragment of
MF-. By inspection, D(A) and V ( A ) c a n be regarded as PT-terms and
hence we have:
4. P R O P O S I T I O N . The second diagonalizalion lemma holds for FT.
Of course, we can expand P T with the notion of natural number, some
sort of number-theoretic induction schema and axioms for successor, zero,
plus, times and additional number-theoretic operations. The resulting
theory would have a non-trivial mathematical content; yet, in comparison
with M F - and its extension, P T has models in which - is decidable and is
compatible with the axiom that everything is a number. Thus it could be a
basis for applicative refinements of the theory of abstraction (cf. Ch. XIV).
Appendix II- on the explicit abstraction theorem
We give details for the proof of the explicit abstraction theorem 9.13; then
we conclude with a few comments about the applications of explicit
abstraction and related problems.
1. LEMMA (Combinatorial operators)
We can define ELF-terms Prod(a, b), V, Y n, Sap, Sap-, RE(n, k, a) (for
l < n,k), LE(n,k,a), (for l < n, l < k< 3), Ins(k,i,a) (for l < k,
i e {2,3}), which satisfy, provably in MF-:
78
1.1.
Extending Operations with Reflective Truth
Vu(uT1Prod(a , b)~-~ 3x3y(u = (x, y) A xqa A yr/b));
W(u V"
1.2
[Ch.2
3x1...
=
V u ( u q S e p ( a ) ~ 3 x 3 y 3 z ( u = (x, y, z) A (x, (y, z))r/a));
Vu(urlSep-(a ) ~-, 3x3y3z(u = (x, (y, z)) A (x, y, z)qa)).
1.3. Expansion to the right:
Vu(urlRE(n, k, a ) +--,
+--+3 X l . . . ~ x n ~ Y l . . . ~ y k ( t t
-- ( X l , . . . , X n ,
Yl,...,yk)
A (Xl,...,Xn)r]a)).
1.4. Expansion to the left:
Vu(urlnE(n, k, a) ~-*
+-~3Xl""" 3 X n 3 Y l " " 3Yk(U -- ( Y l " " " Yk' X l " " " xn) A (Xl, . . . , xn)r]a)).
1.5. k-Insertion:
Vu(urlIns(k, 3, a)~-, 3X3Xl...3xk3Y3Z(U = (X, X l , . . . , x k , Y,Z ) A (x,y,z)rla));
Vu(urlIns(k, 2, a ) ~-. 3X3Xl. . . 3xk3Y(U = (X, Xl, . . . , xk, y ) A (x, y)rla)).
P R O O F . 1.1. Choose:
Prod(a, b) - C o n v ( E x p ( a ) ) N Exp(b); Y - dom(nD), Y k+l - P r o d ( Y k, Y).
1.2. Choose S e p ( a ) = Cyc2(Conv(a))
Cyc2(t) = Cyc(Cy
and S e p - ( a ) = C o n y ( e y e ( a ) ) ,
where
(t)).
1.3: by iteration of R E ( n , 1, a) = C o n v ( E x p ( a N Vn)).
1.4: we inductively define
F(1, a) = Exp(a); F(2, a) = Sep(Exp(a));
F(n+3, a ) - S e p ( F ( n + 2 , Sep-(a))).
Then we verify by induction on n:
urlF(n , a) ~ 3v3x1. . . 3Xn(U = (v, Xl , . . . , Xn> A (Xl,... , Xn)rla ).
Finally choose LE(n, k, a) - V k+n N F(n, a).
1.5: let
G(a)-
Sep(Cyc2(Sep(Exp(Cyc(a))))).
Then we recursively define:
Ins(1,3, a) = G(a); I n s ( k + l , 3 , a) = G ( I n s ( k , 3 , a)).
The case of 2-insertion is similar. 0
We can find ELF-predicates
-~n(i), such that:
2. L E M M A .
~ee.(i, j, k), QD.(i, j), N.(i),
Appendix II
II.A.2]
2.1
l~ppn(i,j,k )
2.2
~Dn(i,j )
-
2.3
Nn(i ) -
{(Xl,...,Xn).Nxi};
-
{(Xl,...,Xn).
x i -
79
xjxk};
{(Xl,...,Xn)" x i - xj};
2.4
-[n(i) - { ( x l , . . . , x n ) " Txi};
(all the variables are distinct and 1 <_ i, j, k <_ n; in 2.1, 3 <_ n and in 2.2,
2_n).
PROOF. 2.1" we proceed by induction on n___3. If n - 3, NPPn(i, j, k) is
obviously obtained from /~PP by permutation and hence Cyc and T r a n s
suffice. If n > 3, let u, v, w denote distinct variables in the set {xi, xj, xk}.
Case a)" u, v, w already occur in the list Xl,...,Xn_l; then we apply the
induction hypothesis and expansion to the right.
Case b): u - Xn, but v, w r {Xn_l}. It is enough to observe that IH applies
to T r a n s { ( x l , . . . , X n ) : x i - x j x k } .
Case c)" v - xn, w - Xn_ 1. Apply left expansion and possibly insertion to
/~PP, whenever u r Xn_ 2.
The proof of 2 . 2 - 2.4 are similar. ['1
3. LEMMA. Let r be a term of s
For every n, i, such that 1 < i ~ n,
and x i not occurring in r, we can effectively find an elementary term ~r~n(r)
with the free variables of r except X l , . . . , X n , such that, provably in MF-:
{(Xl,...,Xn)"
X i -- r} -- gin(r ) .
PROOF. Induction on r. If r is a constant or a variable v q~ { x l , . . . , X n } , we
apply right and left expansions to the singleton {r}. If r is a variable
v C {Xx,...,Xn}, by assumption v - x k with k r
and hence lemma 2.2
applies. If r - rlr2, we use IH, dom, n and lamina 2.1. E!
Remarks and problems
(a)
The collection ELP (see 9 . 1 1 ) o f elementary terms is well behaved
with respect to substitution (if t(y) is elementary, z ~ {y}, z free for y in t,
then t[y : - z ] is elementary), and renaming of bound variables; this should
be contrasted with the collection of elementary formulas. By the theorem
9.13(ii) we can avoid the syntactical notion of elementary formula in
defining { x ' A } , if A is elementary in y.
(b)
Add a new unary predicate E C l ( x ) " - " x is an explicit class" to the
language of M F - with the axioms: 1) E C l contains DD, /~PP, N; 2) E C l is
closed under the combinatorial and basic operators of 9.11; 3) E C l is closed
under join; 4) E C l is the C_-least such property (i.e. the schema saying
80
Extending Operations with Reflective Truth
[Ch.2
that any "externally definable" property with the same closure conditions of
E e l , contains E e l ) . Note that 4)immediately implies Vx(ECl(x)---, C l ( x ) ) .
Call the resulting extension of MF- (MFc, etc.) EMF- (EMFc, etc.). Show
that all these extensions are consistent with the construction principle CP"
3y(ECl(y) ^ 9
-
CP can be regarded as a sort of abstract Suslin-Kleene theorem
(Moschovakis 1974). Prove that EMFc+CP is a conservative extension of
MF c and hence of OP.
(c)
Consider the language s
expanded with the new unary predicate
C1, the binary relation E, and f~PP, ~D, N, singleton { - } , - , f 3 , dora,
T r a n s , Cony, Cyc, E x p and E (all regarded as primitive individual
constants). Let if be the first-order extension of OP in the modified
language, which includes, besides the number-theoretic induction schema,
finitely many axioms, stating that: 1) C1 contains g~PP, nD, N; 2) Cl is
closed under singleton, complement, domain, intersection, transpose, cycle,
converse, expansion and join, and it satisfies the intended equivalences for
each of the given operations (e.g. for E x p we require:
Cl(a)----~Cl(Exp(a)) A V x ( x E Exp(a)~---~3y3z((y,z) - x A z~Ta)).
Then ~f is a possible reformulation of Feferman's system for explicit
mathematics E M 0 + J + "every operation is total" (Feferman 1979), in a
language, which avoids the use of countably many constants.
Appendix HI" independence of truth predicates from the encoding of the
logical operators
The notion of truth we introduced and axiomatized in w clearly involves
the use of the special terms
I D "- )~xy.(1,(x,y));
T R "- ~x.(2, x); N A T "- ~x.(3, x);
(1)
N E G "- Ax.(4, x); A N D "- )~xy.(5, (x, y)); A L L "- Ax.(6, x);
they represent the basic predicates and logical operations in 2.op. However,
this choice is immaterial for the development of the theory MF-. This fact
can be seen as follows. We choose an expansion 2.+ of the basic language s
in which ID, TR, N A T , N E G , A N D , A L L are new primitive individual
constants. The new constants are supposed to satisfy the set LOG of finitely
many axioms, obviously suggested by 7.1.1:
LOG1
L l x - L2Y --+ L 1 - L 2 A x - y,
where Li, L 2 E L O G 1 - { N A T , N E G , T k , ALL};
II.A.3]
LOG2
Append& Ill
81
GlXy - G2x'y' ---,. G 1 - G 2 A x - x' A y -- y',
where G1, G 2 E LOG 2 - {ID, AND};
LOG3
--1 L l X - L2Yz , for L 1 E LOG 1, L 2 E LOG2;
LOG4
-- L 1 - L2, where L1, L 2 are distinct elements of LOG 1 (.J LOG 2.
Now let MFL- be the theory which is obtained from MF-:
1) by replacing the terms ID, N A T , T R , N E G , A N D , A L L with L+primitive constants ID, N A T , T R , N E G , A N D , ALL (respectively)in the
truth axioms T.1-T.6 of theorem 7.10;
2) by adding the axioms LOG.
The reader can check that, if the axioms LOG are assumed, all theoretical
developments of part A through part C can be carried out without difficulty
in MFL-.
PROPOSITION ("Change-of-basis")
1.1. We can define a term ~+x and a map A ~ ( A ) + from the set of Lformulas into the set of L+-formulas such that:
(i) (A)+ is obtained from A by replacing every subformula of the form
Tt with T(O+t), and hence (A)+ - A if A does not contain T;
(ii) if MF- F- A, then MFL-F-(A)+ (A arbitrary).
1.2. Conversely, we can define a map A ~ ( A ) _ from the set of L+-formulas
into the set of L-formulas such that:
(i) if MFL- F- A, then MF- F- (A)_ ( A arbitrary);
(ii) A - (A)_ if A is an L-formula.
PROOF.
Ad 1.1. By applying the fixed point theorem 2.3 and definition by
cases on N, we can find a term (I)+, which satisfies the following recursive
conditions"
O_l_X- ID((x)2)l((X)2)2 if (X)l --]-;
O + x - T k ( O + ( ( x ) 2 ) ) i f (x)l - 2 ;
O+x - N A T ( ( x ) 2 ) if (X)l - 3;
(b+x - NEG(r
if (X)l - 4 ;
O+x - AND(O+((z)2)I)(r
) if (Z)a - 5;
O + x - ALL(~u.O+(((z)2)u)) if (X)l - 6 .
82
Extending Operations with Reflective Truth
[Ch.2
Then we define A~A+ by induction on A:
(i) (Nt)+ - Nt and ( t - s)+ - ( t - s); (Tt)+ - T(cI,+t);
(ii) (~A)+ - -~(A+); (A A C)+ - A+ A C+; (VxA)+ - Vx(A+).
The claim is checked by a straightforward induction on the length on the
derivation of A.
Ad 1.2. (A)_ is obtained from A by replacing the constants ID, T/~,
NAT, NE(~, AND, ALL with the corresponding terms ID, NAT, TR,
NEG, AND, ALL. gl
A second route for ensuring the independence of the truth theory on the
choice in (1) is (roughly) suggested by the analogy with recursion theory
and acceptable Gbdel numberings (see Rogers 1967, Odifreddi 1989). One
might introduce a notion of acceptable logical basis and show that the truth
theories do not depend on the choice of a particular basis.
PART B
TRUTH AND RECURSION THEORY
"Niemand kann eine unendliche Menge anders beschreiben als dutch A ngabe
von Eigenschaflen, welche fiLr die Elemente der Menge charakteristisch sind;
niemand eine Zuordnung zwischen unendlich vielen Dingen stiffen ohne
Angabe eines Gesetzes, d.h. eine Relation, welche die zugeordneten
Gegenstande miteinander verkn~pft." (H.Weyl 1918).
This Page Intentionally Left Blank
CHAPTER 3
INDUCTIVE MODELS AND DEFINABILITY THEORY
w
w
w
w
Inductive models and the induction theorem
The envelope of an inductive model
The uniform ordinal comparison theorem for inductive models
Applications of the uniform ordinal comparison theorem
We know from w that the minimal fixed points of a simple monotone
operator yield natural set-theoretic models of MF-, the so-called inductive
models. On the other hand, there exists a close relation between inductive
models and elementary inductive definability on combinatory algebras: not
only inductive definitions are crucial for the semantics of reflective truth,
but the language of reflective truth offers a natural framework for an
intensional approach to non trivial recursion-theoretic facts.
The two-sided link with generalized recursion theory is first illustrated
by the equivalence theorems of this chapter. First of all, the fixed points of
elementary inductive definitions in the sense of Moschovakis (1974) are
naturally represented by terms of our language, as soon as we work within
inductive models (w167
Second and more important, we can prove in
w a uniform ordinal comparison theorem for inductive models, which
readily implies that total predicates on a given model . ~ of O P - ( = OP
without N-induction) coincide with the collection of hyperelementary
subsets of 31~ (w
The recursion-theoretic investigations naturally lead to an extension of
the basic system M F - with simple axioms on a suitable approximation
operator 7r; 7r uniformly splits each non-empty property into a C_-chain of
subclasses, satisfying certain simple compatibility and well-foundedness
conditions. The 1r-axioms are still conservative, even in presence of a full
generalized induction schema, as far as number-theoretic induction holds for
total predicates only (see the general conservation theorem of 15.5). We
shall see that ~r-axioms are quite effective for obtaining consequences, that
usually require ordinal-theoretic arguments. In a sense, this is not
surprising: 7r-axioms are directly responsible for interpreting fragments of
set theory in Ch.V.
86
w
Inductive Models and Definability Theory
[Ch.3
Inductive models and the induction theorem
Ah is a fixed model of O P-, whose universe is the set M. As in Ch.II, s
(.~op(J~)) denotes the language of MF (of OP without the truth predicate T
respectively) with distinct individual constants, for each element of M. For
the sake of simplicity, we stick to the notations and conventions of w If I'
is the monotone operator of w O(At,) is the C_-least fixed point, which is
generated from below by transfinite iteration of I'"
(i) O(.AI~,O) - O;
(ii) O ( . A ~ , a + l ) - F(O(dll~,a));
(iii) O(dtb, A)-a ~ ,xO(ylb'a) (~ limit).
Once dtb is fixed or clear from the context, we set 0 " - O ( d l ~ ) and
O(a) :-O(Ml~,a). If S C_ M, we identify (31,,S) with S; thus we simply
write SI= A instead of (MI~,S)I= A. As usual, if tin a closed term of s
.Al~(t) is the unique value of t in Eft,.
12.1. LEMMA. If A is an arbitrary sentence of s
(,)
O ( a + l ) [= T A implies O(a)l= A, for every a.
PROOF. It is enough to check (,), by induction on the logical complexity
of A. The case where A is an e-atom is trivial. If A =--,Tt, we have, by
assumption and the inversion lemma 7.4(ii)-(iii), Al~(-~t)E O(a), whence
Jtt~(t) ~ O(a) by consistency, i.e. O(a)l=-~Tt. Let A = B A G . Again by
assumption and inversion, we have Mb([B]), Mh([C]) E O(a) C_ O(a + 1),
which implies O(a + 1)[=TB and O(a + 1)[=TC, whence we conclude by
IH O(a)[= B A C. The remaining cases are similar. Vi
12.2. (i) Below we need to consider the expansion of s
by a new unary
predicate symbol; we put s 1 6 3
(P unary predicate
symbol distinct from T). Then a realization of s
over Jft, is given by
any pair ($1,$2) of subsets C_M: it is understood that S 1 is the
interpretation of T, while S 2 is the interpretation of P.
(ii) Let A(x,P) be any formula of s
if B ( x ) i s a formula of
s
A ( x , B ) i s the s
obtained by replacing each occurrence
of Pt in A with B[x " - t ] (we assume that the substitution is legitimate).
Then, if S C_ M, we obviously have (induction on A)"
12.2.1. FACT
(i) If A(x,P) is a formula of s
P) with the free variables shown,
lhen (S, {a" a E M and SI= B(a)})l= A(a, P) iff S]= A(a, B).
(ii) If A(x,v) is an operalor (in the variable v) of s
10.3.1),
III.12]
Inductive Models and the Induction Theorem
87
and A(x,P) is the s
obtained from A(x, v) by replacing
each subformula of the form trlv with Pt, then A(x,P) is positive in P and
T (i.e. its negation normal form contains an even number of negations in
/ ont of atom of
fo m Tt, Pt). Co,v
ly
of
which is positive in T and Q, determines an operator A(x,v) of s
obtained by replacing Pt with toy (v being a fresh variable).
Clearly if A(x, v)is an operator and B(x)is a formula, it is natural to write
A(x,B) also for the result of replacing each occurrence of "tTlv" by B(t).
12.3. DEFINITION
(i) If B(x)is a formula of s
and A(x, v)is an operator of s
ClosA(B ) "- Vx(A(x, B) ~ B(x)) ( - "B is A-closed");
(ii) If B ( x ) - xrIb, we simply write ClosA(b)(- "b is A-closed").
12.4. THEOREM (Generalized induction). If A(x,v)
s
and I(A) is the fixed point term of 10.1, lhen
is an operator of
01= ClosA(B ) implies O1= Vx(xqI(A)~ B(x)).
PROOF. Assume Ol=ClosA(B ). We verify by transfinite induction on a"
for every a e ~ , O ( a ) I = a r i I ( A ) i m p l i e s Ol=B(a ).
(1)
If a = 0 or a is a limit, (1) is trivially true. Let (1) hold for a, and assume
that O(a + 1)r=aqI(A), i.e by (/?)-conversion, O(a + 1)I=TA(a,I(A)). But
Lemma 12.1 implies O(a)l= A(a,I(A)); if P is a new predicate symbol and
we set
I ( A , a ) := {a: a C ~1~ and O(a)l=a~I(A)} ,
we obtain by 12.2.1"
(O(a), I(A, a)) ]= A(a, P).
By IH, since A depends positively upon the interpretations of T and P,
(O, {a: C 31o and O I=B(a)})I=A(a,P);
0 I=A(a,B)(by 12.2.1);
hence, by A-closure of B, O [= B(a). [7
12.4.1. GID ( = generalized inductive definition) is the schema:
ClosA(B ) ~ Vx(xqI(A)-~ B(x)),
where B is an arbitrary formula of s and A(x, v) is an operator of 2..
88
[Ch.3
Inductive Models and Definability Theory
Theorem 12.4 and 7.10 imply:
12.5. COROLLARY. MF + GID is consistent.
w
The envelope of an inductive model
We fix a structure ./~ with universe M, which is a model of OP-. If S C_ M,
each closed term t of s
naturally defines a subset of M in the structure
(dtt, S), namely the set:
t(S) "- {a" a E M and S[=arlt}.
13.1
If X C_ M and X -
t(S), for some closed term t of s
we say that X is
representable in (31~,SI (or simply in S), and we put"
13.2.
ENV(At~,S)"- {X" X C_M and X is representable in (31~,S/}.
Trivially S E E N V ( ~ , S ) . We call ENV(.AI,, S) the envelope of (.AI,,S); as
usual, once ~ is clear, we simply neglect its explicit mention and we speak
of the envelope of a given S C_ M. The envelope of O(.A~) has a natural
characterization in terms of elementary inductive definability in the sense of
Moschovakis (1974). The argument is the recursion-theoretic pendant of the
generalized induction theorem 12.4. We give the basic definitions, suitably
adapted to the present context.
13.3. DEFINITION
(i) If P is a predicate symbol distinct from T and A ( u , P ) i s a formula
of s
which only contains the free variables
shown, A(u,P) is positive elementary (in P) iff A belongs to the smallest
class of s
which includes formulas of the form t - s, Nt,
-,t - s, -,Nt, Pt, and is closed under A, V and Qx (Q - 3, V).
(ii) If A ( u , P ) i s positive elementary in P, then A(u,P) defines a
monotone operator r A 99 ( M ) - , 9(M) ( 9 ( M ) - power set of M), namely,
if S C_ M and we interpret P by S,
FA(S ) "- {a'a E M and (.Ate, S)[= A(a,P)}.
For simplicity, it is convenient to call A(u,P) itself an elementary positive
operator on dtt~ (thus identifying the formula with the operator it defines).
(iii) We write I A for the smallest fixed point of F A (in short, the fixed
point of the given operator; so I A satisfies FA(I A)-C_ I A and I A C_X,
whenever FA(X ) C_ X).
(iv) A set X C_ M is inductive on ~
iff there are a positive elementary
III.13]
89
The Envelope of an Inductive Model
operator A and an element f of M such that, for every a E M ,
(f "M a) E I A ( ' M is the application operation of Jtl~).
(v) I N D ( 2 ~ ) ' -
aEX
iff
the collection of sets C_ M, which are inductive on dtl,.
(vi) X C_ M is coinductive on M1, i f f - X ( - the complement of X in M)
is inductive on Jtl~.
(vii) X C_ M is hyperelementary on ~ iff X is both inductive and
coinductive on Jtt~.
(viii)
HYP(JtI~)'- the collection of subsets C_ M, hyperelementary on ~ .
13.4. THEOREM. Let ag be a model of OP-. Then:
(i)
ENV(O(31~))-IND(~);
(ii)
O(Jtt~) E IND(31,)- HYP(31,).
PROOF. (i) IND(31~)C_ E N V ( O ( ~ ) ) . It clearly suffices to check that, if
A(u,P) is an arbitrary positive elementary operator on dtl,, then I A is
representable in O(Mt~). Consider the operator A(u,v) in the language
s
obtained from A(u,P) according to 12.2.1 (ii) (v fresh variable). Now
O(dtl~) is a model of M F - (by 7.10) and M F - p r o v e s by 10.4 that there is an
abstract I ( A ) = IxvA(x, v)such that:
O(all,) ]= Vx(xr/I(A) ~ A(x, I(A))).
(1)
Hence, if we let I(A)(O(.Ytt,))'-I (recall the notation of 13.1), and we
notice that the only T-atoms occurring in A(x,I(A)) have the form tTlI(A )
by assumption on A, we get by 12.2.1 (i),
O(all,)[= A(a, I(A)) iff (0, I)l = A(a, P),
(2)
which implies by (1),
rA(I)-I.
(3)
By minimality of IA, I A C_I. Conversely, it is straightforward to check, by
induction on a with 12.1-12.2"
O(a)l-ar/I(A ) implies a E I A (for a E M).
(4)
Thus I C_I A and I A - I; hence I E ENV(O(all,)). In order to complete the
proof of the theorem, we show"
ENY(O(ag)) C_IND(all,).
If X- t(O(all,)), for some closed term t of 2,(all,) and f have, for every a E M:
alg(t)E M, we
90
Inductive Models and Definability Theory
[Ch.3
a E X iff O(-~)l--a~t iff ( f . M a) G O(Jtt~).
Hence X is inductive on RAt,, since O(atl~) is the fixed point of the positive
elementary operator r(u, P ) o f section 7.3 (v).
(ii) If M - 0(31,) E IND(31~), {a E M:O(Ml~)l=-,arla}={a E M: O(31~)]=a~t},
for some closed term t, and we can derive a contradiction ~ la Russell. V!
13.5. DEFINITION. Let S C M:
(i) S E C ( S ) " - {X" X C_31~ and X - t(S), for some closed term t of
s
such that S[=Cl(t)} ( - " the section of S");
(-
(ii) S E C + ( S ) " - {X" X C_M and X , - X
"the +-section of S").
E ENV(S)}
We immediately obtain, with theorem 13.4:
13.6. LEMMA
If S is consistent and S E FIX(F, Jtt,) (cf. 7.5), SEC(S)C_ SEC+(S) and
SEC+(S) C ENV(S). In particular, SEC+(O(Jlg))- HYP(Mg).
From the ordinal comparison theorem of the next section, it will follow that
We conclude with a prooftheoretic application of 13.4-13.6: as far as we restrict number-theoretic
induction to total predicates, i.e. classes, we obtain a theory of abstraction
and truth, whose arithmetical content does not exceed that of OP, and
hence of Peano arithmetic, even if we add the generalized induction
principle of w
SEC+(O(Jtl~))-SEC(O(JIg))-HYP(~).
13.7. THEOREM. MF c -F GID is a conservative extension of OP, i.e. if A is
a formula of s
and MF c + GID F- A, then OP ~- A.
(For the definition of MFc, see 10.7; as to GID, see 12.4.1).
PROOF. The argument uses well-known results from general model theory:
(i) every structure Ml~ for a given elementary language ,Lop(.J~ ) h a s a n
elementary extension, which is recursively saturated (see Chang-Keisler
19903);
(ii) if Nl, is recursively saturated, HYP(Jtl~)-DEF(JtI~)(here
X E DEF(31~) iff X - {a'a E M and att~[= B(a)}, for some B(x) of .~,op(~l~)
(see Barwise 1975, Barwise-Schlipf 1975).
In order to prove the theorem, we assume that dlt is a model of OP + A (A
sentence of .Lop(dig)). By 7.10, 12.4, O(Ytt,)is a model of M F - + GID + A,
plus N-induction for s
Assume:
O(~)I=Cl(t ) A Vx(x~t ~ ( x + 1)r/t)A 0r/t and X - t(O(Mg));
then, by lemma 13.6, X E HYP(Jt[~). By (i), it is not restrictive to suppose
III.14]
The Uniform Ordinal Comparison Theorem
91
that ~ is recursively saturated; so by (iN) X E D E F ( ~ ) . By hypothesis on
t and N-induction for Lop(Ml~)-conditions , we have O(~4i~)]=Vx(ix-~ xrlt ).
In conclusion, O(~t,)]= MF c + GID + A. F1
Similar results hold if we add the enumeration axiom EA of w or
extensionality for operations Ext op, and we replace class N-induction by the
weaker 3(+)-N-induction of 4.13. Let:
MF1 := M F - + 3(+)-NIND.
Then by 4.15 and the appendix to Ch. I, we immediately have:
13.8. THEOREM. Let Ax be either EA or Ext op.
(i) If f is a combinator such that MF c + GID + Ax F- f " N ~ N, then f
defines a number-theoretic function, which is still provably recursive in
Peano arithmetic.
(ii) If f is a combinator such that MF 1 A- GID + Ax F- f " N ~ N, then
f defines a primitive recursive function.
We do not know what happens if both EA and Extop are present;
perhaps, it might be useful to check that CTM(w) (cf. w is a model of
OP + Ext op + EA, provably in (some conservative extension of) MF c + GID.
w
The uniform ordinal comparison theorem for inductive models
We further investigate the recursion-theoretic structure of O ( ~ )
model of OP-), in order to prove, in the notation of w
HYP(~I~) - S E C + ( O ( ~ ) ) -
SEC(O(~I~)).
(.Ab fixed
(,)
The main result of this section, from which (,) easily follows, is the
extension to O(31~) of a fundamental theorem, originally stated for Kleene's
recursion in higher types by R. Gaudy (see Hinman 1978). Let b E M: since
O(.At~) is inductively generated by the operator F of 7.3, we apply
backwards to b the reductive clauses which specify F, just as in the standard
generation of a semantic tableau. Except for the trivial case where no
F-clause is applicable, the reductive procedure produces a "search tree" T(b):
r(b) starts with a "root" (labeled by) b and v ( b ) i s well founded (in the
usual sense) iff b C O(.At~), i.e. O(.A~)I=Tb. Therefore we can naturally
associate to b an ordinal, called I b I, whenever O(.At~)l= Tb: it is the rank of
the "direct" (cut-free) proof of Tb. The remarkable fact is that if
O(.AI~)I=Ta or O ( ~ ) 1 = Tb, then the ordering relation between l a I and I bl
is representable in O(3t~): this is the key property for obtaining (,) and
selection, reduction and separation principles for ENV(O(.A~)).
92
Inductive Models and Definability Theory
[Ch.3
14.1. Preliminary notions and conventions.
(i)
E(z) := 3 z 3 y ( ( - , P F O R ( z ) A z = (-,x)) V z = [Nz] V z = [-,Nz] V
v z = [~ = y] v z = [ ~ ( ~ = y)]);
S(z) := 3x(z = [-,--z] V z = [Tx] V z = [-,Tz]);
~ ( z ) := 3~3y(z = ~ ( w ) v
H(z) := 3~3y(z = ( w ) v
z = ~(~ ^ ~));
z = (~ ^ y));
(for the definition of P F O R ( z ) : = "z is a p-form", cf. 7.3).
(ii)
If b E M,
we say that
b is in E - f o r m (in S-, E-, H - f o r m ) i f f
3(6 I- E(b) (Jli~l= S(b), E(b), II(b)respectively).
(iii) According to 7.3, if b, c E M, we adopt the abbreviations be, -,b, Vb,
b A c, instead of the proper Jt6(Ap(bc)), .)/b(NEGb), 2~(ALLb), ~6(ANDbc)
(respectively); furthermore, id(b,c), nat(b), t r ( b ) s t a n d
for Jll~([b=c]),
~l~([gb], .Al~([Tb])(in the given order).
(iv)
We stipulate that
b<<cholds
forb, c E M i f f
either c - Vd and b - da, for some a E M, or
c = -,(Vd) and b - -,(de), for some a E M, or
c = (e A f ) and b = e or b = f, or
c =--,(e A f ) and b = -,e or b = - ' S , or
c = tr(d) and b - d, or
c = -,(tr(d)) and b = -,d, or
c = -,-,d and b = d.
(v)
If c = - , - , d , t r ( d ) , - ~ t , ( d ) ,
we put
PRD(c) = d = " t h e unique b << c".
(vi)
If b << c holds, we say that b is a predecessor of c.
Obviously, there is a formula B(z,y) of ,Lop which defines <<, uniformly in
every model of O P - .
14.2. D E F I N I T I O N
(i) Let 8+ be the successor cardinal of ~ (8 being the cardinal of M). If
b E M and b ~ O(Jtt,), we put: I bl " - a + .
(ii) Assume b E O(J16); then we define (by induction on the generation
of O(Jlt~)):
if b is in E-form, I bl - 0;
if b is in S-form, I bl = IPRD(b) I + 1;
if b is in H-form, I bl = s u p { I c I + 1: c E M and c << b};
The Uniform Ordinal Comparison Theorem
III.14]
if b is in E-form,
(iii)
I bl
93
- i n f { ] c I + 1" c E M and c << b}.
Icl);
b<_c'-(bEO(J~)and
]b] _<
b<c'-(bEO(J~)and
]b I < I v ] ) .
Obviously, c < b iff c E O(.flt,) and not ( b < c). I bl is called the norm of b.
It is straightforward to verify, by definition of norm and O ( ~ ) :
14.2.1.
<,
]b I < 5 + iff b E O(~t,).
< are well-founded relations on Eft, and we have, for every b, c E M:
14.2.2.
b < c iff b E O(~t,) and (if c E O(.At,), ]b I < ]c ]);
b < c iff b E O(~t~) and (if c E O ( ~ ) ,
I bl
< I c I).
14.2.3. LEMMA
(i) The predicates E, S, E, II are pairwise disjoint: if W r V and
W, V E {E, H, E, S}, then for every b E M, r
~(W(b) A V(b)).
(ii)
For every b E M,
~ [ = (PFOR(b) A-~E(b)) ~ (S(b) V II(b) V E(b)).
PROOF: apply 7.1.1 and the definition of the logical combinators before
7.1.[3
14.2.4. LEMMA. Let b, c E M and assume
.AhI= E(b) V E(c) V--,PFOR(b) V ~PFOR(c).
Then:
b < c iff O(.Ab)]= Tb A (E(b) V (Fc A E(c)) V --,PFOR(c));
c < b iff O(~1~)I= Tc A F(Tb A (E(b) V (Fc A E(c)) V -~PFOR(c))).
(1)
(2)
P R O O F . The restriction axioms R E S , T.1 and T.3 of 7.10 easily imply, for
every d E M"
O(.Ah)I- E(d) A -~Td ~ Fd.
(,)
Ad (1). Assume b _< c. Then O(ait,)l--Tb and hence b is in p-form. If b is an
E-form or c is not in p-form, then the right hand side of (1) holds trivially.
Else, c is in E-form and b is not in E-form and we must have I bl > 0,
whence [c I > 0, which implies O(a?b)l=-~Tc (by 14.2 (ii)), i.e. O(all,)l= Fc
with (.). Assume the right hand side of (1). Then clearly b E O(dtl~) and b
must be in p-form by R E S . If b is in E-form, [b[ - 0 and trivially b < c. If
94
Inductive Models and Definability Theory
[Ch.3
c is in E-form and O(~)l-Fc, o r c is not in p-form, then c ~ O(Jll~) by
consistency of O(~1~): hence b __ c (by 14.2.2).
Ad (2)" similar argument, using the assumption from right to left. [3
14.3. T H E O R E M (Uniform Ordinal Comparison). There exists an operator
G(u, v) in lhe language 2., such thai if.At, i= OP-, then for every b, c E M:
b <_e iff O(.At~)I= (b, c)rlI(a);
c < b iff c G O(Jft~) and O(.At,)I = (b,c)-OI(G).
(,)
(**)
(I(G) - the fixed point predicate of 10.4)
PROOF. We describe a set of inference rules for deriving statements of the
form b _< c. In each rule, the conclusion will depend positively on b'_< c',
where b', c' are immediate predecessors of b, c in the sense of <<. G(u,x)
will be easily assembled by formalizing the _<-rules. (,)-(**) are verified by
transfinite induction.
Initial Rules. These rules handle the cases where b, c E M and
Jtt~l= E(b) V E(c) V-~PFOR(b) V-~PFOR(c).
(1)
b E O ( ~ ) , b is in E-form
b _ c (c arbitrary)
1.1
1.2.
b E O(Mt~), (-~c)E O(Mt~) and c is in E-form
b<c
;
1.3
b E O ( ~ ) and c is not in p-form
b<c
"
By 14.2.4, I.l-I.3 suffice.
Inductive Rules (under the assumption that (1) does not hold). By 14.2.3 (ii)
Jtl~l= S(b) V ~(b) V II(b) and ~At~l=S(c) v ~(c) v II(c). Hence, we have to
specify nine rules; each rule is labeled by a two-letter word W , V where W,
Y E {II, E, S}. W , V denotes the case where W(b), Y(c) hold in ~ . We
adopt the abbreviations:
(3u << b)(...):= 3u(u ~ b A...) and (Vv ~ c ) ( . . . ) : = Vv(v <~ c---,...).
[S*SI b, c are in S-form"
PRD(b) <_PRD(c) .
b<c
III.14]
IS,HI b is
The Uniform Ordinal Comparison Theorem
95
in S-form and c is in H-form:
(3c' << c)(PRD(b) < c')
b<c
]H,S] b is in H-form and c is in S-form:
(Vb' << b)(b' < PRO(c)).
b<c
m
]S,~] b is in S-form and c is in ~-form:
(Vc'<< c)(PRD(b) <_c').
b<c
]E*S] b is in ~-form and c is in S-form:
(3b'<< b)(b' < PRO(c)) .
b<c
]H,H] b, c are in H-form:
(vb' << b)(3c' << c)(b' < c').
b<c
[~,E] b, c are in E-form:
(3b' << b)(Vc' << c)(b' < c')
b<c
[H,E~] b is in H-form and c is in E~-form:
(vb'<< b)(Vc' << c)(b' < c')
b<c
[El,HI b is in E]-form, c is in H-form:
(~b' << b)(~c' << c)(b' < c')
b<c
This completes the list of the inductive inferences.
Obviously, to each inference V , W (where V , W E {S,H,Z}) we can
associate a T-positive formula Y,W((b, c), v) of s operative in v, which
states that b is in V-form, c is in W-form and formalizes the general clause
embodied in the premise of V,W, by replacing each condition b ' < c' with
(b',c')qv. Let G(x,v) be the T-positive formula, operative in v, which
describes the initial rules and the V,W-rules; then we can find a fixed point
I(G) (by 10.4) such that, for every b, c E M:
O(.Ab)]= (b, e)qI(G) iff O(J~I,)I-G((b, e),I(G));
(2.1)
Inductive Models and Definability Theory
96
O(Jtt~)l= (b,c)-ffI(G)
iff
0(.~)1= FG((b,c),I(G)).
[Ch.3
(2.2)
Then
b < c implies
O(jll~)l=(b,c)r/I(G);
(3.1)
(b,c)-ffI(G).
(3.2)
c < b implies O(~l~)l=
Indeed, if b, c satisfy (1), (3.1)-(3.2) simply follow from the lemma 14.2.4,
(2.1)-(2.2) and the fact that the conditions E(u), S(u), II(u) and E(u) are
mutually incompatible. Otherwise, we can assume that b, c are not in
E-form and we establish (3.1)-(3.2) by transfinite induction on
- min{[b[, [c 1}, whenever b is in V-form and c is in W-form
(V, W E {S,E, II}). In particular, (3.1)-(3.2) can be reduced to an inductive
verification on a that if b is in V-form and c is in W-form, then
b < c implies O(dlt,)[=
c < b implies
V,W((b,c),I(G));
O(JII~)I=F(V,W((b,c),I(G))).
We only discuss two significant cases; the remaining ones are obtained by
straightforward arguments.
E,E" b, c are in E-form, e.g. b - - ~ ( V e ) , c - - ~ ( V d ) . If b < c holds, then
b E O(.At~) and inf{l~(ey) l + l ' y E M} <_inf{l--,(dy) l + l ' y E M}. Since
O(~6) is a model of T-axioms and O(J?6)l= F(Ve), we can also assume that
there exists a y E M such that O(Jll~)[=Fey and [-.(ey)[ <_ [-,da [, for
every a E M. By IH,
o ( ~ ) 1 - 3~(~ << b A
Vv(v << ~-~ (~, v),Tz(e))),
which implies O(.AI~)I= E,E((b,c),I(G)).
Hence by (2.1) and logic, O(.hl~)l= (b, c)r/I(e). This takes care of (3.1). As to
(3.2), if we assume c < b, we must verify O(.At~)l=F(e((b,c),I(e))); since
O(Jli~)i= E(b) A E(c), it is enough to check:
O(Jll~)l=
F(qu(u << b A Vv(v << c ~ {u, v)rlI(e))), or equivalently,
O(dtt,)]=
Vu(u << b ~ 3v(v << c A (u, v)-flI(G))).
By assumption, for some x E M , O(.~)l=F(dx
hold for arbitrary y E M, whence by IH
O(~hl~)l= 3v(v << c A Vu(u <<
(4)
) and I~(dx) l < I-,(ey) l
b-->(u, v)-ffI(G))),
(5)
which implies (4).
II,E. We assume b is in H-form and c in E-form.
Ad (3.1). If b <_ c, then b' < c' holds for every b' << b, c' << c, whence by IH
o(..,r
wvu(= << b A y << c--, (=, y)rlI(C)) , i.e. o(~)1= u,z((b, c), I(G)),
III.15]
Applications of Ordinal Comparison Theorem
97
which yields by (2.1) the required conclusion.
Ad (3.2). By assumption we have, for some c' <:< c and some b' << b, c' < b'.
It follows by IH that O(Al~)[= 3x3y(x << b A y << c A (x, y)-~I(G)), which
readily implies the validity of F(II,E((b,c),I(e)))in O(Jtt,). By (2.2) we
obtain O(.Al~)[=(b, c)-~I(G).
We then proceed by observing that, for every ordinal 7:
O(Jtt~,7)[= (b, c)rlI(G) implies O(.A1,)[= Tb.
(6)
O(Ml,,7 ) is the 7th-stage of O(3t~) (see 7.6 or w
( 6 ) i s checked by
transfinite induction on 7. If 7 - 0, the claim holds, as the premise is false,
and, if 7 is a limit, we use IH. Assume O(.At~,fl + 1)1--(b,c)~?I(G); by lemma
12.1 we have:
o( u,
=
G((b, c), I(e)).
(7)
As above, we must distinguish several cases according to the form of b, c. If
b (or c ) i s in E-form, (7)implies O(Jft~)]-Tb (by (1)). In the remaining
cases, IH applies and it always entails a sufficient condition to assert
O(Jtl~)[-Tb (e.g. if b is in H-form, we have from (II,S), (II,II), (II,E) that
every b' << b holds in O(Al~), which implies the conclusion with T-axioms for
A and V).
Then we prove"
O(Al,)l= (b, c)qI(G) implies b _< c;
(8.1)
O(At)[= (b,c)-~I(G) and O(Jft~)[= Tc imply c < b.
(8.2)
Ad (8.1). Assume O(.At~)[=(b,c)riI(e). Were b~_c false, then by (6)
O ( ~ ) [ = T b and [c [ < I b[, whence O(At~)]=Tc, i.e. c < b, which implies
O(Ah)[= (b,c)-~I(e) by (3.2), against consistency of O(uit~).
Ad(9.2). Assume b < c is false and O(.At~)[=(b,c)VI(G), O(Ml~)l=Tc. Then
[b[ _~ [ c [ a n d hence O(Al~)[=Tb; by definition, b~_c and by (3.1),
O(Jtl~)[= (b,c)~7I(e): contradiction!
(8.1)-(8.2), (3.1)-(3.2)prove the theorem. [7
w
Applications of the uniform ordinal comparison theorem
We conclude the recursion-theoretic analysis of w by characterizing the
section of an arbitrary inductive model. We then introduce an
approximation operation ~r, which splits each non-empty property in a
C-chain of subclasses, and we show that zr satisfies six simple conditions,
the 7r-approximation axioms. As we shall see in the next chapter, 7r-axioms
98
Inductive Models and Definability Theory
[Ch.3
suffice to derive non-trivial ordinal-free consequences of the ordinal
comparison theorem. However, we easily have that r-axioms are still
conservative over MF c + G I D (hence over OP), by an obvious extension of
the model-theoretic argument of w
The basic step for proving the full characterization theorem is a
separation lemma for disjoint predicates, which is easily implied by 14.3.
15.1. LEMMA (Separation). We can find a closed term ~x)~y.SEP(x,y)
such that if dtt~l= OP-, b, c E M and O(.AI~)]=Vu(-,urlb V ~uzlc), then
O(.~t~) [= Vx(xr/b --, x~SEP(b, c)) A Vx(xr/c ~ z-~SEP(b, c)).
PROOF. Choose S E P ( b , c ) ' - { x ' ( b x , cx)rlI(G)}, where I ( G ) i s
the
predicate, whose existence is ensured by the ordinal comparison theorem. If
d e M and O( )l=dnb, we have by assumption O(~l,)l=-,(dr/c); by
definition 14.2, bd <_ cd trivially holds and hence O(.~)l=(bd, cd)nI(G) by
14.3 (,), i.e. O(A,)I= dnSEP(b, c). On the other hand, if O(Jtt,)l= dr/c,
then cd < bd holds and hence by (**) of 14.3, O ( ~ ) l = d ~ S E P ( b , c ) . [:1
15.2. THEOREM. Let ~
be an arbitrary model of OP-; then
HYP(.AI~)- SEC(O(Jtt~)) - SEC+(O(3b)).
PROOF. By lemma 13.6 it suffices to verify SEC+(O(JII~)) c_ S E C ( O(.?$ ) ).
Let X C M and assume that
X-
{a" a E M and O(Mt,)l= a t / b } , - X -
{a" a E M and O(Jtt~)l= at/c},
for some b, c e M. By 15.1 and consistency, O(Ml~)l=d~SEP(b,c ) iff
O ( ~ ) l = d q b , for every d c M; but S E P ( b , c ) i s
a class in O(.&):
O(J$)l=~drlSEP(b,c ) implies O ( ~ ) l = d ~ c , whence O(Jtl~)l=d-~SEP(b,c )
again by 15.1. VI
15.2.1. REMARK. If ~ is countable, S E C ( O ( J t b ) ) - A~(Mt~), the collection
of subsets of M, which are definable by formulas B Y A ( Y , x ) , V Z B ( Z , x ) ,
where 3Y, VZ range over subset of M, and A, B are formulas in the
language .Lop possibly containing atoms Yt, Zt, Xt. This follows from the
generalized Suslin-Kleene theorem of Moschovakis (1974).
15.3. DEFINITION
7r "- l y l x . { u " u - x V (yu, yx)rlI(G)}.
Henceforth, it is convenient to adopt the more perspicuous abbreviations:
15.3.1.
xu <_ayv "- (xu, yv)rlI(G);
15.3.2.
yv < G xu "- vr/y A (xu, yv)-~I(G).
Applications of Ordinal Comparison Theorem
III.15]
99
The intuition behind 7r is that, if x is in y, then 7ryx defines those u
which satisfy yu < G yx, i.e. which fall under the property y at a level not
higher than ]yx I (see 14.2); it turns out that ~yx is a class in O(dtt~).
15.4. THEOREM (Approximation). Let J~ be a model of OP-. Then O(atb)
verifies the following sentences:
YxVy(xrpryx);
7F.2.
w v v ( ~ v ~ c t ( ~ v ~ ) ^ ~v~ _c v);
7r.3.
YxYy(~xrly --, y C_ ~yx);
~r.4.
VyVuVv(u~Iryv A vrly --~ lryu C_ ~ryv);
7F.5.
~.6.
v v w ( ~ , v ~ 3z(z,Tv ^ Vv(~,v ~ ~vz c_ ~vv)))
PROOF. lr.1 holds trivially. As to the other sentences, we first observe:
O ( ~ ) 1 = WVyW(x~y - ~ ( u ~ y x ~ yu <_ a yx)).
(1)
(r
by abstraction and definition of ~r. =~: if x~ly, <_ G is reflexive, and the
conclusion follows by abstraction).
Ad 7r.2. Assume brla, d~Trab hold in O(dit~), where a, b, c, d E M; for
typographical reasons, we keep using the same symbols for elements of M
and their names. By (1), O(dtt~)l=ad _< Gab, hence by 14.3 (.), O(dlla)l=d~a ,
i.e. O(dlt)l= 7tab C_ a. Assume now:
O(Mt~) 1= brla A --,drprab.
(2)
o(J~)l=-~d = b;
(3)
O(jtl~)l_.~a d <_ Gab.
(4)
By definition of 7r,
(4) and 14.3 ( , ) i m p l y O(Mg)l=-~drla , or l abl < l a d l . In both cases (with
(1), 14.3 (**)), we have
O(Mt~)]= ab < Gad.
(5)
Hence from (5) and (3), O(a?t~)]= d-OTrab, which implies that 7tab is a class in
o(.~).
Ad 7r.3. Assume
O(dtt~)l--~drla A brla.
Then
(6)
l abl < l a d l ( b y 14.2(i)), i.e. trivially l abl <_ lad I, which yields,
100
[Ch.3
Inductive Models and Definability Theory
by (6) and 14.3(**), O ( ~ ) l = a b <_ Gad, whence O(~)l=boTrad. Since b is
arbitrary, O(.At,)]= a C_fred.
Ad 7r.4. Assume
O ( .At~) l= d rlTra b;
(7.1)
O(-ah)l= bqa;
(7.2)
O(.~)l- crprad.
(7.3)
By (7.1)-(7.3), (1) and ordinal comparison, we obtain
]ac ] < ]ad], whence
]ad] < ]ab]and
[acl < labl.
(8)
By (7.2) and 14.2.1, we must have
O(~)1= crla.
(9)
From 14.3 (,), (9) and (8), O(.At~)[=crprab, whence O(~)l=Trad C 7rub.
Ad 7r.5. Assume
O(.At~)l= brla A crla;
( 10.1 )
O(.A~)l=dqTrab A--,drprac, for some d E M;
(10.2)
O(.Al~) l= erlTrac.
(10.3)
By (10.1)and (10.3), (1)and 14.3 (,),
l ae I < I ac land O(A,)[= er/a.
(11)
From drprab, (1)(10.1)we infer
O(Jfl~)]=dqa and l adl <_ l abl.
(12)
But ~drprac, (1), (12) and (10.1) imply ]ac [ < lad], whence
lac] < lab].
(13)
(13) and (11) entail Lae [ < lab I, i.e. with 14.3 (,) O(Jtt~)]=e~?Trab.
Ad 7r.6. Let O(~t~)]= crla, for some c E M. Then the set of ordinals
{lad[
9d e M, O(~)[=dT/a and [ ad ] _ ]ac ]}
is non-empty, and it possesses a least element 5. Choose any d such that
-lad[
and suppose O(.A~)]=bqaAerlrad; then by ordinal comparison
and the choice of d, lee I < lab] and O(Ml~)]=erla (because O(.At,)l=brla),
which yields O(.Ah)[=erlTrab. Since b, c are arbitrary, it follows that
O ( ~ ) l = Vv(vrla ~ 7red C ray). i"1
Let IIAX be the list 7r.1-r.6 of the approximation theorem 15.4. Then
III.15]
Applications of Ordinal Comparison Theorem
101
we immediately have, with the conservation results of 13.7-13.8:
15.5. T H E O R E M (General Conservation)
Let Ax = EA or Ext op"
(i) If A is a formula of s
such that M F c + G I D + I I A X ( + A x ) ~-A,
then A is already provable in OP of+ Ax).
(ii) If f is a combinator such that M F + G I D + I I A X ( + A x ) ~ - f : N ~ N ,
then f defines a number-theoretic function which is provably recursive in
Peano arithmetic PA.
(iii) If MF I + G I D + I I A X ( + A x )
F-f:N~N
and f is a combinator,
then f defines a primitive recursive function (MF 1 being the subsystem of
w
15.6. FINAL REMARKS
(i) 15.5(i) can be strengthened to a proof-theoretic equivalence by
using a refinement of the techniques of part D (see Cantini 1992).
(ii) Set /r := )~y)~x.{u: (yu, yx)~lI(G)}. If ~r.i is any sentence of IIAX
(where i E {1, ...,6}), let ~.i be the sentence, which is obtained from ~r.i by
replacing everywhere 7r with ~. Then we obtain"
if ~
is any model of o e - , O(~1~) verifies ~.2--~.6;
O ( ~ ) 1 = VyVuW(,ny A u~Ty A ~ryu C_ r
~ u~7~ryv).
(~r.7)
Clearly #.4 and #.7 imply:
vyw( ny
un y );
VyVuW(vuy ^
( yu c
(,)
M F - + IIAX is interpretable into M E - + {/r.2-~.7}.
(**)
Using (,), we can show:
(**) is made precise as follows. First of all, let us consider 7r and ~r as new
primitive symbols. Then we define a translation ^ from the language of
M F - + I I A X into the language of {/r.2-/r.7}: if A is a formula of
M F - + {Tr.l-Tr.6}, A ^ is the formula obtained by replacing each occurrence
of 7r with the term 7r' := )~y)~x.~ryx U {x}. Then, for each i E {1,..., 6 },
M F - + {/r.2-#.7} ~ (Tr.i)^.
(***)
The verification of (***) is trivial for i - 1,2,3,4; it makes use of/r.7 in the
cases of 7r.5 and 7r.6. A consequence of (**) is that we can work with an
operator, which satisfies r
instead of 7r.1, without weakening the
approximation structure.
This Page Intentionally Left Blank
CHAPTER 4
TYPE-FREE ABSTRACTION
WITH APPROXIMATION OPERATOR
w
w
w
w
Approximating properties by classes
The approximation theorem for extensional operations and the fixed
point theorem for monotone operations
Topology displayed: basic definitions
The representation theorem for explicitly CL-continuous operators
Appendix: alternative proofs
In chapter III we semantically justified the introduction of a suitable
approximation operator 7r with a simple axiomatic description, and we
proved that the resulting extension of the minimal framework MF c - to be
called P W c b e l o w - i s conservative over OP, even in presence of a
generalized induction schema GID for fixed points of positive operators (see
15.5). We now investigate in some detail the extended axiomatic system
P W c + GID and we show that it has natural non trivial consequences.
As we expect from the basic intuition underlying 7r, the relevant
consequences are recursion-theoretic and show that the chosen system yields
a kind of axiomatic intensional theory of inductive sets over a combinatory
structure. In w we obtain principles of approximation for properties by
classes and we get typical results from definability theory. Incidentally, we
obtain a strong form of separation for disjoint properties, which is a basic
tool for proving the consistency of an interesting property theory, due to
Myhill-Flagg (1987). In w we exploit the local structure of properties and
we prove a generalized continuity theorem for extensional operations, an
abstract
version of the Myhill-Shepherdson theorem.
Of course,
"extensional"
operations are those operations which preserve the
membership relation r], associated to reflective truth. As a consequence
(together with generalized induction), we can prove uniform internal version
of the fixed point theorem for monotone operators; it also turns out that
extensional, positive, monotone and (generalized) continuous operators all
coincide, provably in P W c + GID.
The axiomatic results can be rephrased in denotational style: in w we
introduce a natural topology, the class topology on the power set of M
(M = support of a given OP-model 3t~), and we show that non-trivial facts
Type Free Abstraction with Approximation Operator
104
[Ch.4
about "constructive objects" of the space can be adequately dealt with in
the untyped language of reflective truth. Among others, we characterize the
"explicitly" open sets of the space and the explicitly continuous operators of
the space into itself (w167
we conclude with an analogue of the Kleene
"first recursion theorem" for explicitly continuous operators.
The class tdpology is an analogue of the positive information topology (in
the sense of Scott-Ershov), where the role of "finite" (or compact) elements
is played by the hyperelementary subsets of M, i.e. the extensions of total
predicates in the inductive models.
w
Appro~dmating properties by classes
In Ch. III, w
we d e f i n e d - w i t h i n inductive m o d e l s - a n operator 7r, which
uniformly assigns to every property a chain of subclasses. 7r has its source in
this informal idea: in order to verify, by "predicative means", that x
witnesses a property b, one employs a portion of b, which is a class and can
be determined, uniformly in b and x, by looking at the "search tree", which
is naturally associated to (the presentation) of b. Formally, the basic
consequence is that we can compare two given elements x, y falling under a
property b, according to their "order of generation", and this order is
well-founded. A major corollary is that there is an operation ~, which
associates to any given non-empty property b a collection of witnesses of b,
which is a class (CL-compactness). Below, we explicitly define the
generation order and we collect together a few basic facts, inspired by
generalized recursion theory.
16.1. D E F I N I T I O N
(a)
P W c "- MF c + {Tr.l-Tr.6)
(NB: MF c is based on class N-induction of 10.7).
P W is an acronym for prewellordering ; r . l - l r . 6 are the sentences of the
approximation theorem 15.4, here restated for reader's sake:
7r.1. VxVy(xri~yx);
~r.2.
7r.3. VxVy(-~x~Ty --, y C_ ~yx);
7r.4. VyVuVv(uri~yv A vriy --~ ~yu C_ ~yv);
7r.5. VyVvVu(ur]y A v~?y ~ ~yu C_ ~yv V ~yv C ~yu);
7r.6.
vyw( y- 3z(z y ^
c
Approximating Properties by Classes
IV.16]
105
As usual, P W - is the subsystem of PW c without the number-theoretic
induction axiom, while PWp is P W - + P-NIND ( - the property induction
axiom of 10.7).
(b)
x < z y "-- xrlz A Trzx C rrzy;
(c) x < z Y " - x~l z A lrzx C 7rzy, where z C b "- z C_ b A 3x(xrlb A - , x T l z ).
16.2. LEMMA. The f o l l o w i n g f o r m u l a s are provable in PW-:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
T(x< zy)~x<
zY;
~X<zX;
x < zY A y < zw--+x < zw;
x < z Y "--+xrlz;
x <
x rlz
xrlz
xrlz
xrlz
yrlz
xrlz
x~z
z Y --+x -< zY;
A -~y rlz -+ x < z Y;
A yrlz ---+x < z Y V y < z x;
--+ x <_ z x;
V yrlz -+ x < z Y V y < z x;
A F ( x < z Y) ~ Y < z x ;
V yrlz---+ P r o p ( x < z Y ) (where P r o p ( z ) " ---+V y ( y < z x ---+-, x~rrzy);
3v(v
Tz V Fz);
z ^ Vv(v z--+ v < z v)).
PROOF. (i) =:>: trivial by T-logic. r if x < z Y, then xrlz, whence Cl(~rzx)
(by axiom rr.2). It follows that r r z x C ~rzy implies T ( r r z x C rrzy), i.e.
T(. < . v).
(ii)-(v): trivial by definition 16.1.
(vi): from xrlz and -~yrlz, we have z C_ rrzy (rr.3)and rrzx C_ z (~r.2). But
y~rrzy (~r.1); hence (rrzx C rrzy) A x~z.
(vii)" it is a restatement of rr.5.
(viii)" trivial.
(ix)" apply (vi)-(v), (vii).
(x) ==>: assume y r l z A F ( x < zY)" If-,xrlz, y < z x (by (vi)); if xrlz , then
~rzx C_ rrzy or 7rzy C rrzx (7r.5). But x ~ z implies - , F ( x r l z ) , whence it follows
3u(urlrrzx A u-~Trzy); then rrzy C ~rzx, i.e. y < z x. r
9 if y < z x is assumed,
3 u ( u ~ r z x A u-~rrzy) and yrlz hold (rr.2); hence yrlz and F ( x < z Y)"
(xi)" by (ix)-(x), (i).
(xii): assume xrlz and x~Trzy, y < zX; then yrlz and ~rzy C 7rzx. Hence
lrzx C_ lrzy (7r.4)" contradiction!
(xiii): it is a reformulation of 7r.6. V!
16.3. T H E O R E M ( C L - c o m p a c t n e s s ) . We can f i n d a closed t e r m ~ such that
P W - proves"
(i)
~zC_z;
106
Type Free Abstraction with Approximation Operator
(ii)
(iii)
[Ch.4
3x(xr/z) ~ Cl(~z) A 3u(u~I~z);
3 x 3 y ( x - e Y A ~ , ~ x - ~y) (so ~ is non-extensional).
PROOF. We define ~ "-Av.{x" Vy(x < v Y))" Clearly, if u ~ z is assumed,
then u~Iz holds (by 16.2 (ii)-(iv)). It is also clear that 16.2 (vi)-(v) entail
Vy(x < zY) iff x~Iz and Vy(yrIz---. x < zY)"
(1)
We assume that z is non-empty and we verify, for every x,
FVy(x < z Y) iff -~Vy(x < z Y)"
~ : trivial by T-logic. r
(2)
suppose that there is an x such that:
3y--,(x <_ zY);
(3.1)
-~FVy(x < z Y)"
(3.2)
Fix an element u of z; then (3.2) and 16.2 (xi) imply x < z u, i.e. x~lz. Hence
(3.1) and 16.2 (ix)-(x) yield 3yF(x < z Y), whence FVy(x < z Y)" by classical
logic we are done. By (2) and 16.2(i), ~z is a class; moreover, if z is
non-empty, ~z is non-empty, by (1) and 16.2 (xiii).
We conclude by verifying (iii): we show that, if V x V y ( x - e y ~ ~ x - e ~Y)
were assumed, then there would exist a class b - e r - {x"-~xrIx}, which is
impossible by 9.3. Choose
W ( x ) "- {y" y - 0 V (y - 1 A x~lr)}, V(x) "- {y" y - 1 V (y - 0 A xrIr)},
and let b - {x" ~ W ( x ) - e~Y(x)}. Since W(x), Y(x) are non-empty, ~ W ( x )
and E.U(x) are non-empty classes by (ii) and hence b is a class. Assume
-,xrir: then W ( x ) - e {0} and V ( x ) - e {1}. Since ~ W ( x ) a n d ~V(x) are nonempty and ~W(x) C_ W(x), ~V(x) C_ U(x), then - , ~ . W ( x ) - e ~ U ( x ) ; hence
b _C r. Conversely, let xrir: then W ( x ) - e U ( x ) - e {0, 1}. So, if ~ preserves
= e, ~W(x) - e ~V(x) i.e. x~b: Hence r C_ b: contradiction. 0
As a consequence of 16.3 (iii), the existence of an approximation operator r,
satisfying 7r.1-r.6, is inconsistent with
EXT(~r) "- VyVzVx(x~Iy A y -- e z ---*7ryx -- e 7rzx).
CL-compactness implies a useful result:
16.3.1. COROLLARY (CL-Reflection Schema). Let A ( x , y ) be an arbitrary
formula. Then there is a term )~u)~v.p(A, u, v) such that, provably in P W - :
Cl(a) A Vx~a.3y~Ib.TA(x, y) ---,
3c(c - p(A, a, b) A Cl(c) A c C b A Vx~Ia.3y~Ic.A(x, y)).
PROOF" choose H ( A , x , b ) -
{y" yrlb A A ( x , y ) } and define
p(A, a, b) "-xUa~H(A, x, b).
Approximating Properties by Classes
IV.16]
107
Assume Cl(a), YxTla.3yqb.TA(x,y ). Then we have, by abstraction 9.2:
Vxrla3y(yT1H(A , x, b)).
By 16.3, since H(A, x, b) C b and yT1H(A , x, b) ---. A(x, y), we conclude:
YxTla. (Cl(~H(A,x,b)) A ~H(A,x,b) C_ b);
(1)
YxTla. 3yTI~H(A , x, b). A(x, y).
(2)
If we set c := p(A,a,b), then (1), 9.14.6 and the assumption Cl(a) imply:
Cl(c) ^ c c_ b;
from (2) and the choice of c, it follows Vx~la.3y~c.A(x, y). H
We now use 16.3 to give natural uniform counterparts of the classical
reduction and separation theorems.
16.4. DEFINITION. RD((y',z'),Iy, z)) ( = l y ' , z ' )
conjunction of the the following sentences:
reduces (y, zl) is the
(i) y' C_yAz' C_z;
(ii) y U z C _ y ' U z ' ;
(iii) -~3x(xTly' A X~lZ').
16.5. THEOREM (Reduction). We can find closed terms R1, R 2 such that
P W - proves:
YyYz.RD((RlYZ , R2Yz),(y , z)).
B
PROOF. Define d ( y , z , x ) " - {i" (i - 0 V i - 1) A (x,i>71(y @ z)} (for @, cf.
9.14.2). Then we choose:
m
RlYZ "- ix" O~(d(y,z,x))}
n2Yz "- ix" l~/~(d(y, z, x)) A "=OTl~(d(y,z, x))}.
Conditions (i) and (iii) of
assume XTl(yU z). Then for
a non-empty class C_ {0, 1}
done. Else, 0 ~ ~(d(y, z, x))
i
16.4 are immediately checked by 16.3. As to (ii),
some i C {0,1}, iTId(y,z, u); hence ~(d(y, z, x)) is
by 16.3 (ii). If O~l~(d(y, z, x)), xTIRlYZ and we are
and hence 17/~(d(y, z, x)), whence xTiR2Yz. 0
16.6. COROLLARY (CL-Separation).
We can find a term C S P ( y , z ) such
that:
P W - F- Vx(xr/y V XrlZ) ---, (Cl(CSP(y, z)) A
A Vx(=x~ly ~ x~ICSP(y, z)) A CSP(y, z) C_ z).
(.)
PROOF. We reduce the pair (y, zl: then Vx(xrlRlYZ V xrlR2yz ) and RlyZ ,
R2Yz are disjoint. If =xTly and -~xTiR2Yz , we should get XqRlYZ C_ y:
Type Free Abstraction with Approximation Operator
108
[Ch.4
absurd. Therefore we have Vx(~x~ly ~ xqR2Yz ) and R2yz C z. Put:
d(y,z,x) "- {u" (x, u)rlRlyz 9 R2yz};
then we get" Vx3!u.(x, u)~d(y, z, x).
We choose CSP(y,z) "- {x" lri~d(y, z, x)}. Since ~d(y,z,x) is a class for
every x, CSP(y,z) is a class too, and it satisfies (,). O
An interesting consequence of CL-separation is the "A-comprehension"
schema, which corresponds to hyperarithmetical comprehension in secondorder arithmetic:
16.7. COROLLARY. Let A(u,x), B(u,y) be formulas elementary in x, y
respectively. Let A(A, B) "- Vu(3x(Cl(x) A A(u, x)) ~ Vy(Cl(y) ---+B(u, y))).
Then:
PW-
F A(A, B ) ~ 3z(Cl(z) A Vu(u~lz ~ 3x(Cl(x) A A(u,x))).
16.8. DEFINITION. z weakly separates the pair (x,y) of disjoint properties
(i.e. x A y - e 0), iff x C_ z and y C_ - z .
We now prove in the extended axiomatic framework the separation lemma
of 15.1:
We can define a term SEP(x,y) such that,
if x n y = e 0, then SEP(x, y) weakly separates (x, y).
16.9. PROPOSITION ( P W - ) .
PROOF. Define
SEP(x, y ) : - {u: ur/x ~_ uyy}
where uyx ~_ u~Iy := (u, x)~?E A 7rE(u, x) C_1rE(u, y) and E := {(u, y): uyy}. If
u~x, then -~uyy by assun~ption: hence by 16.2(v), u~x~_uyy, i.e.
u~SEP(x,y). If u~y, then F(u~x < u~y) (by 16.2(vi), (x)), which implies
u-~SEP(x, y). 0
16.10. COROLLARY (in P W - , Effective Inseparability). If (x,y) is any
pair of disjoint properties such that r'-{u'--,uTlu } C x, - r C y, then
-~SEP(x,y)~l(xUy) (where S E P is given by 16.9).
We conclude the section with a nice application of reduction and
7r-axioms. Each c trivially defines a pair (d,b I of disjoint properties with
d-ec
and b - e - C
( - c ' - { x ' - ~ X r l C } ; - e is 7/-extensional equality);
conversely, we may wonder whether to each pair (d,b) of disjoint properties
we can associate a property c satisfying d - e c, b - ~ - c . We call such a c,
if it exists, an exact representation of the pair (d,b). The answer to the
problem is positive and shows that the present language is complete with
IV.17]
Approximation Theorem for Extensional Operations
109
respect to exact representability of pairs of disjoint properties. The result
generalizes a well-known theorem for recursively enumerable sets of
numbers, due to Putnam, Smullyan, Shepherdson (see Smullyan 1961).
16.11. THEOREM.
P W - proves:
We can define a closed term A y A z . E R ( y , z ) ,
such that
y gl z - e 0 ~ ( E R ( y , z) - e Y A - E R ( y , z) - e z).
PROOF. Define:
~(u, z ) ' - {(v, u). v~y v (v, ~)~u};
~2(v, z ) . - {(v, u) . v~z v (v, u)~u}.
Below we simply write r 1 for rl(Y,Z ) and r 2 for r2(Y,Z ). The definition of
R1, R 2 (see 16.5) and CL-compactness imply:
Vu(uTlr 1A-~urlr 2 ---.urlRl(rl, r2));
(1)
Vu(u~lr 2 A -~u~lr1 ---, u~lR2(r 1, r2) ).
Since R l ( r l , r2) N R2(rl, r2) - e 0, we find c - S E P ( R I ( r l , r2), R2(rl, r2) )
by 16.9, such that:
Vu((u~rl ^ - u ~ , 2 ) - ~ u~c);
Vu((~2 ^ - ~ 1 ) ~
(2)
~Vc).
Let E R ( y , z) "- {v" (v, C)~lC}; we claim
~,y~(v,c)~c.
(3)
(3) ::~" assume v , y and ~(v,c),c. Since-~v,z, we get ~(v,c)yr 2. But v , y
implies ( v , c ) , r 1" hence by (2), (v,c),c and the conclusion follows by
tertium non datur.
(3) r "let (v,c)~c and ~v~y. Then (v,c)r]r2; also, by consistency, ~(v,c)~c,
whence-~(v,c)yr 1. By (2)we have (v,c)~c: contradiction.
The verification of Vv(vyz ~-, (v, c)-~c)is similar. [:]
w The approximation theorem for extensional operations and the fixed
point theorem for monotone operations
We consider the following generalized continuity problem: if we have the
value of an operation f on a property a, under which condition can f a be
approximated by the values of f on the subclasses of a ?
The answer turns out to be natural and simple: f must be extensional in
the sense of 7/ and the resulting continuity theorem is the analogue of the
Myhill-Shepherdson theorem in a general setting. A remarkable consequence
110
Type Free Abstraction with Approximation Operator
[Ch.4
the notions of extensional operation, monotone operation and
operation induced by a positive operator (in the sense of 10.3) essentially
coincide (modulo r/-extensional equality). In particular the minimal fixed
points of monotone operations are already generated by positive operators.
is that
We derive the main result within P W - , as a corollary of a generalized
Rice-Shapiro theorem; the neat exposition of Fitting (1981) is our guideline.
17.1. D E F I N I T I O N
(i) An operation f is extensional 2 iff x - e Y implies .fx - e fY, for
every x, y.
(ii) z is extensional I iff x - e Y and yr]z imply xriz.
17.2. L E M M A ( Upward closure). If z is extensionall, then z is C_-upward
closed. Formally:
M F - k- EXtl(z ) ~ VxYy((x C_ y A X~lz) --~ yTIZ)
(where EXtl(z ) is the obvious formula representing extensionalitYl).
P R O O F : assume x C y and x~lz. By the fixed point theorem for properties
10.4, we find a term I depending on x, y, z, such that:
Yu(u~lI +-+(UTIX V (u~ly A DIz))).
If -~ITiz, then by (,), I - e x
, i.e. ITIz. Hence ITIz , which implies I (because x C y), i.e. by extensionality, y~Iz. [3
(,)
eY
17.2.1. R E M A R K . Observe that M F - K Y z Y x ( E x t l ( z ) A -,x~Iz--+-~O~lz). If z
is an extensional class, so is the complement - z ; now either 0~/z or 0r/(-z),
whence either z - e V or z - e 0; so 10.15 is a corollary of 17.2.
17.3. L E M M A ("Rice-Shapiro.generalized"). If z is extensional 1 and x is in
z, then there is a class u in z such that u C_ x. Formally:
P W - t- VzYx(EXtl(z ) A xTIz---, 3u(Cl(u) A u~Iz A u C_ x));
(indeed, u is given uniformly in z, x by a term of the language).
P R O O F . Fix x in z and define t " - x | z (or, which is equivalent, x | 7rzx).
By the fixed point theorem for properties, we can find a term J, which
depends on x, z, such that:
Yu(mlJ +-+3y(yTIt A (u, O)Tprty A ~(J, 1)~/Trty)).
(1)
right to left, we apply 7r.2 to show that -~(J, 1)~vrty implies
(J, 1)~ 7rty). But (1), the definition of t and 7r.2 again entail
(From
jc_ .
(2)
IV.17]
Approximation Theorem for Extensional Operations
111
Then we obtain
j~z.
(3)
Indeed, if-~J~z holds, we have by r.2"
Vu(u~J ~ 3y(y~t A (u, O)yrty)).
(4)
Moreover, if uyx holds, we have (u, O)yt and (u, O)yrt(u, O) (by r.1), i.e.
3y(y~t A (u,O)~rty), whence by (4), uyJ. By (2), x - e J is true; since x~z
and z is extensionall, J~z.
If d - - {~. 3y(y~t(J, 1)^ (~, 0)~ty ^-~(g, 1)~ty)}, w~ claim:
d is a class such that d - e J"
(5)
Indeed, as to the classhood of d, let A(u) be the defining condition of d. We
show that -~TA(u) implies T~A(u). By T-axioms, this amounts to check
that for arbitrary y, (J, 1)~rty follows from the assumptions"
-~yTITrt(J, 1) V -~(u, O)~prty V -( g, 1)~ 7rty;
(6.1)
~y~ 7rt(J, 1) A -~(u, 0)~ 7rty.
(6.2)
Since t is non-empty by (3), ~t(J, 1) is a subclass of t by ~'.2. Hence by (6.2)
that rty is a subclass of t (again by ~'.2). By the
definition of class, (6.1)-(6.2) transform into
y~t, which implies
-~y~lTrt(g, 1) V -~(u, O)~vrty V (J, 1)Tvrty;
y~prt(J, 1) A (u, O)~prty,
whence (J, 1)yTrty.
As to the claim d _C J, assume uyJ. Then (u,O)rITrty, for some y in t such
that (J, 1)~Trty (by (1) and 7r.2). Since J is in z, (J, 1)zIt and hence
7rty C_7rt(J, 1) (by 7r.1, 7r.5); hence y is in r
1), i.e. u is in d.
That d C_ g follows from 7rt(g, 1) C_t and g~z. Extensionality 1 of z and (2)(3)-(5) yield (d~Iz A Cl(d) A d C_x). [3
17.4. COROLLARY ( P W - )
(i) Every infinite property contains an infinite class.
(ii) CL-compactness again: there is a term Ax.~x such that
~z c z ^ (3~(~z)-~ cl(~z) ^ ~u(u,~z)).
PROOF. (i): let INF "-{x" 3f" N ~ x } be the property of being infinite
(here f" N~--, x "- f" g---. x and f is 1-1). Then INF is extensional I and we
can apply 17.3.
(ii) c "-{x" ~U(UTIX)} is extensional1; hence by 17.3 there is a term r
such that
Type Free Abstraction with Approximation Operator
112
9~z ~ C l ( r
z)) ^
3~(~r
z)) ^ r
Choose (hint by Minari): ~ z - {x "XrlZ A xrlr
[Ch.4
z) c_ z.
(,)
Then apply (,). F!
17.5. T H E O R E M ("Myhill-Shepherdson"). Let f be extensional 2. Then for
every a, f a - e U {re: Cl(c) A c C_ a). Formally:
P W - F- V f V x (Ext2( f ) ---, f x = e U { f y : Cl(y) A y C_ x}),
(~h~r~
E~t2(]):= WVy(~
= ~V-~ f~ = ~fy).
PROOF. Define c ( f , u ) : = {y: urlfy}, where u is in f x . By assumption on
f, c(f, u) is extensional 1. If urlfy , for some subclass y of x, then yrlc(f, u)
and by lemma 17.2 Xrlc(f, u), i.e. urlfx. If u~fx, then x~c(f, u): hence 17.3
yields a subclass y of x such that urlfy. E]
17.5.1. REMARK. We already know that every operator A(x,a) gives rise
to an extensional 2 operation f A a ' - {x" A(x,a)} (see 10.3-10.4). By 17.5,
the converse also holds: for every extensional 2 operation f, we can find a
positive operator A$(x,a) such that for every a, f a - e {x" Ay(x,a)}.
Of course, we have a uniform choice: A I ( x , a) - 3c(Cl(c) A c C_ a A xrlf c).
It is easy to check that the two operations f~--~AI and A~--~fA are inverse to
each other.
17.6. DEFINITION
(i) M o n ( f ) "- VaVb(a C b---, f a C fb); if M o n ( f ) is assumed, we say
that f is C-monotone, or, simply, monotone.
(ii)
f-clos(x) "- f x C_ x (in words: x is f-closed).
If B(x) is an arbitrary formula,
f-clos(B) "- VwVu((ur/fw A Cl(w) A Vv(vriw---, B(v)))---, B(u)).
(iii)
I(f)"-
I(AI); (here I ( A ) : - fixed point of A, see 10.1).
17.7. FACT
(i)
(ii)
By 17.5 an operation f is extensional 2 if.]:f is monotone.
If f is monotone and B ( u ) " - u r / x (with u, x distinct variables),
17.5 implies
f-clos(x) ~-, f-clos(urlx ).
An important application of the continuity theorem and generalized
induction (12.4.1) is a uniform version of the fixed point theorem for
monotone operations:
IV. 18]
Topology Displayed
17.8. T H E O R E M ("Knaster-Tarski"). P W - + G I D
113
proves:
Mon(f)-~ f ( I ( f ) ) C_I(f) A (f-clos(B)--, Vu(ur/I(f)--, B(u))).
In particular, P W - + GID F Mort(f)--, (f-clos(x)--, I(f) C_x).
PROOF. If f is monotone, 17.5 and u~f(I(f))imply:
3w(w C I(f) A Cl(w) A ur/fw).
(1)
By choice of I ( f ) , we can conclude:
urII(f).
(2)
If B is f-closed, B is by definition Af-closed and hence the conclusion
follows by GID. The second part is a consequence of 17.7 (ii). 0
17.8.1. REMARK. The fixed point operator I, which can be explicitly
defined by the term Af .Y(Av.(x" 3c(Cl(c) A c C_v A xrlfc))), is extensional
too, in the sense that it preserves the natural pointwise equality on
operations. Set f U g "- Vx(fx C gx): then I(f) C_I(g) (apply GID).
w
Topology displayed: basic definitions
We make explicit the topological flavour of the previous results on
extensional operations. Thus we try to build a bridge between the present
intensional approach and denotational semantics, which relies on
topologically restricted notions of partial function and functional.
After the definability results of Ch. III (see 13.4), we know that the
syntax of s is strongly adequate for representing inductive sets. But there is
a more recondite adequacy, which is embodied in the results of w
specifically in the MS-theorem 17.5. We prove that, if the power set of the
ground combinatory algebra ~ is endowed with a suitably natural
topology, then "explicitly continuous" operators, mapping the power set of
M into itself, can be adequately mirrored by our logical approach.
We work in a fixed model ~ of O P - , while O(~6) is the least inductive
model of P W - + GID. Of course, M "-I.At, I is the support of Jl~, while
~P(M) is the power set of M.
18.1. DEFINITION. If b E M, E(b)"- {a" O(atb)l=ar/b } - "the extension of
property b". By 13.4 and 15.2, we have I N D ( . ~ ) - {E(a)'a E M} and
H Y P - {E(b)" b is a class in O(.AS)}.
All the notions introduced depend on ~
(e.g. E "-Ealt, , IND(alI~)). Since
114
Type Free Abstraction with Approximation Operator
is fixed, we can simply omit the explicit indication of ~
we shall speak of classes instead of M-classes.
[Ch.4
and M. Also,
CONVENTION: X, Y, Z range over ~P(M), P, Q, R range over families of
subsets of M. We freely use formulas of the language of P W - to state facts
about O(.)tl~.); e.g. " assume arlb" means " assume O(Ml~)l-ar/b" , etc...
We induce a topology on 9 ( M ) and, by relativization, on I N D . For the
topological notions, see any good reference (e.g. Kelley 1955).
18.2. DEFINITION. If O ( ~ ) l - C l ( e ) ,
a basic open (set).
V(e) . - { x c_ M" E(e) C_ X } is called
18.3. LEMMA
(i) The family {V(e)" O(J~)l-Cl(e)} is a basis for a topology CI-~T,
the class topology over J?l~.
(ii) (~P(M),Cl-~') (and hence the subspace ( I N D , Cl-~)) is a To-space ,
which is not T 1.
(iii) For any class e, if V(e) is covered by a family of basic opens, then
V(e) is already covered by a basic open of the given family itself"
if V(e) C_ U {V(ek)" k E I}, then V(e) C_ V(ej), for some j E I.
(iv) P C_ ~ ( M ) ) is open in the class topology iff
1. P is upward closed (i.e. Y E P and Y C_ X imply X E P);
2. whenever X E P, then there exists a class e such that E(e) E P
and E(e) C_X .
The proof is an easy exercise. Clearly (~P(M),CI-~) induces a sort of
generalized Scott topology on inductive sets (under the Kreisel-Sacks's
analogy "finite" - "to be a class"; see Kreisel-Sacks 1965). It is also useful
to regard the subspace I N D as an analogue of f-spaces in the sense of
Ershov(1977), where the subset of finite elements is just the collection
{E(c)" c is an M-class}.
We now state a simple proposition relating classhood to a constructive
abstract version of the compactness property; not surprisingly, the proof
requires CL-compactness (16.3).
18.4. THEOREM (CL-finiteness). Let X E I N D . Then X - E(c), for some
M-class c iff whenever X C_ E( U { f i" irlI} ), then also X C_E( U { f i" ir/Io} )
for ~om~ U-~ubcla~ 10 of I (I, f arbitrary).
P R O O F . =~. By assumption X - E(c) and we have Vxrlc.3irlI.xrlfi. Set
A(x, i ) " - i r l I A xrlf i and choose I o - p(A, c, I) by C/-reflection 16.3.1.
Topology Displayed
IV. 18]
115
Then I 0 is a subclass of I which meets the condition of the lemma.
r : assume that X := E(c) satisfies the right side of the lemma; trivially
E(c) = E( U {{x}:xrlc}) and hence for f := )~x.{x} and some M-class b with
E(b) C_ E(c), E(c)C_ E( U {{x}:z~b}), i.e. E(c) C_ E(b), for some M-class b,
i.e. X - E(b). B
Let P S ( X ) "- {E(a)" Cl(a) and E(a) C_ X}. Clearly:
P S ( E ( a ) ) - {E(c)" cTlP+(a)},
where P + ( a ) " - {z: Cl(x)A z C_ a} (see 9.15). It is immediate to verify:
18.5. LEMMA
(i) PS(X)
c-directed: if z, Y c PS(X),
such that Z C W and Y C W.
(ii) If Y E I N D , Y - U P S ( Y ) .
(iii) V(b)n V ( c ) - V(e), for some class e (b, c classes).
(iv) I N D N Y(O) - I N D (0 - empty property).
W
PS(x)
I N D can be naturally viewed as a sort of constructive "ideal completion" of
H Y P (in the sense of lattice theory). In particular, ideals in H Y P which
are explicitly presented by means of arbitrary operations, are already
principal ideals in I N D . The point is made precise below.
18.6. DEFINITION. If f" I---,CL shortens VxrlI.Cl(fx), we let"
I D E ( f ) "- {c" Cl(c) A 3I(Cl(I) A f" I ---, CL A c C_ U {fi" i~I})}.
18.6.1. FACT. The following is provable in M F - :
crlIDE(f ) ~-~Cl(c) A 9I(Cl(I) A f" I ~ e L A c C_ U {fi" irlI}).
The
I D E ( f ) represents an ideal in the lattice with support
{E(a)'a is a class in O ( ~ ) } (and with set inclusion as ordering);
term
HYP-
the definition is justified by the following:
18.7. P R O P O S I T I O N
(i)
I D E ( f ) is C-downward closed:
yrlIDE(f ) A el(x) A x c_ y ---,xrlIDE(f);
(ii)
I D E ( f ) is closed under U over its subclasses:
el(b) A b C_ I D E ( f ) ~
(iii)
( U b)~lIDE(f);
Constructive completeness: we can find a term L such that
E(IDE(f))-
E(P+(L(f))).
116
Type Free Abstraction with Approximation Operator
[Ch.4
PROOF. (i) is trivial.
(ii) Assume that b is a subclass of I D E ( f ) . Then by definition
b C_ C L -
{x" C/(x)} and Vxyb.3cyVL.A(x,c),
where A(x, c) "- f " c ~ CL A z C U {fi" iyc}.
Since A(x, c)is quasi-elementary in x, c, we have by 9.6 (i)"
Vzyb.VcyCL.(A(x, c) ~ T A ( z , c)).
Then one can apply CL-reflection 16.3.1 and there exists a class of classes e
such that
Vxyb.3cye.A(x,c).
(1)
By 9.14 d - LJ e is a class and U b C_ U {fi" iyd}. Indeed:
uy U b =:~3xyb.uyx;
:=~3x3c(xyb A eye A uyx A x C U {fi" iyc});
(by (1))
==~3cye.uy U {fi" iyc};
:=~3cye.3iyc.uy f i;
==~3iyd.uyf i.
(iii) Let D ( f ) " - U {c" Cl(c) A f" c ~ CL} and L ( f ) " - U {fi" i y D ( f ) } .
I D E ( f ) C f + ( L ( f ) ) is immediate by definition of L ( f ) and I D E ( f ) .
As to P + ( L ( f ) ) C_ I D E ( f ) , assume el(d) and d C U { f i ' i y D ( f ) } . Then by
CL-finiteness d C U {fi" lye}, for some class c C_ D ( f ) . But f" D(f)---,CL
and afortiori f" c ~ CL; hence d y l D E ( f ). 0
18.8. DEFINITION
(i) Put ~f(b)"-{E(a)" ayb}. Then P C Z2(M)is an RS-family iff for
some extensional 1 b, P - ~f(b); (in short P E RS; R S - Rice-Shapiro).
(ii) Let O ( b ) " - U {Y(e)" e E E(b), e class}. Then P C Z2(M) is explicitly
open in the class topology iff for some b E M, P - O(b) (b is called index of
the open set).
(iii)
D(b) "- {x" 3c(Cl(c) A cyb A c C x}.
(iv)
E C L - O P E N "- {O(b)" b E M}; R S "- {~f(b): b E M, b extensional1}.
RS-families are important because they characterize the explicitly open sets
of I N D .
18.9. LEMMA
(i) D(b) is exlensionall: x - - e Y A y y D ( b ) ~ xyD(b);
( i i ) ) ~ x . D ( x ) is extensional 2" x - e Y ~ D(x) - e D(y);
(iii) ~(D(b)) is an RS-family;
IV.19]
The Representation Theorem for CL-continuous Operators
117
(iv) if b is extensional1, then Y ( b ) - Y(D(b));
(v) O(b) - O(D(b)) and O(b) M I N D - Y(D(b)).
PROOF. (i)-(ii) follow from the fact that the formula defining D(b) is
positive in x and b; (iii) is immediate from (i), while (iv)is a restatement of
lemmas 17.3 and 17.2. Proposition (v) is a straightforward application of
the definition of O and D(b). [-1
18.10. DEFINITION
cr: E C L - O P E N ~ R S and w" R S ~ E C L - O P E N are the maps defined by:
cr(O(b))- Y(D(b)) and w(Y(b))- O(b).
18.11. THEOREM. RS-families are exactly the intersections of explicitly
open sets with I N D . More precisely:
(i)
if b E M and b is extensionall, then O(b) M I N D - Y(b);
(ii) for every b E M, ~o(~r(O(b)) - O(b));
(iii)
if b E M and extensional1, then ~(w(Y(b)))- Y(b).
PROOF. (i)is a consequence of 18.9 (v)-(iv).
(ii): by definition of w, e and 18.9 (v).
(iii)" by 18.9 (iv). El
Since I N D C Y(V) and the universe V - { x ' x - z} is an extensional 1
class, then we have that I N D U{Y(b)" b is extensionall}; on the other
hand, Y(b M c) C Y(b)f3 Y(c)" hence R S is a basis for a topology on I N D , we
simply label RS-topology. Once we observe that the intersection of any
basic open of C l - ~ with I N D is an RS-family and we keep in mind
18.11 (i), we obtain:
18.12. COROLLARY. The RS-topology coincides with the class topology on
IND.
w
The representation theorem for explicitly CL-continuous operators
As usual, an operator F" ~ ( M ) ~ ?P(M) is continuous with respect to the
class topology Cl-~T iff the inverse image of a basic open of Cl-~f is open in
Cl-~f .
19.1. DEFINITION
(i) F - ~ ( M ) - - - , ~ ( M ) i s CL-continuous iff 1) F is C_-monotone;
2) whenever x E F ( X ) ( X C_ M), then z E F(E(c)), for some class c
with E(c) C_ X;
118
Type Free Abstraction with Approximation Operator
(ii)
[Ch.4
F : ~ P ( M ) ~ ~P(M)is E C L (explicitly CL-continuous)iff
F - F I where F I ( X ) "- U { E ( f c) " Cl(c) A E(c) C_ X};
f is called an index for F.
N O T A T I O N . E C L : - s p a c e of explicitly continuous operators (from Z)(M)
to ~ ( M ) ) .
ECL-continuous operators are CL-continuous; as soon as we consider
the subspace I N D with ECL-operators restricted to I N D , CL-continuity
and continuity in the class topology yield the same notion.
19.2. LEMMA
(i) Assume that F" ~ ( M ) - - - ~ ( M )
is continuous with respect to the
class topology. Then F is CL-continuous.
(ii) If F is ECL-continuous, then F is CL-continuous and the
restriction of F to I N D is continuous in the class topology relativized to
IND.
P R O O F . (i)" straightforward with 18.3 (iv).
(ii) Let F - F f"
a E F(X)::~ for some class c with E(c) C_ X, a E E(fc),
=V there are classes c, d with E(d) C_ E(c) C_ X, a E E ( f d),
=:~ for some class c with E(c) C_ X, a E F(E(c)).
Monotonicity is also immediate by definition. As to the second claim, if e is
a class, f is a index for F and F is restricted to I N D ,
F - i ( U ( e ) n I N D ) - {E(a)" 3b(E(e) C_ E(b) A F(E(a)) - E(b)};
= { E ( a ) ' e C_ U {fc" Cl(c) ^ c C_ a}} (by ECL-continuity);
=
where J(e) "- {x" e C_ U { f c" Cl(c) A c C_ x}}. But g(e) is extensional a and
hence F - I ( V ( e ) n I N D ) is an RS-family, i.e. open in the relativized class
topology by 18.11. [3
We now see that the restrictions of ECL-operators to I N D have the
expected representation, as effective counterparts of RS-continuous
operators on I N D ; we write F [ P for the function F, restricted to the
domain P.
19.3. DEFINITION. An operator F" I N D ~ I N D
is effective (in short
F E E F F ) iff F ( E ( a ) ) - E(fa), for some extensional 2 f and every a E M.
F is "generated by f" and simply denoted by E l ; thus E I ( E ( a ) ) - E ( f a ) .
IV.19]
The Representation Theorem for Ct-continuous Operators
119
E F F will denote the the space of effective operators.
19.4. LEMMA. Assume F E E F F . Then F is RS-continuous and hence
CL-continuous in the class topology relativized to I N D .
PROOF. Observe that if F = E , f is extensional 2 and b is extensional1,
F - l ( { E ( a ) : a E E ( b ) } ) = {E(c): (fc)rIb } and {c: (fc)rIb } is extensional1;
then apply 18.12. E!
19.5. DEFINITION.
f*a "- U { f c" c C_a A Cl(c)}.
19.6. LEMMA
(i) f* is extensional2; hence E l , E EFF. f*
e
*a
(ii) If f is extensional2, f*a - efa; hence
a - (f*)
(a arbitrary)
(iii) F I - F I , .
PROOF. (i) f* is extensional 2 because f*a depends positively on a.
(ii) fa C_f*a is simply the main theorem 17.5, while f*a C_f a follows by
C_-monotonicity of extensional 2 operations (17.7).
(iii) 9by definition of F f and f*. [3
We now see that the restrictions to I N D of explicitly CL-continuous
operators are effective operators; viceversa, each effective operator has its
source in a uniquely determined element of ECL.
19.7. DEFINITION
(i) If E I E E F F , let C(EI) .-- F I.
(ii) If F$ C ECL, let ~(Fi) "-- E I , .
19.8. T H E O R E M
(i) Let F y E ECL. Then ~(F f ) E E F F and
$(Fi)- F$[IND(ii)
FI,[IND.
Let f be extensional 2. Then
C(EI) E E C L and r
EI.
(iii) Moreover"
if f
(e(Es) ) -ES;
if F$ E ECL, C($(FI) ) - F I.
(,)
(**)
PROOF. (i) ~ ( F I ) E E F F by 19.6(i). Moreover, F I ( E ( a ) ) - E ( f * a ) i s
immediate by definition of f* and explicit CL-continuity with index f. On
the other hand, F y , ( E ( a ) ) - E ( f * * a ) - E(f*a) by 19.6 (ii).
Type Free Abstraction with Approximation Operator
120
(ii) Let G ' - E ( E f ) .
Then we have G E E C L
e ( E ( a ) ) - E(fa) is a restatement of theorem 17.5.
(iii)
Let
EI,-E
F'-EIEEFF:
$(C(F))-Ef,
I by 19.6(ii). Let F s E E C L :
as
then
by
C(F)
lemma
[Ch.4
19.2(ii).
has index
C($(FI))-FI,-F
f;
but
I by
19.6 (iii). [3
If F is a monotone operator from ~ ( M ) to ~P(M), let ](F) denote the
(set-theoretic) least fixed point of F. Then we obtain the fundamental:
19.9. T H E O R E M ("First Recursion Theorem"). If F I E ECL, then
~(FI)- E(I(f)).
PROOF.
Let F ] be ECL: then
a E F$(E(I(f))) =:r O(,Al~)]=Ay(a,I(f))(CL-continuity);
=:r O(MI~)I-- arlI(f), i.e. a E E(I(f)) (by fixed point 10.1);
hence E ( I ( f ) ) i s F/-closed and ~(FI) C_E(I(f)). On the other hand,
{a E M" (O(.AI~),~(F$))I=A$(a,X)} C_~(FI).
But E ( I ( f ) )
is the
theorem of w
C_-least As-closed subset of i
by the induction
hence E(I(f)) C_~(F$). [3
Theorem 19.9 is a completeness result: as far as we consider continuous
operators, which are explicitly given via internal maps of the underlying
combinatory algebra, the restriction to the subspace I N D is unessential and
we are still able to capture all the explicitly "recursive" objects we have in
the full space. Theorems 18.11 and 19.9 establish the adequacy of a
s y n t a x - t h e language of operations and reflective t r u t h - to the semantics
of ECL-opens and ECL-operators. This should add some evidence in favour
of the approximation axioms as reasonable choices.
19.10. R E M A R K
(i)
Define:
FUN*(c)(X) "- {a E M" for some class d, with E(d) _C X, (d, a) E E(c)}.
It is easy to see that {FUN*(c)" c E M} - { F ] ' f E M}; so one has an
alternative definition for explicitly CL-continuous operators. Also, if
F, G" ~ ( M ) ~ ~(M), F C_G "- F(X) C_G(X) for every X C_ M.
Finally, we define:
IV.19]
The Representation Theorem for CL-continuous Operators
Vl(c)
" - {F" (F" ~P(M)~ ~ ( M ) ) and
121
FUN*(c) C_F};
then CLF-% "- { V i ( c ) ' c class} is a basis for a topology on {F" F operator
from ~ ( M ) to ~(M)}. One checks that ECL and E F F are homeomorphic
as function subspaces via $ and C.
(ii) Define:
Graph+(f) "- {(c,x)" Cl(c) A x~lf c};
Fun+(a) "- )~u.{x" 3y(y C_u A Cl(y) A (y,x)~a)};
a,b "-(Fun+(a))b.
Then:
if f C_ g, then Graph+(f) C_ Graph+(g);
if f is extensional2, then for every a, f a -
e (Graph+(f))*a.
I N D becomes a )~-model if we interpret application via 9 and )~-abstraction
via Graph + .
(iii) By 17.8.1, the internal fixed point operation I preserves the natural
pointwise equality on operation. This suggests the definition of a hierarchy
of extensional operations, in analogy with the classical hierarchy of
hereditarily effective operations HEO (see Troelstra 1973). One may wonder
whether this hierarchy mirrors the corresponding natural hierarchy based on
class topology at the ground type ~ ( M ) - {X" X C_ M} and extending
upwards via pointwise convergence.
Perhaps, it might be useful to generalize to the present theory Ershov's
notion of numeration. Ershov's basic idea is to consider enumerated sets,
i.e. pairs (X,v), where X is any set and v is a map from natural numbers
onto X. Enumerated sets with an appropriate notion of (computable)
morphism give rise to a category, which forms a suitable environment for
developing a theory of partial computable functionals in higher types.
In this respect, the reader, who seeks for possible generalizations, could
profitably read the interesting paper of Ershov (1985): there, the author
constructs a theory of E-predicates of higher types over any admissible set
f~. In Ershov's approach, sets in gi play the role of classes and one can
introduce a topology on E-definable families of E-sets of f~, which appears
an analogue of the class topology. The topological structure is then applied,
in order to define morphisms between enumerated sets (in the sense of f~),
and to induce an appropriate notion of approximation.
122
Type Free Abstraction with Approximation Operator
[Ch.4
Appendix: alternative proofs
The proof of the generalized Rice-Shapiro, given in w
depends upon
axioms ~'.1, ~'.2 and ~'.5. However, during the final proof reading of the
book, P. Minari found a slick variant 3 of the term J of 17.3. The resulting
argument relies on 7r.2 and r.3 and 3 satisfies an additional condition.
Alternative proof of 17.3. By the fixed point theorem for properties, we can
find 3 such that, if we set J "- 3zx and E - {(x, y)" xqy}, then
Vu(u~J +-+(u,x)~rE(J,z)).
(1)
Assume E x t i ( z ) and x~]z. If-~J~z, by ~.3, E C_ 7rE(J,z); hence, if ur]x,
(u,x)~E C_ r E ( g , z ) .
So x C_ J, which implies g~z by 17.2. By classical
logic, we conclude:
JrIz.
(2)
By ( 2 ) a n d ~-.2, 7rE(J,z)is a class: hence
cl(g).
(3)
Again (2) and 7r.2 imply 7rE(J,z)C_ E. If u~Ig, then {u, x)TprE(J, z)) and
hence uTIx; hence
gC_x
(2)-(4) w ify
VzW(E
g(z) ^
(4)
^
^
c_
We can also prove, recalling that that J "- 3zx:
EXtl(z ) A J~z -~ x~Tz.
Indeed, assume EXtl(z),-~x~z and J~z. Then by (4) J C_ x, whence xr]z by
extensionality I of z: contradiction. Q
It is also clear that CL-compactness becomes provable in
MF-+{Tr.2, r.3}. The separation property 16.9 can be proved on the
ground of 7r.2 and 7r.3.
PROPOSITION ( M F - + {~r.2, Tr.3}). There is a term S E P ( x , y ) such that,
if x N y -- e 0, then S E P ( x , y) weakly separates (x, y).
PROOF. Choose S E P ( x , y) "- {u" (u, x}~prE(u, y}}. [3
Since the developments of the chapters to come, only depend upon class
compactness and exact representation 16.11 (which only requires 16.9 and
CL-compactness), we might replace P W - b y M F - + {~'.2, ~'.3}.
However, the proofs of w167 illustrate a different way of handling stage
arguments, that might perhaps be useful elsewhere. Moreover, the definition
IV.A]
Appendix
123
of ~, given in 16.3, is easier to understand, because it reflects a natural settheoretic interpretation of ~a, as the collection containing the elements of a
with minimal stage.
The previous remarks naturally raise the following problems:
1.
2.
3.
4.
find models of 7r.2 and 7r.3, which are not inductively generated;
prove 7r.2 and 7r.3 from some of their notable consequences (say class
compactness) and MF--axioms;
prove that the system of the r-axioms is independent.
find significant consequences of 7r.4, 7r.5 (if the previous problem has
positive solution), which cannot be obtained by 7r.2, 7r.3.
This Page Intentionally Left Blank
CHAPTER 5
TYPE-FREE ABSTt~CTION, CHOICE AND SETS
w
w
w
w
Choice principles and the distinction between operations and
functions
Admissible hulls: elementary facts
A model of admissible set theory
The boundedness theorem
This part concludes the axiomatic investigation of the recursion-theoretic
properties of inductive models for reflective truth. We are interested in
relating the present framework, extended with approximation axioms and
generalized induction, to classical systems, namely theories of sets. Thus the
main bulk of this chapter is devoted to the construction of a model for
admissible set theory within PW c + GID.
In w we investigate choice principles, under the assumption of the
enumeration axiom 4.14, and we observe that they have a problematic role
in the present theory. Inconsistencies underline the difference between the
intensional notion of operation and the set-theoretic one.
By the results of w167
an ilerative notion of set can be adequately
defined within the theory, extended with approximation axioms and the
generalized induction principle GID. Indeed, we uniformly associate to each
property U a structure AD(U), the admissible hull of U, which is a model of
admissible set theory with urelements in U.
In the final section w we show that two natural ways of characterizing
the minimal fixed point of a monotone operator are extensionally equivalent
("boundedness" theorem), and it is quite surprising to see that the
equivalence result holds in a proof theoretically weak system. The theorem
exploits the generalized continuity property of 17.5 and the uniform
Knaster-Tarski theorem from the previous chapter, together with the theory
of ordinals available from inner set-theoretic models.
The boundedness theorem should add some evidence in favour of the
present axiomatic choice: the extended theory is natural, because there is a
certain harmony between set theory and recursion theory, predicative and
impredicative definitions.
Type-free Abstraction, Choice and Sets
126
w
[Ch.5
Choice principles and the distinction between operations and functions
In this section we investigate the problem of consistently refining the
approximation operator 7r of w to a choice operator, in presence of suitable
well-orderings of the universe, in particular when the enumeration axiom of
w is assumed. This problem leads to consider how far the operational
axioms can proceed in witnessing "implicitly defined" functions. We shall
see that it is essential to distinguish between operations and functions-asgraphs, and to take into account the extensional character of the operations
involved.
If the approximation operation 7r is applied to properties of natural
numbers, 7r can be obviously strengthened to a selection operation and the
same happens if the enumeration axiom EA (cf. 4.14 and following) is
assumed. We recall that EA is the sentence 3 f V x 3 y ( N y A f y = x) and that
EA is consistent with PW c (consider the inductive models O(CTM) or
O ( R E ) , where CTM and R E are introduced in Ch. I).
20.1. DEFINITION
(i) Let w encode a binary relation, i.e. let w be a property of ordered
pairs. We keep using the infix notation x -~ w Y in place of Ix, y)~lw.
F i e l d ( ~ w ) is the term {x" 3z(x ~ w z V z ~ wX)} representing the field of
-~w, while the x-segment of -~w determined by x is defined by the term
(ii)
LO( ~ w ) "- VxVyVz(--,(x -~ w x) A
A(x -~ wY A Y ~ wZ----, x -~
^ Co..(
)),
where Corm( -~ ,,,)"- VxYy(xTIField( ~ ~) A
A yrlField( -< w ) ~ ( x
LO(
-< wy V x -
y V y -< wX)).
w ) states that -< w is a linear ordering.
(iii)
Progr(b, ~ w ) "- Vx(xrlField( "< w ) A Vy(y -< w x -~ yrlb ) --, x~b).
Progr(b, ~ w ) i S
to be read "b is progressive" (relative to "~w)"
We also define:
T I ( -~ w , b ) " - Progr(b, "~ w ) 4 Field( ~ w) C_ b.
(iv) A linear ordering "~w is called a pseudo-well-ordering- in symbols
P W O ( ~ w ), and, in short, -~ w is a p w o - iff Vb(Cl(b)-~ T I ( ~ w, b)).
(v) A pwo -~ w is acceptable iff -~ w is a class.
(vi) In P W c the standard ordering < of N is a pseudo-well-ordering; if
we assume the axiom EA and f is any surjective operation from N onto
Choice Principles, Operations and Functions
V.20]
V - {x" x - x} given by EA, we can define x < $ y "- Ix I i <
I xly - the least k in N such that f k - x.
127
l yls,
where
20.1.1. CONVENTION. It is understood that all the constructions below,
which rely upon EA, will be uniform in some fixed witness E for EA, such
that
V x 3 y ( N y A E y - x).
For simplicity, we leave dependence on E, usually implicit.
Of course, if E satisfies the enumeration axiom, it is easy to get"
20.1.2. FACT. < E is an acceptable pwo, provably in P W c + EA.
On the other side, we can verify:
20.1.3. F A C T (provable in P W - ) . -z, w is an acceptable pwo iff ~ w
linear ordering such that
(i) {(x,y)" x -< wY) is a class;
(ii) every non-empty class C_ Field("<w) has a -< w-least element.
is a
For the proof of 20.1.3, one applies closure of C L under complement, the
assumption that -4 w is a class and elementary comprehension from 9.7.
20.2. LEMMA (Selection, provable in P W - ) .
We can define a term 5(a, -< w) such that, if '<w is an acceptable pwo and
a C _ F i e l d ( - < w ) , then S(a, -<w)C_a and, if a is non-empty, 5 ( - < w , a ) is a
class, which contains exactly one element of a (i.e. x ~ 5 ( - ~ w , a ) ,
YrlS( < w, a) imply x -- y).
PROOF. Set
CL-compactness (16.3) and elementary comprehension 9.7 complete the
proof. 0
If we apply the selection lemma, binary relations can be uniformized via
functions-as-graphs, provably in P W c + EA.
20.3. DEFINITION. Let:
F u n r e l ( f ) "- C l ( f ) A Vu(u~lf --~ 3x3y(u - (x, y))) A
A
wvvvz(( ,v) f A
-
z);
F u n r e l ( f ) is read " f is (or encodes) a functional binary relation ".
20.4. T H E O R E M . We can find a term Sel(r, b) "- Sel(r, b, < E) ( cf. 20.1.1
for notations) such that, provably in PW c + E A , if b is a class and r is a
Type-free Abstraction, Choice and Sets
128
[Ch.5
binary relation defined on b, then Sel(r,b) is a functional relation which
uniformizes r on b; formally
cz(b) ^ w(~ob~3y((~, y)o~)) --.
--, s~z(~, b) c_ ~ ^ Vu,Tb.~!y((u, y),~sd(~, b)).
P R O O F . Define U(r, x) := {y: (x, y)rlr) and choose
Sel(r,b) := { u : 3xqy(x~lb A yrl~(U(r,x), < E)A u = (x, y))}.
(,)
In (,) < E is the pseudo-well-ordering which exists by EA, while ~ is the
operator of lemma 20.2. [-i
It is natural to ask whether the theorem 20.4 can be strengthened. If
Vxrlb.3y.A(x,y ) is assumed, is there a choice operation g such that A(x, gx)
for every x in b ? The answer is easily seen to be negative in general.
20.5. THEOREM. (i) The choice schema ACv(oP) over the universe V
Vx3yA(x,y)---, 3 f V x A ( x , f x ) (A in the language of OF)
is inconsistent with O P - ( (ii)
OP without N-induction).
The choice axiom for classes Cl-AC(op)
CZ(a) ^ CZ(~) ^ W~a. 3y <~, Y ) ~ ~ 3gW~a.((~, g~/~)
is inconsistent with any theory, which includes elementary comprehension
EC and O P - .
PROOF. (i) By intuitionistic logic plus ( x - y V - - x - y), we can prove in
OP-:
Vx3!y((y - 1 A x -- 0) V (y -- 0 A-~x -- 0)).
(1)
Were a choice operation f available in O P - , we should have:
V x ( ( f x - 1 A x -- O) V ( f x -- 1 A-~ x -- 0)).
(2)
But (2) yields a "global test for zero" and it leads to an inconsistency via
paradoxical combinator (see 3.9).
(ii): apply (i) and elementary comprehension 9.7. I-1
20.5.1. REMARK. The argument of (i) essentially depends on classical logic
(i.e. the decidability of - ) and on the use of type-free operations; in fact,
Barendregt (1973) shows that ACv(oP) is consistent with combinatory logic
based on intuitionistic logic. Part (ii) also holds without classical logic: the
trick can be found in Beeson (1985), where a survey of consistency results in
constructive type-free systems is given (see also Troelstra-van Dales 1989).
The
inconsistency
of theorem
20.5
has,
however,
a
conceptual
V.20]
Choice Principles, Operations and Functions
129
significance: it underlines the limits of the interplay between type-free
operations and type-free predicate abstraction. Nevertheless, we can produce
choice principles, consistent with the present framework, as soon as we
consider extensional operations.
First of all, recall that m, n, k range over number-theoretic variables
(hence VnA and quA stand for v x ( g x - - , A ) , 3 x ( N x A A)).
20.6. D E F I N I T I O N
(i) E x t 2 ( f ) "- YxYy(x - e Y ~ f x - e fY);
2 - E x t ( f ) "-- YxYyYzVw(x - e z A y -- e w ~ f x y -- e f zw).
(Here
x
-
~y " - V u ( u ~ x ~
w
2-Ext(f)"-f
variables).
(ii)
is
u~y)); E x t 2 ( f ) " - f is extensional 2 in the sense of
2-extensional or extensional as a function of two
E x t - A C is the axiom 9
Vb3hV f (Ext2( f ) A Vxrlb.3y(Cl(y ) A xrlf y) ~ Vxrlb(Cl(h f z) A xrlf (h f x)) ).
(iii) Ext-DC is the axiom:
3hV f ( 2 - E x t ( f ) A V n V x ( C l ( x ) ~ 3y(Cl(y) A nTlfxy))--~
---, Vx(Cl(x)--~ (h f xO - x A Vn(Cl(h f xn) A n~f(h f xn)(h f x(n+l)))))).
20.7. T H E O R E M ("Extensional choice"). P W c + EA F Ext-AC.
P R O O F . Assume that f is extensionM 2 and let Vxyb.3y(Cl(y)A xrlfy ). Put
t(f, ~)-- {y. cl(y) ^ ~fy}.
Then, by assumption, we have Yxrlb.3y(yrlt(f,x)); thus, if we choose
r ( f , x) "- 5(t(f, x), < E), where < E is the pwo induced by EA on the
universe, lemma 20.2 implies in P W c + EA:
r ( f , x) C t(f, x) A Yxyb. ( C l ( r ( f , x ) ) A 3!y(yyr(f,x))).
(1)
Hence r ( f , x ) is a non-empty class of classes, for every x in b, and if we
choose h f x U r ( f , x ) we obtain Vzrlb.Cl(hfx) by 9.14. Observe now that,
for x in b:
yrlr(f ,x)---~y -- e h f x
(2)
(apply the fact that r ( f , x ) has a unique element). If xr/b, ( 1 ) - ( 2 ) i m p l y
xrlfy for some y - e h f z , hence E x t 2 ( f ) yields xrlf(hfx). 0
20.8. T H E O R E M ( "Extensional dependent choices"). P W p + EA F Ext-DC.
PROOF.
Assume the antecedent of Ext-DC, and let f be 2-extensional.
Type-free Abstraction, Choice and bets
130
[Ch.5
Then, if R(n, x, f ) " - {y" Cl(y) A ml(fxy)}"
VnVx(Cl(x) ~ 3y(y~ln(n,x, f))).
Then we can find S ( n , x , f ) " - 5 ( R ( n , x , f ) ,
PWc + EA:
ww(c~(~)
-~
(1)
< E), such that, provably in
(S(n, ~, f) c n(~, ~, f) ^
A Cl(S(n,x, f ) ) A 3!y(Cl(y)A yrlS(n,x, f)))).
(2)
Then we can find, by primitive recursion on N (cf. 3.2) with parameters x,
f, an operation h such that"
hfxO-x
and h f x ( m + l ) -
U S(m, h f x m , f )
(3)
(U -generalized union of 9.14). By N-induction for properties, it easily
follows with (2), if x is a class:
V m ( C l ( h f x m ) A S(m, h f x m , f ) C CL);
(4)
(4) and the uniqueness requirement of (2) also yield:
Vy(y~lS(m, h f xm, f) ~ y - e h f x(m+l)).
(5)
Hence by property N-induction, definition of h, (5) and 2-extensionality, we
can conclude"
hfxO - e x A Vn(u~lf(hfxn)(hfx(n+l))). D
(6)
The uniform choice axioms Ext-AC and Ext-DC entail two schemata for
elementary conditions, which are extensional in the relevant parameters.
These schemata are significant for interpreting fragments of second-order
arithmetic (see Ch. VIII).
20.9. DEFINITION
(a) A formula A of s is elementary extensional in X l , . . . , x n iff A
belongs to the least class of formulas inductively generated by means of A,
-~, Vy (y distinct from Xl, ...,xn) from atoms of the form t = s, Nt, t~xi,
provided X l , . . . , x n do not occur in t, s (compare with 9.5).
(b) Let z x := {u: (x,u)~z}. E A C ( = the elementary choice schema):
Cl(b) A W,b. 3y(Cl(y) ^ A(~, y))-~ 3z(CZ(z) ^ W,b.A(~, ~ ) ) ,
for A(x, y) elementary extensional in y;
(c) EDC( = the elementary dependent choice schema)):
VnW(Cl(~) -~ 3y(Cl(y) ^ A(~, ~, y))) -~
-~ W(Cl(~)-~ ~z(Cl(z) ^ Zo = ~ ^ VnA(~,z~, z~+i))),
Admissible Hulls
V.21]
131
for every A(u, x, y) elementary extensional in x, y.
20.9.1. FACT. If A(U, X l , . . . , x n ) is elementary extensional in X l , . . . , x n and
= e is extensional equality with respect to y (see 9.11), we can prove in
pure logic, for 1 _<i _<n:
A(u,
Xl, . . . , Xn)
A x i - - e Y i --o
A(u, Xl, . . . , X i _ l ,
Yi ' Xi+l"
" " Xn)"
20.10. C O R O L L A R Y
(i) P W c + E A F EAC;
(ii) P W p + EA F EDC.
P R O O F . (i) Assume Vx~lb.Jy(Cl(y) A A(x, y)), where A(x, y) is elementary
extensional in y; then f y := {x: A(x,y)} is extensional and by Ext-AC
there is an operation h such that Cl(hx), whenever xTIb, and such that
A(x, hx). Choose z = E(b,h): then z is a class by join principle 9.9 and
V x A ( x , % ) (use extensionality of A in y).
(ii): similar argument (apply Ext-DC). 0
20.11. FINAL REMARK. It may be of interest to restrict the investigation
to the special class b - N. Then we can apply 5.13 and we can show that
MF c is consistent with the full N-choice schema ACN:
Vx(xT1N ~ 3yA(x, y ) ) ~ 3fVx(x~lN---+ A(x, f x)) (A arbitrary).
The reader can readily verify in MF c + AC N"
20.11.1. Let
P d ( N ) ' - {{x" g x - 0}" gr/2N}. Then for every A, there exists
an element b of Pd(N) such that Vx(x~N --, (x~b ~-, A(x))).
20.11.2. For every a C N there is a class b in Pd(N) such that a - eb"
w
Admi~ible hulls " elementary facts
In the previous sections, we pointed out the fundamental aspects of a theory
of abstraction in a world endowed with an approximation structure. We
now apply the strengthened machinery to the reconstruction of standard
concepts, namely sets.
Henceforth, the general frame theory is P W c + G I D , which is
conservative over Peano arithmetic. Our basic aim is to define models of
admissible set theory with urelements (or atoms). Indeed, we shall introduce
an extensional operation AD such that AD(U) is a canonical admissible
structure over U; AD(U) is called the admissible hull of U. If U is a class (in
our sense), U will correspond to a set in AD(U). If U = N, we have the
Type-free Abstraction, Choice and bets
132
[Ch.5
counterpart of the "next admissible structure" above natural numbers in the
sense of Barwise (1975) (see also Barwise, Gandy, Moschovakis 1971,
Moschovakis 1974).
The idea is that atoms are just pairs Ix, 0) with x in U, while U-sets
have the form (x, 1) where x is a class C_ AD(U). Thus AD(U) can be
regarded as the solution of natural inductive conditions. As we shall see, the
construction is monotone in U, and AD is actually functorial: every
injective operation f from U into W can be canonically extended to an
embedding AD(f) of AD(U) into AD(W).
21.1. D E F I N I T I O N
(i)
Let P A R ( a ) " - ( a -
((a)l , (a)2)): we define
AU(X , v) "- P A R ( x ) A [((x)2 -- 0 A (X)lrlV) V ((x)2 -- 1 A
A Cl((X)l ) A Vu(u~(x)l V urlv))].
(ii)
Clearly, A u ( x , v)is an operator in v and we can choose, by 10.4:
AD(U) "- Ixv.Au(x , v) (in short I(Au);
U-AT "- {x" PAR(x) A (X)lrlU A (x)2 -- 0);
U-SET "- {x" xrlAD(U) A (x)2 -- 1}.
If x is in U-AT (U-SET, AD(U)), we say that x is an U-atom (U-set, Uobject).
21.2. LEMMA (MF-)
(i)
(ii)
Cl(U) ~ Cl(U-AT) A (U-AT,-f )rlAD(U).
Vx(x~?AD(V)~ PAR(x) A (((X)lr]U A (x)2 -- 0) V (Cl((X)l) A
A Vy~(x)l. yrlAD(U)A (x)2 -- 1))).
(iii)
Vx(xrlAD(V) ~ (xrlU-AT Y xrlU-SET));
(iv)
~(xrlU-AT A xrlU-SET).
(i)-(iv) are straightforward by choice of AD(U) and 10.1.
We now proceed by inductively defining an equality relation - u
on
- u is extensional on sets, together with its dual "internal"
version ~ u 9We put:
AD(U);
El(u, v,z) "-- Vx(x~(u)i V 3yrl(V)l.(x,y)rlz);
E(u, v, z) "- El(u, v, z) A El(v, u, z).
Admissible Hulls
V.21]
133
Clearly E(u, v,z) determines an operator in z; if (U)l , (v)l are classes,
we can prove by T-logic:
21.3. E(u, v, z ) ~
(Vxrl(U)l.::lyrl(v)i. (x, y>yz) A (Vx~7(v)i.::iyr](U)l. (x, Y)~TZ).
Let us define the following formula, which uniformly depends on U and
is an operator in z"
B U (u, z) :-- P A R ( u ) A ( ( ( u ) I , U - S E T A (u)2,U-SET A E((u)a, (u)2, z)) V
Y ((u)I~IU-AT A (u)2~TU-AT A (U)l -- (u)2)).
Then we find a fixed point I ( B u ) : -
I u z . B u ( u , z ) (by 10.1).
21.4. D E F I N I T I O N
(i)
u - U v := (u, v)rlI(B U ); E U (u, v) := E(u, v, I ( B U ));
(ii)
u ~ u v := uoAD(U) A v~?AD(U) A F(u - u v);
(iii)
D E u ( u , v ) := ::txr]('a)i.~y(y~(v)i V x ~. uY) V
V 3x~(V)l.Vy(y"~(u)i V x ~. U Y)"
21.5. L E M M A ( M F - )
(i)
(ii)
-~(U--uvAu~uv);
u -- U v ~ (u~U-SET A vrl U - S E T A E U (u, v)) V
V (u~U-AT A v~U-AT A u = v).
(iii)
u ~ U v ~ (u~AD(V) A v~AD(U) A (u)2 :/: (v)2) V
V (mTU-AT A vrlU-AT A u :/:: v) V
V (u~U-SET A v y U - S E T A D E U (u, v)).
P R O O F . (i): by T-consistency.
(ii): apply 10.1, 21.3 and the fact that, if u is an U-SET, (U)l is a class.
(iii) :=V" By T-logic, F ( u - U v) requires distinction of some cases. We only
consider the case where u and v are U-sets such that F ( E u ( u , v)) holds. To
be definite, we also suppose Vy(y~(v)l V F ( x - u Y)), for some x in (u)1. By
hypothesis on u and v with 21.2(iii), (V)l is a class and x is in AD(U).
Hence i f - ~ y ~ ( v ) l holds, we also have y ~ A D ( U ) a n d F ( x - u Y ) ,
which
imply x ~ u Y, i.e. D E u ( u , v).
r
Assume that u, v are U-sets and D E v ( u , v ). T h e n a fortiori
F(u~AD(U) A (u)2 -- 0), i.e. F(u~U-AT), which entails:
F(u~U-AT A vrIU-AT A u - v).
(1)
134
Type-free Abstraction, Choice and Sets
[Ch.5
But D E U (u, v) yields F E U (u, v), i.e.
F ( u r l U - S E T A v r l U - S E T A E u ( u , v)).
(2)
(1)-(2) imply u ~ u v. The other cases are left to the reader. I3
21.6. PROPOSITION (MF- + GID)
(i) arlAD(U) --~ a - ua;
(ii) - a ~ u a;
(iii)
(a - u b ~ b - Ua) A (a ~ U b ~ b ~ Ua);
(iv)
a r l A n ( U ) A b ~ I A D ( U ) ~ a - u b V a ~ u b;
(v)
(vi)
a-u
bAb-U
a - ub ^ b
c~a-yc;
uc-
a
uc.
PROOF. (i): apply the generalized induction schema for A D ( U ) to the
condition x - u x and 21.2.
(ii): a ~ u a implies arIAD(U), whence a - u a by (i), which contradicts
21.50).
(iii): trivial by inspection of 21.5 (iii).
(iv): we apply GID on A D ( U ) to the formula
C(a) "- V b ( b q A D ( U ) ~ a - u b V a ~ u b).
Thus we assume A u ( a , C), b~IAD(U) and we verify a - u b or a ~ u b. We
repeatedly use 21.5 (ii)-(iii). If a is an U-atom and b is an U-atom, we have
a - u b or a ~ u b, according to a - b or a # b ; i f a i s an U-atom and bis
an U-set (or symmetrically), we get a ~ u b. If a, b are U-sets, then (a)i ,
(b)l are classes and the following conditions hold:
Vx~I(a)I.VV(vrIAD(U ) ~ x - u v V x ~ u v);
(1)
Vyrl( b)l. YrlAD( U ).
(2)
If---,a- u b holds, we have by 21.5 (ii) -,Eu(a,b); for instance, suppose that
there is xT/(a)l such that, for every y, either -~yr/(b)l or - ~ x - u Y" In the first
case we obtain y~(b)i (classhood of (b)l); if y is in (b)l , - - , x - u Y holds,
whence x ~ u Y by (1)-(2). As a consequence, n E u ( a , b ) holds and finally
a~ub.
(v)-(vi): argue by GID, 21.5, (iii)-(iv)above. [3
21.7. DEFINITION (Extensional membership and its dual relation).
X C U Y "-- ~V(V -- U X A V r / ( y ) i A (Y)2 -- 1);
x -~ u Y "- xrlAD(U) A Vv(v ~ u x V @(Y)I V (Y)2 ~ 1).
V.21]
Admissible Hulls
135
If we systematically apply 21.6 (e.g. you need 21.6 (iv) to check (v) below),
we have that - u
is a congruence with respect to E U, ~ U and that
Axy.[x E u Y] is a propositional function, if it is restricted to elements of
An(u).
21.8. L E M M A ( M F - + GID).
(i)
~(xEuYAx~uY);
(ii)
x~7(y)l A (Y)2 -- 1 A ( x ~ A D ( U ) Y y ~ A D ( U ) ) - ~ x E U Y;
(iii)
xEuYAy--yZ--,XEuZ;
xEuYAX--uZ~ZEuy;
(iv)
x -E u Y A Y -- u Z --) X -E u Z;
x -E u Y A X -- u Z --, z -E u Y ;
(v)
(vi)
xT1AD(U) A yT1AD(U) ~ x E u Y V x -~ u Y;
y~IAD(U) A a E U Y ~ yTIU-SET"
As one might expect, - u and E U-extensional equality coincide on U-sets:
if we define x C u Y " - Vw E U x. w E U Y, we can prove:
21.9. L E M M A ( E x t e n s i o n a l i t y )
M F - + GID ~ ( x ~ ? U - S E T A y ~ ? U - S E T A x C u Y A y C_ u x) --~ x -- u Y.
P R O O F : let w in (x)l; then w is in A D ( U ) and hence w =_ U w (21.6(i)),
which implies w E U x. By assumption w E u Y, whence w~/(y)l; with similar
arguments, we obtain Y u r l ( Y ) l . 3 V ~ ( X ) l . U - u v and hence E u ( x , y ). Then
x - u Y follows from 21.5 (ii). F!
21.10. P R O P O S I T I O N ( P W - + GID)
I f :f(V) " - A D ( U ) ,
- u , r u , E U, -E U,
u c
c
(,)
In particular"
Vx(x~AD(U)
~ 3 c ( C l ( c ) A c C_ U A x ~ A D ( c ) ) .
(**)
P R O O F . (**) is an application of the approximation theorem 17.5 and (,);
(,) is verified by GID on the definition of A D ( U ) (also use 21.2(ii) and
21.5 (ii)-(iii)). El
According to 21.10, it is not restrictive to consider only classes of
urelements; if U is not a class, no new set is generated in A D ( U ) , unless it
is not already in some A D ( W ) , where W is a class C_ U. More generally, in
order to check x E uY (where also y is in A D ( U ) ) or x - u Y, we only need
Type-free Abstraction, Choice and Sets
136
[Ch.5
a collection c of U-atoms, which contains x, y and is a class.
21.11. D E F I N I T I O N
(i)
I n j ( f , U, W ) " - f is an injective operation from U into W "= f" V --~ W A VxVy(x~V A yTlV A f x -- f y ~ x -- y).
(ii)
Embed(f, A D ( U ) , A D ( W ) ) " - f embeds AD(U) into A D ( W ) " = w v y ( ( ~ - ~: y ~ f ~ - w f y ) ^ (~ ~ ~: y ~ f ~ ~ w f y ) ) ^
^ W V y ( y ~ A D ( U ) -~ (~ ~ v Y -~ f ~ ~ w fY) ^ (~ -~ u Y-~ f ~ -~ w f Y ) ) ^
A Vx((x~IU-SET ---, ( f x ) ~ I W - S E T ) A (x~IU-AT ---, (fx)~IW-AT)).
(iii)
in(x) "- (x, 0).
Clearly Ax.in(x)is an injective operation from U into A n ( u ) .
21.11.1. R E M A R K . If f is an embedding of A n ( u ) into A n ( w ) , we can
reverse the arrows in 21.11(ii), as a consequence of 21.8(v), 21.6(iv),
21.2 (iii); for instance, we have:
f x - W f Y-* x - U Y , whenever x, y are in A n ( u ) .
Now every injective map f of U into W can be canonically extended to
an embedding A n ( f ) between the corresponding admissible hulls; A n ( f ) i s
completely determined by f and its defining recursive conditions. Indeed, let
R e c u r ( h , U , W ) "- Va(aTIU-SET--~ h a - w ( { h y " y~(a)l},l));
f o g stands for the composition Ax(f(gx)) of f and g; id U is the restriction
of the identity map to U. Then we obtain:
21.12. T H E O R E M ( P W - + a i D ) . We can define
a closed term A f . A D ( f )
such that"
(i)
I n j ( f , U, W) ~ E m b e d ( A n ( f ) , A n ( u ) , A n ( w ) ) A
A Vx(xyU ~ A D ( f ) ( i n ( x ) ) - in(fx));
(ii)
I n j ( f , U, W) A Recur(h, U, W) A Va(a~lU -~ h(in(a)) - w i n ( f a)).
Vx(x~IAD(U) ~ ha - w A n ( f ) ( a ) ) .
(iii)
A D is functorial, i.e. A D preserves identity maps and composition:
Va(aTIAD(U ) ~ AD(id U )(a) - u idAD(U) (a));
Inj(g, U, W ) A I n j ( f , W, Z ) - ~ E m b e d ( A D ( f o g),AD(V), A n ( z ) ) A
A Va(a C A n ( u ) ~
A n ( f o g)(a) - z ( A n ( f ) o An(g))(a)).
V.22]
A Model of Admissible Set Theory
137
PROOF. (i) By the fixed point theorem 2.3 and definition by cases on N,
we can find an operation AD such that
if a E U-AT, (ADf)a - (f(a)l,0/;
(1.1)
if a E U-SET, (ADf)a - ({(ADf)y" yrl(a)l), 1).
(1.2)
Let f" U ~ W be injective; we have to check that A D ( f ) " - A x . ( A D f ) x is
an embedding of AD(U) into A n ( w ) . First of all, GID with 21.2 (ii) and
(1.1)-(1.2) immediately yields that U-atoms (U-sets) are sent by AD(f) into
W-atoms (W-sets). a ~. u b implies AD(f)a ~. w AD(f)b (a, b E An(u)).
In fact, let
C(a) "- Vb(b~?AD(U) A (((a)2 r (b)2) V ((a)2 - (b)2 A a r b) V
V ((a)2 - (b)2 - 1 A DE U (a, b ) ) ) ~ AD(f)(a) ~ w AD(f)b).
By GID, it is enough to prove that C(a) follows from the assumption of
Au(a, C) (A U being the operator defining AD(U)).
If b is in AD(U) and (a)2 r (b)2 , AD(f)(a) ~ u AD(f)(b) holds since AD(f)
preserves "category" by (1.1)-(1.2). If (a)2 - (b)2 - 0 and a r b, then
(a)l :/:(b)l and hence f ( a ) l :/: f(b)l by injectivity, which implies by
definition AD(f)(a)=/= AD(f)(b). Since AD(f)(a) and AD(f)(b) are Watoms, we can conclude AD(f)(a) ~ w AD(f)(b).
If a and b are U-sets and DEu(a , b) holds, let us suppose (for instance) that
for some v in (b)l , either u~(a)l or v ~ u u (u arbitrary).
Since (a)l is a class, C(u) and v ~: u u, for every u in (a)l , which implies
AD(f)(u) ~ wAD(f)(v); hence, if we choose z - AD(f)(v), we conclude
Vy(y~AD(f)(a) V z ~ wY), i.e. AD(f)(a) ~ wAD(f)(b)" The remaining
conditions are left as exercise.
(ii)-(iii)" straightforward by GID and the basic properties of - u
21.5-21.6). Vl
w
(see
A model of admimible set theory
We s h o w - within PW c + G I D - that AD(V), where V is the universal class,
is a model of admissible set theory above the ground combinalory structure.
An admissible set above a given model 31~ of OP is basically a two-sorted
structure with M, the universe of d~, as domain of urelements, while the
collections of sets always includes M itself and is closed under pairing,
union, bounded separation and bounded collection (see below for details).
Sets are well-founded, in the sense that they satisfy forms of E-induction
and number-theoretic induction.
Type-free Abstraction, Choice and Sets
138
[Ch.5
In order to axiomatize admissible structures of this sort, it is convenient
to consider a purely relational variant s of s
the language of OP
without T-predicate (see w which coincides with s
except for replacing
the application symbol Ap with a new 3-ary relation symbol App. Thus the
terms of s are simply individual variables and individual constants of s
In addition to Nt, t = s, s has the new atom App(t, s, r), with the same
meaning as the former t s - r. We then extend s to a new language s
with a binary relation symbol for set-theoretic membership, to be denoted
with E, and two unary predicate symbols Set, Ur, which classify sets and
urelements (respectively). Of course, s has atoms of the form t E s, Set(t),
Ur(t), besides those of s s -terms coincide with s
and formulas
are inductively generated from atoms by means of classical connectives and
quantifiers. We systematically adopt the abbreviations:
VxEt.A:=Vx(xEt~A);
3xEt.A:=3x(xEtAA).
The basic equality = is well-defined on urelements, while extensional
equality is adopted for sets; therefore we can introduce a general equality
relation:
22.1
x _= y : - (Ur(x) A Ur(y) A x = y) V
v (s~t(~) ^ s~t(y) ^ w
~ ~ . u ~ y ^ Vu c y. u c
~).
For convenience, we keep using s
(in short applicative terms), as
metamathematical abbreviations. Applicative terms (denoted by t, s, r) can
be explained away according to the following contextual definitions:
22.2.
(xy = z):-- App(x, y,z);
(~ =
t ~ ) : = 3 u 3 v ( u = t ^ v = ~ ^ 9 = uv);
(t--s):=3u(u=tAu=s);
A(t) := 3u(u = t A A(u)).
22.3
(a) The collection of bounded formulas (of s
the smallest collection %
containing s
and closed under the clauses:
(i) if A is in %, so is -~A;
(ii) ifA, B are in %, so is A A B;
(iii) if A is in %, then so are Vx E t.A and 3x C t.A (t arbitrary term
of s
(b) If B ( x ) i s a formula with only the free variable shown and A is an
arbitrary formula (B, A in Ls) , the relativization A B of A to B is obtained
by replacing each unbounded quantifier Qx occurring in A with Qx B, where
3 x S c "-- 3x(B(x) A C), v x B c "- V x ( B ( x ) ~ C).
V.22]
A Model of Admissible bet Theory
139
22.4. The theory KPU(op) (Kripke-Platek set theory above a model of OP)
It consists of elementary classical logic and the universal closures of the
following:
(General equality)
a -- b--~ A[x "- a] ~ A[x "- b] (A .(.s-atom);
(Ontological axioms)
s~t(~) ~ ~v~(~);
( N ~ - ~ V~(~)) ^ (~ - y-~ g~(~) ^ V~(y)) ^ (~ c y-~ S~t(y));
App(x, y, z ) ~ Ur(x) A Ur(y) A Ur(z);
Ur(c), for each individual constant c of s
VxVy(Ur(x) A Ur(y) ~ 3z(Ur(z) A App(x, y, z)));
VxVyVzVw(App(x, y, z) A App(x, y, w) --. z - w).
(Combinatory axioms)
A Ur, for each axiom A of OP-;
( O P - - O P without N-induction, see w observe that, by the convention on
applicative terms, each axiom of OP can be regarded as a formula of s
(V is a set) 3a(Set(a) A Vx(x E a ~-+Ur(x)));
(Pair)
VxVy3a(Set(a) A x E a A y E a);
(Union)
Va(Set(a)-~ 3b(Set(b) A Vx E a. Vy E x. y E b));
(Bounded separation)
Va(Set(a) ~ 3b(Set(b) A Vx(x C b ~-. x E a A A(x)))),
for all bounded A, in which b is not free;
(Bounded collection)
Set(a) A Vx E a. 3yA(x, y)--. 3b(Set(b) A Vx E a. 3y E b. A(x, y)),
for all bounded A, in which b is not free;
( E-transfinite induction)
Vx(Vy E x. A ( y ) ~ A(x))-~ VxA(x) (A arbitrary);
(Bounded number-theoretic induction)
Vx(N(x) A A ( x ) ~ A ( x + l ) ) A A ( 0 ) ~ Vx(Nx ~ A(x)),
for all bounded A.
140
Type-free Abstraction, Choice and Sets
22.5. Translation of s
[Ch.5
into 2.. We inductively define a map v ' s s---, s
(i) if c is an individual constant of s
(hence of s
(ii)
the map r is the identity map on variables;
(iii)
(t = s) r = t r ~ V - A T A s t a Y - A T A (tr)l = (St)l;
(iv)
(Vr(t))r= tr~V-AT;
(v)
(gt)r=
(vi)
( S e t ( s ) ) r = s r y Y - S E T (see 21.1);
(vii)
( t E s ) r = t r E ys~;
(Viii)
(App(t, r, s) r = trr]Y-AT A s t a Y - A T A
(c) r -
(c, 0);
N ( t r ) l A t~r]Y-AT;
A r r ~ Y - A T A ( t r ) l ( r r ) l = (St)l;
(ix)
(x)
(-~A) r = -~(A)r;
(A A B) r = (A) r A (S)r;
(Vx A) r = V x ( x ~ A D ( Y ) ~ At); (3xA) r = 3x(x~lAD(Y) A A t ) .
In 22.5, A D ( V ) ,
previous section.
V-AT,
V-SET,
EY
are the notions defined in the
We can now state the main interpretation result:
22.6. T H E O R E M . /f KPU(op) F- A ( X l , . . . , X n ) and the free distinct variables
of A occur in the list Xl , . . . , Xn, then
P W c + GID ~- X l ~ A D ( Y ) A . . . A x n r l A D ( Y ) ~ A r ( x l , . . . , X n ) .
In words, A D ( V ) is a model of KPV(op), provably in P W c + GID.
The proof of 22.6 requires two preliminary lemmata.
22.6.1. LEMMA (Soundness of the equality axioms of KPU(op))
If A is a formula of KPV(op), M F - + GID proves:
x y A D ( Y ) A y ~ A D ( Y ) A (x -- y)r ___,(Ar[u ._ x] ~ Ar[u "- y])
(x, y free for u in A).
PROOF: by induction on A. We first assume that A is an atom of s
Let A[u "- x] r - ( x E V a) and (x - y)r.
Case 1" x , y are V-atoms and ( X ) l - (Y)I, i.e. x - y .
Then by 21.5(ii)
x - v Y, whence A[u "- y]r _ (y E y a) by 21.8 (iii).
Case 2: x, y are V-sets and x C_ vY , Y C yX (where in general x C vY "-V U E y X . u E V Y ) . By extensionality 21.9, x - y y ,
whence again by
21.8 (iii) we get x E y a. The argument for A - (a E y x) is similar.
A Model of Admissible Set Theory
V.22]
141
Let A [ u ' - x ] r - - ( ( X ) l ( U ) l - ( V ) l ) and let x,u,v be V-atoms. Then the
assumption ( x - y)r implies ( x ) 2 - (Y)2 and ( X ) l - (Y)I, whence y is a Va t o m and by identity logic ((y)l(U)l - (V)l), i.e. A[u "- y]r.
The remaining atomic cases are left as exercise; the induction step follows
without difficulty with induction hypothesis. I"1
If A is a bounded formula of s let A+r be the s
from A r by replacing each bounded quantifier
which results
Vx E V a with Vx(xrl(a)l A (a)2 -- 1 ---+...)
{respectively 3x E v a with 3x(x~7(a)l A (a)2 - 1 A ...)}.
Then we obtain:
22.6.2. LEMMA. For every bounded s
A, M F - + GID proves:
(i)
XlrlAD(Y ) A...A xn rlAD(V)~ TAr+(xl,...,xn) Y FAr+(xl,...,xn);
(ii)
XlrlAD(Y ) A . . . A XnrlAD(Y)-+ Ar+(xl,...,Xn)*-+ Ar(Xl,...,xn).
PROOF
(i)" by induction on the definition of bounded formula of s If A is an
atom, A r - A+,
r . we must check that each A r is always well-defined on
elements of AD(V). We only consider two cases.
If A has the form App(x,y,z), assume that x,y and z are in AD(V) and
-~T(App(x,y,z)r). Then, by T-logic and definition of the r-translation, we
must have:
-,x~?V-AT Y --,yrlY-AT Y --,z~?V-AT Y
-~(X)l(Y)l
-
(Z)l ,
which implies, again by T-logic,
F(x~AD(V) A yrlAD(V) A zr]AD(V)
If
A
-
A (X)l(Y)l
-- ( Z ) l ) "--
F(App(x,y,z) r.
(x E y), x, y are in AD(V) and -~T(x E v Y), we have, for arbitrary
v:
-~v - - V x V - ~ v r ] ( y ) i
V-~(Y)i
-- 1.
I f - ~ ( Y ) 2 - 1, also F ( ( Y ) 2 - 1) and hence F(x E v Y) by T-logic. Therefore
we can assume that y is a V-set and hence that (Y)I is a class. As a
consequence, ~vy(y)l implies F(x E v Y)" Otherwise, if vy(y)l and ( Y ) 2 - 1,
v is in AD(V); hence - w - v x and vyAD(V)imply by 21.6 (iv) F ( u - v x),
i.e. F(x E V Y)"
Let A be of the form VxEy. B(x,a) and assume that y and a are in
AD(V), and
m
-,TA~+(y, a) - ~T(Vx(x~?(y)l A (Y)2 - 1 - , Br+(x, a))).
(,)
Type-free Abstraction, Choice and Sets
142
[Ch.5
By (.), T-logic and the fact that (Y)I is a class, there is an x such that
(Y)2- 1, -~x~(y)i and -~TBr+(x,a). Then xrl(y)i and hence, as y is a V-set,
also xrlAn(V). Since ariAn(V), we have by IH FBr+(x,a), and by T-logic
we can infer FAr+(y,a). If A is a conjunction or a negation, the conclusion
is an immediate consequence of the induction hypotheses and T-logic.
(ii) is trivial in the atomic case. Again, let A be of the form
Vx E y. B(x). First assume that (Vx E y. B(x)) r - Vx E Y Y" Br(x) where y is
in A n ( v ) and let xrl(y)l , ( Y ) 2 - 1. Since x is in A n ( v ) , x - v x holds
(21.6 (i)) and hence x E v Y (definition 21.7). By assumption, we have
Br(x). Conversely , assume (Vx E y. B(x)) r+ and let v such that v - V x '
v~/(y)l and ( y ) ~ - 1 ; then we get Br+(v)and by IH also Br(v), whence
Br(x) by the equality lemma 22.6.1. V!
P R O O F of Theorem 22.6. The verification of the ontological and
combinatory axioms is straightforward by definition of the r-translation
and by the basic properties of AD(V), V - S E T and V-AT (see 21.2(iii)).
The equality axiom is already justified by 22.6.1.
Pairing axiom. Let x, y in AD(V) and choose b - ( { u ' u - x V u - y } , l ) .
Clearly (b)l is a class of elements of AD(V) and hence b is in AD(V)
(21.2 (ii)). On the other hand, x, y are in (b)l and x - y x, y - y Y, whence
X C y b and y E y b.
Union axiom. If a is a V-set, choose b - ({u" 3y~i(a)l.x~(y)l A (Y)2 -- 1), 1).
Clearly b is a V-set; if x E V Y and y E y a, it is immediate to see that there
is a v - y x such that vy(b)l (apply the definition of E V, 21.8 (iii)).
Bounded separation. If c is a V-set and A(x) is a bounded formula of s we
choose b - ({x-xr](C)l A A~(x)}, 1).
(b)l is a class by lemma 22.6,2 (i) and it only contains elements of AD(V)"
hence b is a V-set. If x is in AD(V), x E y b ~ x E y c A A r ( x ) holds by
n
22.6.2 (ii) and 22.6.1.
Bounded collection. Let A be bounded and assume
Vx E y a . 3y(y~An(V) A At(x, y)), where a~iAn(Y).
(1)
We can also suppose that a is a V-set; hence
k/xr] (a)l. 2y(yrlAD(Y) A At(x, y));
by 22.6.2 (i)-(ii),
Vx~ (a)l. 3y(y~iAn(V) A TA~(x, y)).
(2)
As a consequence of CL-reflection 16.3.1, there exists a class b C AD(V),
such that, again by 22.6,
A Model of Admissible Set Theory
V.22]
143
Vxr/(a)l. 3yrlb. At(x, y).
Then, by construction, c -
(b, 1/ is a V-set such that:
Vx E V a. =ty E yc. Ar(x,y).
Bounded number-theorelic induction. Let A be a bounded condition of s
such that (A(O) A V x ( N x A A ( x ) - - , A ( x + I ) ) ) r" then by definition of r_
translation with V-AT C AD(V), we obtain
Ar((O, 0)) A Yx(xrlY-AT A N(X)l A Ar((x, 0 ) ) ~ A~((x+l, 0)))
(3)
Consider c - ({x" xrlV-AT A i ( x ) l A A~F(x)} , 1); then (c)l is a class by
22.6.2(i) and every element of (c)1 is a V-atom. Hence we can apply
number-theoretic induction in MFc to the class {x" (x, 0)rl(c)l} and we
easily obtain, with ( 3 ) a n d 22.6.2 (ii),
Vx(Nx --, (x, 0)~(c)1),
which immediately implies (Vx(Nx--, A(x)) r.
of U
a
oS
holds by 21.2 (i) (choose b - (V-AT, 1)).
E-Transfinite
induction
schema.
(3b(S
Assume
t(b)^
e
(Vx(Yy E x. B ( y ) ~ B ( x ) ) ) r,
where B is arbitrary; then we have"
for every x in AD(V), Yy(yTIAD(Y)~ (y E v x ~ B r ( y ) ) ~ Br(x)).
(4)
We apply GID to AD(V); if we put C ( x ) " - x~lAD(Y)~ Br(x), it is enough
to verify that A v ( x , C) implies Br(x) under the assumption xTIAD(V ) (here
A y is the operator of 20.1). If ( X ) l - 0, x is a V-atom and hence x is in
AD(V); by definition 21.7, we also have -~y E y x and (4) trivially yields
Br(x) and C(x). If (x)2 - 1, (X)l is a class and Yy(yrl(X)l --~C(y)) holds, we
also have Yy E v x. Br(y) (apply 21.6(i), 21.7, 22.6.1); hence by (4) Br(x)
holds. [1
The previous theorem easily implies a kind of conservation result of
KPU(op) over OP. To this aim, we identify in the set theoretic language of
KPU(op) the formulas, which correspond to usual s
22.7. DEFINITION. FORop(s
is the smallest collection of s
which is generated from atoms of the form Ur(t), Nt, App(t, r,s), t - s by
application of-~, A and quantifiers restricted to urelements Vx(Ur(x)~...).
22.7.1. FACT. If A E FORop(s
to a formula A r~ E s
A r is provably equivalent (say, in ME-)
Type-free Abstraction, Choice and bets
144
[Ch.5
PROOF" by definition 22.5 , observing that V-AT is definable is fop" F1
Hence, if we apply 22.6 and the general conservation theorem 15.5, we
obtain:
22.8. C O R O L L A R Y . / f A E FORop(Zs) and KPV(op) F- A, then OP F- A ~-0.
22.9. REMARK
(a) 22.8 can be refined and one can show that KPU(op) and OP have
the same proof theoretic strength. As to related results, Jiiger (1984) shows
that the theory KPU r of admissible sets above natural numbers, with
number-theoretic induction and E-transfinite induction restricted to sets, is
conservative over Peano arithmetic.
(b) The AD-construction can be used to calibrate the strength of
several intermediate options, insofar as we strengthen N-induction and we
restrict GID. For instance, if we replace number-theoretic induction for
classes with the corresponding axiom for properties, AD(V) validates the
existence of an infinite ordinal and a schema of N-dependent choices for
El-formulas (El-formulas have the form 3xA, where A is bounded). If we
only accept the schema GID for conditions of the form x~la, AD(V) verifies
E-induction only for El-conditions (for more information, see Jiiger 1980,
1982, 1986; Simpson 1982; Cantini 1982, 1983, 1985; Rathjen 1992). If we
extend OP with the enumeration axiom, AD(V) verifies a choice schema for
relations defined by bounded conditions.
w
The boundedness theorem
In classical mathematics, there are two equivalent ways of handling
inductively defined sets. According to the Frege-Dedekind impredicative
style, if we are given a monotone operator F" ~ ( U ) ~ ( U )
(where
a 2 ( U ) - power set of a given non-empty set U), I(F), the set inductively
generated by F, is identified with M{X C U" F(X)C_ X}. On the other
hand, I ( F ) can be obtained from below by iterating F on ordinal numbers,
i.e. I ( r ) - u {I(F, ~) . ~ E ON}, where I ( r , ~) - r ( ~ u<~,I(r' /3)).
This situation has a satisfactory counterpart in P W - + G I D ,
as a
byproduct of the continuity theorem, the generalized induction principle
GID and the fact that a non-trivial theory of ordinals is available in the
system, by the main result of the preceding section.
First of all, we single out the collection ON of ordinals within the
model AD(O) of pure admissible set theory (see 21.5-22.6). Without
145
The Boundedness Theorem
v.23]
urelements, a simpler definition of the notion of set suffices (see 10.2):
23.1. LEMMA. We can find a term I S such that M F - +
GID proves:
V x ( x ~ l l S ~ C l ( x ) A Vy(y~lx -~ y~lIS));
(i)
(ii)
if B ( x ) is an arbitrary formula,
V x ( C l ( x ) A Vy(y~x ~ B(y))--~ B ( x ) ) . ~ V x ( x ~ I S ~ B ( x ) ) .
Now let K P - be the set theory, which includes extensionality, pairing,
union, full E-induction schema, A0-separation and Ao-collection schemes
(in the pure set-theoretic language with only membership as primitive
predicate). By a straightforward adaptation of 21.4-21.7 and 22.6, we can
find a term - , which inductively defines extensional equality on I S : - is
the least relation such that, provably in P W - + GID:
23.2.
x - y ~ ( x ~ I S A y ~ I S A Vu~x. 3v~y. u -- v A Vu~y. 3v~x. u -- v).
If x E y := 3u(uyy A u -- x), we can rephrase 22.6 as follows:
23.3. P R O P O S I T I O N . The structure
provably in P W - + GID.
(IS, E,-)
is a model of K P - ,
Now let:
T r a n s ( a ) := Vu E a. Vv E u. v E a;
O r d ( x ) := x y I S A Vu~x. Vv~u. v E x A Vu~x.Vvyu. Vwyv. w E u;
ON "- {~" O~d(~)).
23.4. LEMMA. P W - + GID proves:
(i) x T l O i ~-~ ( x y I S A T r a n s ( x ) A Vy E x. T r a n s ( y ) ) ;
(ii)
(iii)
x ~ O N A y -- x ~ y y O N ;
x~ON A y E x~yrlON;
T I ( B , O N ) , for arbitrary B, where T I ( B , O N ) " -
:= W ( ~ O N A Vy ~ ~. B(y)-~ B(~))-~ W ( ~ O N - ~ B(~));
(iv)
(v)
~ O N A y~ON-~ (~
- ~) A (~ -- y V 9 C y V y e
~);
if Ua "-- {x" 3yrla. x~y}, then:
Cl(a) A Vx(x~la ~ x ~ O N )
( U a)~7ON A V x ( x e a---, x E ( U a) V x -- Ua);
(vi)
if a + l "- a U {a},
a~ON---~(a-~-l)~Og A V x ( x ~ O g A a E x---, x - ( a + l ) V ( a + l ) e x).
Type-free Abstraction, Choice and Sets
146
[Ch.5
PROOF. (i): Ord(x)---,TOrd(x), because the negative occurrences of ~ in
Ord(x) are applied to classes (being x in IS). By definition of E , - and
21.8, the right member of (i)is equivalent to Ord(x).
(ii) See 21.8.
(iii) Assume:
(1)
Vx(x~ON A Vy C x . B ( y ) - , B(x)).
It is enough to verify
w(.
Is
c(.)),
(2)
where C(x) "- xrlON ---,Vu(u - x ~ B(u)). Then, using ON C_I S and
21.6(i), (2)implies the required conclusion Vx(x~ON---,B(x)). In order to
check (2), we apply 23.1. So, we assume:
(3)
Cl(x), Vyrlx. C(y), xrlON and x - u,
and we prove B(u). Since u~ON by (3) and 23.4(ii), and ( 1 ) i m p l i e s
Vy C u . B ( y ) ~ B ( u ) , it is sufficient to prove Vy E u.B(y), or, which is
equivalent, since x - u, Vy C x.B(y). Let y C x, i.e. y ' - y , for some y'rlx.
By (3), it follows C(y'), i.e.
y'~ON ---,Vu(u - y'---, B(u)).
(4)
Now, as xrlON, then yrlON and hence y'rlON by 23.4 (ii). Thus by (4) we
obtain Vu(u - y ' ~ B(u)); if we choose u = y, since y - y', we can conclude
B(y), and we are done.
(iv)-(vi): exercise (one essentially applies 23.3). 0
23.5. CONVENTION. We temporarily adopt
syntactic variables for ordinals; we also define:
small
Greek
letters
as
:=
VaA(a) := Vx(xrlON---, A(x));
Va < ft. A := Vc~(a < ~ - ~ A);
23.6. LEMMA.
P W - + GID:
(i)
3 a A ( a ) : = 3x(xrlON A A(x));
3a </3. A := 3 a ( a < fl A A).
We can find an operation Rec such that, provably in
if (Reef)( < ~) := {x: 313 < a. xrl(Recf)Z},
(Recf)a = ]((Recf)( < a));
(ii)
Mort(f)---, (a < fl ~ (Recf)a C_ (Recf)fl);
(where Mon( f ) := f is monotone);
(iii)
M o n ( f ) ~ . V a ( H a = f { x : 3/3 < a. xrlH/3})~ V a ( H a = ~(Recf)a).
The Boundedness Theorem
V.23]
147
PROOF. (i): choose R E C := )ff. FP(AhAy. f { x : 3z E y.x,7(hz))), by the
fixed point theorem for operations.
(ii) If c~ < ~, (Reef)( < ~) C (Reef)( < t3) by 23.4 (i) and <-transitivity;
hence Mort(f) and (i) imply (Recf)(~ C_(Recf)~.
(iii): immediate by transfinite induction on c~. 0
23.6.1. REMARK. By (ii) above, Reef is invariani under - " if c ~ - ~,
then (Rec f )a - e (Rec f )~.
23.7. DEFINITION.
R e ( f ) "- {x" 3a.x~(Recf)a).
23.8. T H E O R E M (Boundedness+ Covering). We can define an operation
AcAf . Bound(c, f) such that P W - + GID proves:
Cl(c) A c C_R C ( f ) ~ Bound(c, f)~log A c C_(Recf)(Bound(c, f)).
(,)
PROOF. Let c be a class which satisfies the antecedent of (,). By
CL-reflection 16.3, there is a term S(f,c) representing a class C_ ON such
that:
Yx,Tc. 3a,lS ( f , c). x,7(Rec f )a.
Hence fl(c, f ) " -
U S(f, c)is an ordinal by 23.4 (v) and
Yx~c. 3~ <_~(c, f ). x,7(Rec f )a.
Thus we choose Bound(c, f ) " - ~(c, f ) + 1(= ordinal successor). D
23.9. COROLLARY (Closure). P W - +
GID proves:
Mort(f)---+ f ( R C ( f ) ) C_RC(f).
PROOF. If x,ff(RC(f)), there is a class c C R C ( f ) with xyfc by the
generalized Myhill-Shepherdson theorem 17.5; hence c C (Recf)a, for some
c~, and trivially c C (Reef)( < c~+l); by f-monotonicity and 23.6 (i)"
x,ffc C_ (Recf)(a+l) C_RC(f). [3
23.10. COROLLARY. P W - + G I D ~- Mort(f)---+ I ( f ) - e RC(f).
PROOF: I ( f ) C_R C ( f ) is a consequence of 23.9 and GID.
As to R C ( f ) C_I(f), let C(c~):: (Recf)a C_I(f). If we assume by IH
Vj9 < c~.C(jg), then f-closure of I(f), f-monotonicity, and 23.6(i) imply
C(c~), whence by 23.4 (iii):
V~((Recf)~ C I(f)).
This Page Intentionally Left Blank
PART C
SELECTED TOPICS
"Nut wenn man nicht auf den Nutzen nach aussen sieht, sondern in der
Mathematik selbst auf das Verh~iltnis der unbenutzten Teile, bemerkt man
das andere und eigentliche Gesicht dieser Wissenschafl. Es ist nicht
zweckbedacht, sondern un~konomisch und leidenschaftlich. [... ]
Die Mathematik ist Tapferkeitsluxus der reinen Ratio, einer der wenigen,
die es heute gibt." (R. Musil 1913)
This Page Intentionally Left Blank
CHAPTER 6
LEVELS OF IMPLICATION
AND INTENSIONAL LOGICAL EQUIVALENCE
w
~25.
w
w
~28.
w
Myhill's levels of implication
Formal deducibility based on levels of implication and its
proof-theoretic strength
Introducing an intensional equivalence relation
The infinitary reduction relation =}
The Church-Rosser theorem for =~
A model of type-free logic based on intensional equivalence
We know from part B that the theory PW c + GID of reflective truth,
extended by generalized induction and approximation principles, has several
interesting features from the recursion-theoretic point of view; as to the
relation with classical notions, we saw how to construct an interpretation of
a predicative set theory. We now pursue the opposite aim: we relate
P W c + G I D + E A ( = t h e enumeration axiom) to non-classical systems
which allow type-free abstraction. The results we obtain suggest that the
present framework can profitably be adopted as a metatheory for a variety
of formal systems, whose proof-theoretic strength does not go beyond that
of Peano arithmetic. We consider two systems stemming from intuitions,
which are far from each other and from the theory of reflective truth
introduced in part A.
The first proposal - due to Myhill (1972, 1984) - originally moves from a
reappraisal of Curry's paradox and explicitly proposes a weakening of the
implication introduction rule. According to Myhill's approach, such a rule
should be regarded as an "ideal limit" and should be replaced by an infinite
sequence of distinct approximations. Carrying out the project in details
yields a sort of ramified logic, with respect to the level of implication
assumed. The system turns out to be consistent by essential use of the dual
representability theorem of w (this is the main idea of the Myhill-Flagg
consistency proof).
The second system gives credit to an idea of Behmann (Behmann 1931),
according to which we must ascribe the responsibility for the Russell
paradox to the very definition mechanism. From this perspective, Feferman
152
Levels of Impfication and Intensional Logical Equivalence
[Ch.6
proposed in the seventies a logic based on a concept of definitional
equivalence, which should play for the theory of properties a role, similar to
that of the conversion relation in combinatory logic. Feferman and Aczel
(1980) offered a consistency proof of the related logic of abstraction by
means of a non-trivial adaptation of the Church-Rosser theorem. Rather
surprisingly, we show that a model of an even stronger theory exists jusl
within P W c + GID, plus the enumeration axiom: the proof essentially hinges
upon the boundedness theorem of w and the strong selection theorem of
w
w
Myhili's levels of implication
We are going to prove that the minimal frame system MFc, expanded with
approximation axioms, yields a natural environment for interpreting a
theory of properties, that was proposed by Myhill (1972, 1984) and later
fully developed in Myhill-Flagg (1987). Myhill's system stems from a
criticism of Fitch's "Extended Basic Logic" ( Fitch 1948).
As we know from the general introduction, Fitch's ideas are reflected in our
frame theory MFc; thus Myhill's objections to Fitch are easily rephrased as
pointing out a crucial weakness of the truth predicate T. In essence, the
standard hypothetical reasoning does not work in its full generality with T;
the inference from A---+TB to T(A---~B), which corresponds to the
deduction theorem or to the usual implication introduction rule, is sound
only if we possess the additional information that A is a proposition in the
sense of T (recall 8.6 (iii)). As a consequence, we cannot assert the internal
truth of statements of the form V x ( C l ( x ) ~ TA), since CL is provably not a
class by 9.3. In particular, if we identify Dedekind reals with suitable classes
of rationals, quantification over the reals R will not be preserved in general
by T. In sum, one would like to have at hand some kind of internal
implication.
Such a difficulty led Myhill to explore the introduction of a chain of
implications a D , 2 D , . . . , which reflect various degrees of deducibility in
the context of type-free systems h la Fitch. According to Myhill's proposal,
A 1 D B is true iff B is deducible from A without using any implication
whatsoever; A 2 D B is true iff B is deducible from A, using (at most) the
inference rules for 1 D , and so on.
Flagg and Myhill (1987) propose a system ~oo of illative combinatory
logic, which extends (a classical version of) Aczel's theory of F r e g e ' s
structures by means of 1 D , 2 D '3 D , etc. The main tool for establishing
the consistency of 2oo is given by the theorem 16.11 about the exact
representability of pairs of disjoint properties; so we are naturally driven to
investigate the relation of ~oo with our theories. Indeed, we adopt
VI. 24]
Myhill's Levels of Implication
153
P W c + GID, as a metatheory for developing the semantics of Zoo" In the
following, we introduce a generalized version of Myhill's hierarchy of
implications and we show that the basic construction takes place in
P W c + GID; this additional information will be applied in the next section
to the characterization of proof-theoretic strength of the corresponding
formal systems.
CONVENTION: below we adopt a, b, c as syntactical variables besides the
usual x, y, z.
First of all, recall that we have: 1) a closed A-term of s
for each predicate
symbol and logical operator of s 2) a canonical term [A] with the same
free variables of A, for each s
A. [A] expresses the function defined
by A itself. For convenience, we agree to identify A and [A], when no
ambiguity can arise; thus we still write Nt, t - s, VxA, etc. instead of the
corresponding [-]-terms; we keep using the standard logical symbols -1, A,
V, etc., for the (terms defining in s
the) corresponding operations. Below,
we consider properties R C_ V 2, which represent abstract infinitary
deducibility relations; hence, we shall use the more suggestive notation
F~- R a,
instead of the proper (F, a)71R. We let F, A range over arbitrary properties,
but the informal idea is that F, A are collections of formula-objects, i.e.
denotations of terms of the form [A], where A is an arbitrary formula. U,
{...} denote the corresponding operations on properties (see w If F = 0, we
simply write " k R a "
instead of " F I - R a " ; if F - { a } ,
" a I - R b" is a
shortening for "{a} F R b". We use the abbreviations:
Tr
--
Vb(bnr ~
Tb)
and Fr--
3c(c~r A Fc);
F ~ R A .-- Va(a~lA ---, F t- R a).
24.1. DEFINITION
(i) A d r ( R ) : = n is an abstract deducibility relation iff n satisfies the
following conditions:
Soundness:
VrVa(r k R a ---, ( T F ~ Ta));
T-completeness:
VFVa(FF V Ta ~ F ? R a);
Closure under generalized cut:
VFVAVa((CI(A) A F IMonotonicity:
(ii)
R A A
r
u A I- R a ) ~ r I- R a).
VFVAVa((F k R a A F C A)--~A k R a).
R-implication: if E R is the operation of exact representation, given
Levels of Implication and Intensional Logical Equivalence
154
[Ch.6
by theorem 16.11,
R D "- E R ( R , Ded),
where Ded "- {({a}, b)" T a A Fb}. Henceforth, we let
a R D b "- ( E R ( R , Ded))({a},b).
(iii)
Ded o "- {(F, a)" F F V Ta}.
Observe that Ded o is the C_-least abstract deducibility relation.
24.2. LEMMA (Existence of R-implication). P W - p r o v e s :
A d r ( R ) ~ VaVb((T(a R D b) ~ a F- R b) A (F(a R D b) ~ T a A Fb)).
PROOF" by soundness of R and consistency, Ded and R are disjoint and
hence 16.11 applies. F1
24.3. LEMMA.
CL-compact:
PW-proves
Adr(R) ~ VrVAva(r ~
that every abstract deducibility relation
R
is
a ---+3A(A C F A C l ( A ) A A F- R a)).
PROOF- Prem(a, F- R ) . _ {F" F ~ R a} is extensional by monotonicity;
hence the result follows from the Rice-Shapiro theorem 17.3. F1
Every abstract deducibility relation F R can be extended to a larger
deducibility relation F R which is closed under logical inferences and
natural rules for R D.
24.4. The basic deducibility relation F R
which is closed under the rules below.
~
F R is the least binary relation,
Initial Rules
Hyp"
aqF
FFR ;
, a
Induction Rule C L - I N D R
Lift(R)"
F t- R a
R
a
FF,
for total predicates
F F I~ b-O; F F I~ vx(bx V-,bx);
F F R Nc;
F F-Rbc
for everyn, F U { b n } F R b ( n + l )
Myhill's Levels of Implication
VI. 24]
155
Logical Rules (we recall that the logical primitives are --1, A , V)
[I A ]:
F F- R bl F ~ R b2
*
*
;
F F- R, bl A b2
[I-~ A i]"
FF-R. _~bi ( i - 1 , 2 )
r F- R, _~(bl A b2)
F F- R, bl A b2
[EAi]:
F F- R. bi (i _ l,2) ;
FF-R, ~(blAb2)
[E~ A ]:
Ft- R b
[I-~-~]: r F- R ---~b;
FU{-~b 1}F-R, c
FF- R
,
C
FU{-~b2}F-R, c
FF- R -~-~b
[E-~--]: r F- R b ;
F I-- R Vb
[IV]: for everyc, F F - R b c
*
FF-RVb
;
[EV].
F F- R bc, for arbitrary c
[I-~V]: F F- R --1 bc for some c
FF- R -~Vb
P F- R -~Vb, F U {--,bc} t-- R d, for arbitrary c
[E-~V]:
Ft-Rd
F F- R b
[•
F F- R --,b
r F n. c
F ~ R (bc) V d, for every c
; [vv]:
PU{a}t- R b
Pt--R a
[I R D ]: Fk-R. a R D b ;
[I-, n ~ ]:
[E~ n ~ ]:
Cut:
F t - . R(Vblvd
[E RD]"
*
Ft - R
FkRb.. aRDb;
p~ R
, a
F ~ , R-~ b
F b R , ~(a R D b )
F ~ R , -~(a R D b )
F F R, a
;
FFRA
el(A)
FF-R, ~ ( a R ~ b )
F F - ,R _.b
;
FuAFRb
Ft-Rb
Equality rules
Eq.l:
F~Ra
*
F~ - R
FFRb
, a-b;
Eq.2:
FF_R, a - b
FF - R,
a'-b'
FF- R
,
aa'-
bb'
Levels of Implication and Intensional Logical Equivalence
156
Eq.3:
FF R
. a-b
F~-R
, b-a
;
Eq.4:
[Ch.6
FF R
.
a-b
FF R
, b--c
FF- R
,
a--c
24.4.1. REMARK
(i) F R. differs from Myhill's original notion in the following points:
1) we do allow possibly infinite collections of premises (F is a finite set in
Myhill-Flagg 1987); 2) our cut rule is infinitary; 3) the N-induction rule is
restricted to total predicates.
(ii) If Adr(R) holds, the relation F R
(apply T-completeness of R and Lift(R)):
True:
is closed under the rule below
Ta (or FF)
FF R
,
a
By inspection of the inductive definition 24.4, we can obviously find a
T-positive formula A(R,x, v), operative in v (in the sense of 10.3), such that
its fixed point Ixv.A(R,x, v ) " - I ( A R ) (cf. 10.4)is closed under the clauses
above, provably in MF-; hence, if we let F .~ "- I(AR) , we have:
24.4.2. LEMMA
(i) M F - F VFVb(A(R, (r,b), F . R ) ~ V F R. b);
(ii) ME- + GID F Vx(A(R, x, B) ~ B(F, b))
vrvb(r e R, b---+B ( r , b ) ) ( B
arbitrary).
24.5. THEOREM. (i) PW c + GID F Adr( F R ) ~ Adr( ~- R )
(ii)
Conservation:
pw~ + GID ~- Ae~( ~- R)__, Va(Ta ~ ( ~ R, a ) ~ ( ~- R a)).
(iii)
Consistency: PW c + GID F Adr( F R)___,--,=lb( F- _n b A I-- R --,b)
PROOF. (i) Monotonicity of F- R follows from the corresponding property
of F- R by generalized induction (24.4.2 (ii)). As to T-completeness, it is an
immediate consequence of the rule True of 24.4.1(ii); closure under cut is
immediate by definition of F- R
and 24.4.2 (i) ~
,
It remains to check soundness (again by generalized induction on F R) The
N-induction axiom for classes (see 10.7) and the T-axioms of w take care
of I N D R and the logical inferences, while L i f t ( R ) preserves soundness, as
Adr(R) is assumed.
VI. 24]
Myhill's Levels of Implication
157
Let us consider the rules for R-implication.
[I R D ]: assume F U {b} t - R e and TF. By 24.3, A U {b} F-R c, for some
class A C_ F; but TF implies TA, whence by True and monotonicity,
{b} F- R A. Cut implies b ~ R c and hence T(b R _-3c) (by 24.2).
[I~ R ~ ]" assume F F- R
, a and F F- R
, ~b; then TF ---, Ta A T-~b (by IH). If we
assume Tr, 24.2 implies T(-~(a R D b)).
[E R : 3 ] : assume F ~ R, a, F F-,R a R ~ b and TF; by IH, we obtain Ta,
T(a R ~ b), i.e. by 24.2 a F- R b, which implies by ~- R-soundness Ta ~ Tb,
whence Tb.
[E-~ R D ]-rules are also straightforward by IH and 24.2.
(ii) follows from (i) and application of L i f t and True; (iii)is a consequence
of T-consistency and (ii). [-1
The main consequence of 24.5 is that if we have a deducibility relation
F-R and hence a corresponding R-implication R D, we can conservatively
extend F-R with the rules for R 3 ; but the enlarged deducibility relation
induces a new implication and this suggests a natural iteration.
24.6. PROPOSITION. If A is a formula of Lop and OP t- A, then"
MF c + GID F- Adr(R)---, ( F- R, A).
The proof requires a straightforward metamathematical induction on the
derivation of A in OP; the point is that, if A is an axiom of OP, then
MF c ~- T A and we can apply the rule True of 24.4.1.
24.7. DEFINITION. We introduce a deducibility relation ~- n by induction
on n e w .
(i) Deducibility of level 0: F-0 is the C_-least relation, which is closed
under True and the rules of 24.4, except L i f t ( R ) and the R ~ -inferences.
(ii) F- n + l : = IxvA(~- n,x,v), where A ( ~ n,x,v) results from
formula A(R,x, v) of 24.4.2 by replacing everywhere R with F- n.
(iii)
n D "--R ~ ' w h e r e R ' -
the
~n.
Obviously, if R "- t- n, ~ n + l : _ ~ R.. hence theorem 24.5 (i) ensures
Adr( F- n+l) under the assumption that Adr( F- n). Note also that F- 0 can
formally be defined by the term Ixv.A-(Dedo, x , v), where A-(Dedo, x , v) is
obtained from A ( R , x , v ) by replacing R with Ded o (see 24.1(iii)) and by
simply omitting the clauses for R-implication. Since A-(Dedo, x,v ) is Tpositive and operative in v, we can rely upon the proof of 24.5 (the cases of
R-implication being omitted) and F-0 is an abstract deducibility relation.
Levels of Implication and Intensional Logical Equivalence
158
[Ch.6
Hence, if we put A0(z , v) : - A - ( D e d o , x, v) and An+l(z , v) : - A( F n,x, v),
we can verify by metamathematical induction"
24.8. THEOREM. For each n E w,
(i) PW c + G I D F- Adr ( F n);
(ii) PW~ + GID F Va(Ta ~ ( F n a));
(iii) PW~ + GID F -~3a( F ~ a h F ~ (-~a));
(iv)
(v)
PW c + GID F- vrvb(A.((r, b), e
r
- b);
if B is arbitrary,
PW c + GID F V F V b ( A n ( ( F , b ) , B ) ~ B(F, b)) ~ VFVb(F F- n b ~ B(F, b)).
24.8.1. REMARK. The internal version of 24.8, i.e. V x ( U x ~ A d r ( F - = ) )
can be formally verified in PW + GID, where PW contains the full schema
of number-theoretic induction.
w
Formal deducibility based on levels of implication and its proof-
theoretic strength
We describe a formal sequent calculus Y~oo, which formalizes the deducibility
relations of w and we show that the elementary judgments of ~oo coincide
with the provable sentences of OP, the basic theory of operations of Ch.I.
First of all, ~oo is formalized in the term fragment of the language s 0p for
OP, expanded by a formal deducibility symbol F n, for each n E w. Besides
the usual closed terms representing -1, V, A, - - , N, we fix a closed term
n D , for each natural number n E w. For convenience, we identify n D with
the term introduced in 24.7(iii)(NB" the term belongs to the language
s
The calculus Eoo is a natural deduction system, whose judgements have the
form F F n t, where F is a finite (possibly empty) sequence of s
and
t is a term of s
The inference rules of Ec~ are finite formal counterparts
of the clauses defining k R
"
25.1. Inductive definition of the Eoo-derivability predicate
F F - n t i s Eoo-derivable (in short Eoo F-(r F-n t)) iff r F n t belongs to the
C_-least collection of judgments which contains the axioms (or initial rules)
to be given below, and is closed under the following inferences:
(i) the propositional rules [I A ], [I-~ A ], [E A i] (i -- 1, 2), [E-~ A ],
[I----i, [E--~], [ l ];
(ii) the quantifier rules [IV], [EV], [I--V], [E--V];
VI.25]
(iii)
(iv)
(v)
(vi)
(vii)
159
Formal Deducibility based on Levels of Implication
[Cut] and the structural inferences [K], [W], [C];
rule C L - I N D R of N-induction for classes;
rules [Lift], [Red];
the
the
the
the
implication rules [I n D ], [E n D ], [I-- n D ], [E~ n D ];
equality rules EQ.1-EQ.4.
C L - I N D R , the equality rules and the propositional rules (for A, J_, -~) are
obtained from the corresponding inferences of 24.4, by replacing everywhere
F R with ~ n and by reinterpreting F as ranging over finite sequences of
Lop-terms; the remaining inferences are listed below.
Initial Rules
Hyp/TND:
tFOt;
(K-S):
(Pi):
F~
o K t s - t;
ifA-(t-s),Nt;
~ o Stsr-
F o ((tl, t 2 ) ) i _ ti (for i -
tr(sr);
Nt, Ns, t-s
(D.2):
N t , N s , -~t - s F o D t s r q - q;
(N.1):
N t F o -~(t+l) - O;
(/.2):
F o N-O;
(N.3):
(Id):
S);
1,2);
(n.1):
F o Dtsrq-
F o ~(K-
r;
N t F o N(t+l);
Nt F o PRED(t+I)-
t;
F~
Quantifier rules
rFnsx.
[EV]:
[IV]r ~ ~ w'
(proviso for [IV]: x not free in F, s).
[I-~V]:
r
F n
-~
r F ~ Vs.
FFnst
st
[E-W]:
F F n ~Vs ;
'
F F n-~Vs
(proviso for [E~V]: x not free in F, s, r).
[v v ]:
r F " v ( ~ . s ~ v t)
r F"(w) vt
;
(proviso for [V V ]" x not free in t).
Cut and structural rules
[Cut]:
r F n t r, t F n s
FFns
F~-nt
;
[K]. r , s ~-"t;
F, -~sx F n
Fk-nr
r
Levels of Implication and Intensional Logical Equivalence
160
F, t,~t F- n s
F, t F n s
;
[w]:
[Ch.6
F, t, s, A I- n r
[C]"
F, s, t, A F - n r
Rules for implication of level n
F, t F - n s
;
[I~ n D ]:
[ I n + 1 D]"
rF-n+l
[E n D ]"
F 1- n + l t
r I- n + l t n D s
F F- n + l s
;
r
[E-~n D ]"
Level Rules
[Red]:
t nDs
n + l ~ ( t n D s)
r }-- n+l t
F- n+l t.
F-n t
'
r
r ~- n + l t r ~ n + l -~s
r F-n+l _ ~ ( t n D s )
;
n + l _.~(tn D s)
F I'- n + l "-18
[Lift]"
FFnt
F F" n + l t
Notice that [Red] states a reducibility rule for unconditional statements;
[Red] is not present in Myhill's calculi, but we include it here, since it is
sound under the interpretation of the previous section. Clearly, we can
interpret F ~ n t, where F is the finite sequence t l , . . . , ti, as the formula of
s stating that the abstract deducibility relation F-n of 24.8 holds of the
pair (F,t), (where F is now {x: x = t] V . . . V x = ti} ). Under the given
interpretation, we show:
25.2. P R O P O S I T I O N (Soundness of Eoo ). For each n E w,
(r
t)
implies P W c + GID F (r e " t).
P R O O F : we argue by induction on the definition of E~-derivability. We
systematically apply the closure properties of the corresponding relations
F- n of 24.7.
n = 0. Initial rules: we apply OP-axioms, the rules True, [E-~ A ], [ _1_ ] and
F-~
(hence 24.8 (i)). In particular, T N D follows by True and
axiom T.1 of MF c. The logical inferences and C L - I N D R are disposed of by
means of the corresponding closure conditions for F-0, which hold by
24.8 (iv). The structural rules are sound because F n is extensional in F; the
cut rule of E ~ is a special case of the general cut rule of 24.4.
n = m+l.
As to [Red], assume
Ecx~ F ( ~ m + l t) and P W c + GID F ( F m + l t).
(,)
The second part of (,) implies Tt, provably in P W c + G I D , by theorem
VI.25]
Formal Deducibility based on Levels of Implication
161
24.8(ii). Hence by True, P W c + G I D F ( F
rnt), which is exactly the
interpretation of the conclusion of [Red]. As to the soundness of the
m+l D-inferences, it is again granted by 24.5 and 24.8. V!
25.3. LEMMA. If OP F A, then E ~ F ( F o A).
PROOF" by straightforward induction on the length of the derivation of A;
it is essential to observe that F 0 A V--A, if A is in s
(see T N D ) . [-1
If r is a finite sequence_B1,_..,B i, let T F ' - B 1 A . . . A B i (if r is
empty, we can choose T F " - ( 0 - 0)). Then we conclude by 25.2, 24.8(i)
and the general conservation theorem 15.5:
25.4. C O R O L L A R Y
(i)
If Eoo F (F F n A), for some n e w, then P W c + GID F T r ~ T A . .
(ii) In the same hypothesis of (i), if r U {A} contains only terms of the
form [B], where B is a formula of s
we have:
OP F A F - - . A .
It follows that, if Eoo proves that b is a class, b is a class provably in
P W c + GID; also, 25.4-25.3 entail:
25.5. COROLLARY. If A is a formula of s
OP k- A i f f E ~ F ( F hA).
By the conservation requirement of the previous section (see 24.8(ii))
and by 25.4, we cannot expect new internal truths out of F n; however, we
may expect that the new implications yield useful approximations of
classical set-theoretic objects. For instance, we already know that there is no
way to distinguish extensionally [ C L ~ C L ]
from [V---,CL] (since
M F - F --,FCl(x); see 9.3). However, we can define, if n > 0,
25.6
[a---*b]n "- {f" Vx(xrla n D (fx)~?b)};
Pown(a ) "- {x "Vy(yrlx n D y~Ta)}.
Then, even if a is not a class, we do have, provably in P W c + GID, by 24.2:
25.6.1.
V f(f~7[a--~b]n~-* Vx(x~?a n D ( f x)rlb));
Vb(brlPown(a ) ~ Vx(xrlb n D xrla)).
It is also easy to see that the function space and the power set operator of
25.6 are monotone in n:
162
25.6.2.
Levels of Implication and Intensional Logical Equivalence
[Ch.6
if n < k, Pown(a ) C_ POWk(a ) and [a---,b]n C [a---,b]k.
The previous notions turn out to be interesting, insofar as one can
profitably work with the restricted logic of F n and hence in ~n; but it is
not clear whether these implications can really have significant applications.
However, Flagg and Myhill (1987) show that analysis based on the
hierarchy of implications can partially avoid some pathologies, which are
typical of recursive analysis.
w26. Introducing an intensional equivalence relation
According to Behmann (1931), (1937),(1959), the Russell paradox concerns
definitions, not assertions: the contradiction shows that we can define
notions (hence introduce symbols) by means of the abstraction operator
{x: A}, which cannot be eliminated by replacing the definiendum with the
definiens. The abstraction principle must be regarded as a conversion or
definition schema, which explains the meaning of t E {x: A} as A[x := t];
hence the relation of t E {x: A} with A[x := t] is a sort of definitional
equivalence and not standard logical equivalence. If we adopt this view, the
Russell argument simply teaches us that we can define a term
R:={x:~xEx},
such that R E R
and - ~ R E R become equivalent in
consequence of the given definitions or stipulations; but this fact does not
logically entail R E R ~ (-~R E R). The Russell paradox only follows, if we
know that R E R represents a genuine proposition with a definite truth
value. This situation is familiar from combinatory logic, where we have a
great freedom in defining operations, but we must investigate separately
their specific properties (e.g. convergence on a given domain, normal form,
etc.).
Following this line of thought, Aczel-Feferman (1980) and Feferman
(1984) proposed an extension of classical logic by means of an inlensional
equivalence operator --, where A - B is to be read as "A is equivalent to
B in consequence of given basic definitions". In the extended framework,
the naive abstraction principle can be consistently rephrased in the sense of
Behmann:
AF
Vu(u E {x: A} - A[x := u]) (A arbitrary, u free for x in A).
It turns out that the consistency of AF is by no means obvious and is
obtained in Aczel-Feferman(1980), using an extension of the Church-Rosser
theorem for infinitary calculi. At a first sight, it is not clear how to relate
the Aczel-Feferman logic AF with the present set-up.
In the following, we present a suitable generalization BLc( = Behmann's
logic with class induction) of the AF-logic; we then show in the remaining
VI.26]
Introducing an Intensional Equivalence Relation
163
sections that BL c has a model within P W c + GID + EA, EA being the
enumeration axiom of Ch. I.
The crucial step requires a careful extension to B L c of the AczelFeferman construction and of its underlying reduction relation. As we shall
see, an essential role in our proof is played by the fundamental boundedness
theorem of w and the selection properties of w
In contrast to AczelFeferman's proposal, we formalize Behmann's logic in pure classical
predicate calculus: definitional equivalence is treated as a binary predicate
and not as a logical connective.
We now proceed to a description of the system BLc, which justifies the
principle AF; for convenience, the system is formalized in the extension s
of s
( - the language of OP without T), which contains the new binary
predicates - , C. It is also convenient to include in s
(i)
distinct primitive constants N A T ^, I D ^, N E G ^, A N D ^, A L L ^,
corresponding to the .~.op-terms of 7.1;
(ii)
three new constants E Q ^, ~^, 0 ^ representing the p r e d i c a t e the boolean values "true" and "false" (respectively).
and
s
are inductively generated by application from variables, the
constants of s
and the new constants; s
also include expressions
of the form t C s, t - s
(t, s terms), besides s
s
are
inductively generated using V, -~, A. The [-]-operation of 7.1 which embeds
formulas into terms, can obviously be extended to arbitrary formulas of s
by stipulating:
26.1 [Nt] = N A T ^ t ;
It C s] = st;
It = s] = ID^ts;
[-~A] = NEG^[A] ;
It - s] = (EQ^)ts;
[A A B] = AND^[A][B];
[VxA] = ALL^()~x.[A]).
Thus it makes sense to adopt the standard notation {x: A}, instead of
)~x.[A], for arbitrary A of s
We tend to omit [-], once the given context
is clear and no ambiguity arises. In particular, we write ( x - y ) z,
(x _-- y) ----(u -- v) for (EQ^)xy - z and (EQ^)xy - ( E Q ^ ) u v (respectively);
we read a_--O ^ ( a - - a ^) as "a is false (true) according to the basic
definitions"; we also let Da := (a - O^V a - U^) := "a has a definite truth
value", C l ( a ) : = V x n ( a x ) : - "a is a class". We let A - B stand for the
proper [ A ] - [B].
26.2. BL c contains the system O P - of w (together with classical logic plus
equality formalized in s
number-theoretic induction for classes, axioms
stating that the new constants E Q ^, N A T ^, N E G ^, A L L ^, I D ^, A N D ^, 0 ^, B^
have the obvious independence properties, and a list of proper axioms on
164
Levels of Implication and Intensional Logical Equivalence
[Ch.6
d e f i n i t i o n a l equivalence. T h e a x i o m s are listed below in three groups.
1. I n d e p e n d e n c e axioms:
Costl
L i x = L2y --~ L 1 - L 2 A x = y,
w h e r e L1, L 2 E S Y M B 1 := { N A T ^, N E G ^, ALL^};
Cost2
G l X y = G2x'y' ---, G 1 = G 2 A x = x' A y = y',
w h e r e G1, G 2 E S Y M B 2 := { I D ^, A N D ^, EQ^};
Cost3
-,LlX = L 2 y z , for L 1 E S Y M B
Cost4
--L 1 = L2, where L1, L 2 E S Y M B
a n d L 1 ~: L 2.
2. Equivalence axioms ( a n d the logical o p e r a t o r s ) :
1, L 2 E S Y M B 2 ;
1 U SYMB
2 U {O ^, D^}
is a n e q u i v a l e n c e relation, which p r e s e r v e s itself
E.1
(a - a) A (a = b--~b _= a) A (a -- b A b = c---,a -- c);
E.2.1
a -- a' A b - b' ~ ((a A b) - (a' A b') A (a -- b) - (a' - b'));
E.2.2
( V x ( a x =_ a'x)---, (Va) = (Va)) A (a =_ a'---, (-,a) =_ (~a')).
3. A x i o m s f o r the internal logic of - "
E.3.1
(A-D ^)~A,ifA=Nx,
E.3.2
(A = O ^) ~ (--A), if A = N z , (x = y);
E.4
( ~ a ) =_ n^~-, a ~ O^;
E.5
(a A b) -- I1^~--~(a - I1^ A a -- U^); (a A b) - O^~-~ (a - O ^ V b - O^);
E.6
(Va) =-- n^~--~V x ( a x =_ B^);
E.7
(a -- b) - 0^~-, ( D e A Db A ~(a -- b)).
(x = y), (x -- y), x E y;
(-,a) = O^ ~--~a = D^;
(Va) = O^ ~--~3 x ( a x =_ O^);
Since {x" A} is Ax.[A] a n d [a E b] s h o r t e n s (ba), it m a k e s sense to state:
26.3. L E M M A
(i) BL c k- Va(a E {x" A } - A [ x " - a]) ( A arbitrary);
(ii)
BL c k- ( A - ! ^) ~ A. A . ( A - 0 ^) ~ A
( A arbitrary).
P R O O F . (i) A F is a trivial c o n s e q u e n c e of f l - c o n v e r s i o n , - - r e f l e x i v i t y
[A[x " - t ] ] - [A][x " - t], while (ii) follows by i n d u c t i o n on A. !"1
It is also clear t h a t BL c can
we c o n s i d e r M F c f o r m a l i z e d w i t h
III); if we define T a := (a = g^)
M F c b e c o m e s a s u b l a n g u a g e of s
and
be r e g a r d e d as a n e x t e n s i o n of M F c , once
p r i m i t i v e c o n s t a n t s (see Ch. II, a p p e n d i x
a n d T R := Ax.[x -0^], the l a n g u a g e L of
a n d we have:
The Infinitary Reduction Relation
VI.27]
165
2 6 . 4 . P R O P O S I T I O N . For every formula A, if MF c F A, BL c F A.
P R O O F . It
T.2.1, T.3,
-~(0^ - O^),
transitivity
suffices to check (the translation of the) T-axioms. Axioms T.1,
T.4, T.5 immediately follow from E.3-E.6; E.3 also implies
which yields the consistency axiom T.6 (via s y m m e t r y and
of - ). As to T.2.2, observe:
(~(a -0^)) -I1^ +-~ (a --0 ^) ----O^~-, (a -- O ^) ~-. (~a) ~
I!^
(apply E.4, E.7 and the equivalence
(Da A Db A -,(a -- b ) ) ~ ((a - O^A b - 0^) V (a -
0^
A b _-- O^)).
[]
Conversely, BL c has a model in P W c + G I D + EA. In order to prove
this theorem, we have to give a detailed analysis of a generalized reduction
relation.
w27. The infinitaxy reduction relation ::~
The naive idea is to define a suitable reduction relation a=:~b and to
interpret a - b as "a ::~c and b ==~c, for some c". The basic reduction clauses
embody natural truth and congruence requirements and ==~ turns out to be
confluent (it has the Church-Rosser property; see Ch. I).
Technically, the problem is that if we stick to the obvious reduction
clauses for V, we do get into troubles with the confluence proof, unless we
generalize the language to an infinitary propositional language. This move
causes additional difficulties which are typical of the Church-Rosser theorem
for systems with infinitary terms (due to Girard and Maass). A solution can
be found following the layered reduction technique of Aczel-Feferman(1980),
which makes essential use of the explicit ordinal analysis of inductive
definitions. In what follows, we show that P W c + GID + EA is powerful
enough to grant the existence of a model for a generalization of BL c.
First of all, besides the terms ID, N A T , A N D , N E G , A L L of 7.1, we
define four combinators for interpreting the constants EQ ^, 0 ^, 0^ of the
previous section and a sort of uniform infinitary conjunction q"
27.1.
0"-<0,1);
O'-<0,0>;
D E "- AxAy.(11, (x, y));
n "- Ax. (13, x).
For convenience, in w167
we keep using t - - s for the term D E t s ( D E is
reminiscent of "definitional equivalent"); remind that ~ t : = ( N E G ) t ,
[t = s] := (IDt)s, [Nt] := ( N A T ) s (cf. w
By choice, D E and q satisfy the obvious independence axioms (cf. 26.2)
provably in O P - .
166
Levels of Implication and Intensional Logical Equivalence
[Ch.6
Funcl(f) " - C l ( f ) A Fun(f), where F u n ( f ) i s
27.2. D E F I N I T I O N .
f C_ V 2 A VxVyVz((x, y)71f A (x, z)TIf ~ y -- z) A Vx3y((x, Y)~lf).
( f C V 2 states that f is a property of ordered pairs; see Ch. II, Appendix II,
l e m m a 1). If Funcl(f) is assumed, we say that f is a function-class. To
simplify notations below, when Funcl(f) is assumed, we tend to write
A ( f ( u ) ) instead of the proper 3y((u,y)rlf A A(y)).
27.3. D E F I N I T I O N . P T E R ( x ) ( -
(~
-
o
v
x
v 3y3z(~
-
-
u) 3y3z(~
v
(y
-
-
z)) v ~y(~
x is a pseudo-term) is the formula
[gy]
-
v
x
-
[y
z])
-
(~y)) v 3z(~
-
v
n z).
P T E R ( x ) says that x has the appropriate form, to be processed by means
of the reduction relation below (recall the definition of pseudo-form in w
27.4. (Informal) Definition of the reduction relation :=~
=v is the C_-least reflexive, transitive relation, which is closed under the
following rules:
AT1 9
a ::~n (O), provided a "-[Nt], [t - s] and Nt, t - s are
true (false);
NEG. 1"
from a =r b infer -~a =:~ -% ;
NEG.2"
from a ::r 0 (R) infer -~a :=r
CONJ.I:
from r u n c l ( f ) , runcl(g) and V x ( f ( x ) ~ g(x)) infer
nf~
(0);
ng;
CONJ.2:
from Funcl(f) and Vx(f(x) ~ B) infer ~ f ~ ];
CONJ.3"
from F u n c l ( f ) a n d 3 x ( f ( x ) ~ O ) i n f e r r7 f :=r O;
DE.l-
from a I ~ b1 and a 2 =~ b2 infer (a I - a2) ~ (b I - b2);
DE.2"
from a =:r c and b =:~ c, for some c, infer ( a -
DE.3"
from a =:> O (~) and b ::~U (O) infer (a - b) =~ O.
b)::r l;
27.5. R E M A R K . CONJ.3 makes clear that our interpretation of infinitary
conjunction r3 is non-strict, contrary to what happens in the original
proposal of Aczel-Feferman(1980). We also stress that ~ generates an
equivalence relation - , d e f i n e d
connective on formulas.
on arbitrary objects,
and not a new
Before we proceed to a stage-by-stage definition of :=>, we introduce the
The Infinitary Reduction Relation
VI.27]
167
notion of reflexive transitive closure RT(r) of a given binary relation r.
27.6. LEMMA. We can find a term R T of s such that, provably in
M F - + GID, if r C V 2, then RT(r) C_ V2; moreover"
(i) VxVy((x, y)~IRT(r) ~-*
,-, ((x - y) V ((x, y)rlr) V 9z((x,z)rIRT(r) A (z, y)rlRT(r)))).
(ii)
The principle of RT-induction: for each B(x,y),
[VxB(x,x) A VxVy((x,y)rlr ---, B(x,y)) A
A VxVyVz(B(x,y) A B(y,z)--, B(x,z))] ~ VxVy((x,y)TIRT(r)--, B(x,y)).
(iii) (r q V 2 ~ r C_RT(r)) A (r C r' -~ RT(r) C_RT(r')).
PROOF. The formula
Ar(u,a ) "- 3x3y(u - (x,y) A (x -- y V (x,y)~lr V 3z((x,z)rla A (z,y)rla)))
is T-positive and operative in a. Hence we can choose R T ( r ) " - I ( A r ) by
10.4, and I(Ar) is the C_-least fixed point of Ar(u,a ) by GID. Clearly, if r
is a binary relation, so is RT(r).
(iii) is an immediate application of (ii) and (i). E!
As we can guess from Ch. I (see 4.6), R T preserves confluence, provably
in M F - + GID.
27.7. DEFINITION. C R ( r ) " - "r is Church Rosser" stands for:
c_ y 2 ^ wvyvz((
, y),7 ^
^ (z,
27.8. LEMMA. M F - + GID F CR(r)---, CR(RT(r)).
PROOF. We apply 27.6. Let:
A(a, b) := Vb((a, b')~lRT(r) --, 3d((b, d)71RT(r ) A (b', d)TIRT(r))).
Then we easily obtain A(a, a) and A(a, c) A A(c, b) --, A(a, b) by closure of
RT(r) (see 27.6 (i)). C R ( r ) a n d RT-induction imply:
VaVb'((a, b')rlnT(r ) ---+Vb((a, b)rlr ---,3d((b', d)rlr A (b, d)rlRT(r)))) ,
which yields VaVb((a,b)rlr---,A(a,b)). A final application of RT-induction
yields CR(RT(r)). B
We now show that s - r e d u c t i o n is resolvable into a non decreasing
sequence of approximations (=%) _C ( o 1 ) _ C . . . ( ~ a ) C_...(for a in ON),
which converges to ~ and is suitable for the proof-theoretic verification of
C R ( ~ ) . The basic tool is ordinal analysis of inductive notions.
168
Levels of Implication and Intensional Logical Equivalence
[Ch.6
First of all, we introduce an operator H of 1-step reduction:
27.9. DEFINITION. (i) H(r) "- {(x, YI " H((x, Yl, r)},
where H(u, r) "- 3x3y(u - (x, y) A ((x, y)~r Y Ho(x, y, r))) and Ho(x , y, r) is
the straightforward formalization of the nine inductive clauses defining ::v in
27.4, except, that reflexivity and transitivity are omitted.
(ii) We then define the operator L R of "layered reduction""
L R "- Ar H ( R T ( r ) ) .
We won't bother the reader with the explicit formula for Ho(x,y,r); for
instance, a typical clause corresponding to CONJ.1 has the form:
3f3g(F cl(f) A Fu cl(g) A =
-
n f A y -
Of course, it is essential to observe that
of 10.3. Then we have"
n g A v=(<f(=),
H(u, r) is an operator in the sense
27.10. LEMMA. M F - + GID proves 9
(i)
VrVu(urlU(r ) ~ 3x3y(u - (x, y) A ((x, y)rlr V Uo(x, y, r))));
(ii)
(a C_ V : - ~ a C_ U(a)) A (a C_ b ~ H(a) C_ U(b));
(iii)
(a C_ V 2 ~ a C_ i R ( a ) ) A (a C_ b-~ LR(a) C_ LR(b)).
Note that (i) essentially requires the hypothesis that f, g are classes,
whenever x - q f, y - r3 g; (iii)follows with (ii) and 27.6 (iii).
Since L R is monotone, we can apply the bottom-up characterization of
inductively defined predicates of w
We recall that c~, fl, ~, 5 range over
ordinals, as they are defined in P W - + G I D within the set-theoretic
universes of w167
22-23.
27.11. LEMMA
(i)
We can find a term R E D such that P W - +
GID proves"
V~VxVy((x, y)rlRED~ ~ H((x, y), R T ( R E D ( < c~)))),
(where R E D ( < ~ ) " - {(x,y)" 3fl < (~.(x,y)rlREDfl}).
(ii)
Let ::v "- {(x,y)" 3a.(x,y)TIREDa} "- R C ( L R ) (cf. 23.7).
Then =~ extensionally coincides with I ( L R ) , the C_-least fixed point
of LR:
VxVy(x ~,y ~ (x, y)rlI(LR)).
(iii)
~ < fl---, R E D ~ C_ R E D f l .
PROOF. As to (i) and (iii), choose R E D "- Rec(LR) and apply 23.6 (i)-(ii),
Church-Rosser Theorem for =~
VI.28]
169
27.10 (ii). Part (ii) is a consequence of 23.10. M
27.12. C O N V E N T I O N . We henceforth adopt the following notations:
:::~ "- REDo~ and ---+a "- R T ( R E D (
therefore : : ~ - H ( ~ ) ,
< a));
(x ~ y) ~ 3c~. x : ~ y.
The formal reconstruction of ::~ is indeed adequate to its informal
counterpart of 27.6; this is perhaps better seen if we explicitly state the
relevant introduction rules embodied in 27.11 (i) (from right to left)"
27.13. L E M M A ( P W - + GID)
EXTN"
AT.I.I:
( N x ---, [Nx] :::~c~~) A (x -- y ~ [x -- y] ~,~ D);
AT.1.2:
(-,N~-~ [N~] ~
O) ^ (-,~ - y ~ [~ - y] ~
O);
NEG.I"
a) ~ ( ~ ~
O). ^ .(~ ~ O )
-~ (-,~ ~
9);
NEG.2:
(~~
CONJ.1"
F u n c l ( f ) A Funcl(g) A V x ( f ( x ) ---,~ g(x)) ---+( M f :::~,~ M g);
CONJ.2"
F u n c l ( f ) A V x ( f ( x ) ~ o ~ ~)---* ( ~ f : : ~ U);
CONJ.3"
F~ncl(f) ^ 3~(f(~) ~
DE.l:
(~1 ~ Y i
O) ~ ( n f ~
^ ~2 ~ Y 2 ) - - * (~1 - ~2) ~
O);
(Yi - Y2);
DE.2:
DE.3:
w
The C h u r c h - ~ r
theorem for
In this section we are going to prove:
28.1. T H E O R E M .
P W c + EA + GID F- CR(:::~).
By 27.8 and 27.11 it suffices to verify the Church-Rosser property for ::~a,
where c~ is an arbitrary ordinal. As in the case of finitary reduction (see w
the theorem requires a standard inversion argument, which is here combined
with double transfinite induction on O N ( = ordinals; 23.3). The point is
that if we know the form of x (e.g. x is a truth value O or ~, an infinitary
conjunction, etc.), we can tell which form any given y, such that x:::~ay,
will possibly have.
170
Levels of Implication and Intensional Logical Equivalence
[Ch.6
First, it is convenient to use a stage-by-stage characterization of transitive
reflexive closure R T ( r ) of a binary relation r: since a T ( r ) i s m o n o t o n e in ~,
there is an operation ArAy R T ( r , y ) (by 23.6-23.10), such that:
28.2. L E M M A . P W - +
(i)
GID proves:
VxVy((x,y)rIRT(r,a)~-,((x = y) V ((x,y)r/r) V
V =iz((x,z)r/RT(r, < a ) A (z,y)r/RT(r, < a)))),
where R T ( r , < a) := {(u, v) : 3fl < a. (u, v)r/RT(r, fl)};
(ii)
(iii)
a _</~-~ R T ( r , a) C_ R T ( r , fl);
VxVy((x, y)rlRT(r) ~ 3a((x, y)rlRT(r, a))).
28.3. L E M M A (Inversion). The universal closure of each of the following
sentences is provable in P W - + GID:
SENT-(a):
x=~ay A--PTER(x)---, x = y (see 27.3);
TV-(.):
9~ .
AT-(a):
if A t ( x ) : = 3u3v(x = [Nu] V x = [u = v]),
y ^ *.1{O. 0} --~. = y;
(x ~r y. A At(x)) ~ (y = x) V (y = 0 A T x ) V (y = 0 A T-,x);
NEG-(a):
(-,x=~a y) ---,
[3z(y = -~z ^ 9 ~ , z )
CONJ-(a):
v (y = 0^ 9 ~ ,
( F1f =:ray )---,
-~[3g(ru~cl(g)
^ y = n g ^ v,(f(,)--.,
v ( v . ( f ( . ) ~ . 0) A y = 0 ) v ( 3 . ( f ( . ) ~ .
DE-(a):
o ) v (y = o ^ 9 ~ . 0)];
g(,))) v
O ) ^ y = O)];
(x -- z =:Va Y)
-~ [3*13z~(y = ('1 --- Z l ) ^ ( . - ~ . - 1 ) ^
( z - ~ . Zl)) v
V 3w(x - ~ w h z - ~ w A y = 0) V
v (y = o ^ ((~ ~ o
^ z~.0)v (~~0 ^ z ~
o)))].
P R O O F . Consider the formulas:
D l ( a , x , y ) := ( - ~ P T E R ( x ) A x = y);
D2(a , x, y) := (xq{O, 0} A x = y);
n 3 ( a , x, y) := At(x) A ((y = x) V (y = OA Tx) V (y = 0 A T-,x ));
Church-Rosser Theorem for ~
VI.28]
171
D4(o~,x,y ) "- =:lv(x - ~v A (:=lz(y -- ~z A v ---~otZ) V (y --UA v---~o~O) V
v (y-
o A~-~n)));
D5(a , x, y ) " - q f 3 g ( ( F u n c l ( f ) A [q f -- x A Funcl(g) A y -- V1g A
n w(g(~)-~,
g(~))) v w ( g ( ~ ) - ~ . 0
n 6 ( - . ~. y ) " - 3~3~(~ - (~ - ~ ) n ( ~ 3 z ~ ( y
A y - 0) v 3 ~ ( g ( ~ ) - ~ . O n y -
O));
- (~ - z~)n (u-~. ~)A
A (v--*, Zl) ) V 3w(u--*aw A v--,{~w A y --II) V
V (y -- 0 A ((u--~a O A v--, il) V (u--*aI! A v--~a O ))))).
Clearly the lemma is a consequence of Vc~C(c~), where C(c~)is the formula
and M(oz, x , y ) ' - D i ( o ~ , x , y ) V . . . V D 6 ( o ~ , x , y ) .
We verify Vc~C(c~) by
transfinite induction on c~ ( see 23.4 (iii)). Thus we assume x =r
and
Vfl < c~. C(/3),
(1)
as main induction hypothesis (MIH in short). By 27.11(i), either we have
used the clause EXTN, i.e. x---,ay holds (this always happens if ~ P T E R ( x )
holds), or else H o ( x , y , ~ a ) ( s e e 27.9), and the required conclusion obtains.
{As a typical instance, consider the case where x:::Vo~y has the form
a - b : : : ~ a y , but x:::~ay does not follow by EXTN. Then we must have
applied an inference among DE.I-DE.3" either y - ( x I - Zl) , a---,aXl,
b---~a Zl, for some Xl, Zl, or a "-~otw, b "*or w, y - O, for some w, or else y - O,
a---,a O (0) and b---,a 0 (O); then we are done. Observe that we implicitly rely
upon the fact that a - b is provably different from every infinitary sentence
built up by negation, ~, and basic predicates ; cf. 27.1)}.
Therefore we are left with the verification of
VxVy((x----~{~y)----~ M(oL, x,y)).
(2)
Let us shorten (a, b)~IRT(RED( < c~), 7 ) " - " ( a , b) occurs at the 7-th stage in
the reflexive transitive closure of R E D ( < a~)" with the simpler (a---~a b).
By 28.2, (2) is equivalent to the statement
VTVxVy((x~o, Y)--+ M(o~,x,y)).
1'
(3)
(3) is verified by induction on 7: so we assume (x--,
y) together with
-y
V5 < 7.VxVy((x---,(~sy)--~ M(o~,x,y)) as secondary induction hypothesis (SIH
in short). If (x--,c~ y) holds by reflexivity, then x - y
and M ( o ~ , x , y ) i s
trivial by SIH; on the other hand, if (x,y)~?RED( < c~), we are done by
MIH. Thus by 28.2 (i) we only check M(o~,x, y), under the assumptions:
Levels of Implication and Intensional Logical Equivalence
172
(x
z);
(z
u),
[Ch.6
(3.2)
for some z, 6 < 7- We distinguish six main cases.
1: --,PTER(x) holds. By SIH applied to (3.1), x - z
and hence by (3.2)
(x-~a~ y), which implies x - y again by SIH. Hence Dl(a, x, y).
2: x - 0 ( x O). Then z - 0 ( z - O )
by SIH applied to (3.1), whence y ( y - O) by SIH applied to (3.2). Hence n2(a,x,y ).
0
3: At(x) holds. Apply SIH for (3.1). If x - z, the conclusion follows by SIH
applied to (3.2). If Tx and z - 0 hold, then y - 0 by SIH for (3.2) and we
are done. The remaining case is similar. In all cases, we have D3(c~ , x, y).
4: x -- -~a. By SIH applied to (3.1), we must analyze three subcases.
4.1"
and
and
that
z - ( - ~ u ) , for some u such that a - ~ a u . Then we can apply SIH to (3.2)
we have D4(a,z,y ). If y - - ~ v and u --,, v, we have a - - , , v ; if y - O (0)
u-~a0 (O), also a ~ , 0 (O). In all cases, we conclude to D4(a,x,y); note
we need the fact that -~ O~ is a transitive closure.
4.2-4.3: z - O and a---+a0 ( z - 0 a n d a---+aO). Then (3.2) and SIH imply
y - O ( y - 0), and we are done.
5" x -
R f . By SIH for (3.1), we consider three subcases.
5.1. z - - F1g, where g is a function-class such that f(c)-~ag(c), for every c.
Then we can apply SIH to (3.2). If y -- ~ h and g(c) ~ a h(c), for every c
and some function-class h, we get f(c)--,,h(c) by transitivity for arbitrary
c, i.e. n5(~,x,y ). If z - - 0 and f(c)--,aO, for every c, we have y - 0 b y SIH
for (3.2) and we are done; the case z - O is analogous.
5.2-5.3. If z - 0 a n d f(c)---+aO, for every c (or z some c), we argue as in 5.1.
O and
f (c) ---+aO, for
6. x - ( x 1 - x 2 ) . By SIH applied to (3.1) we consider three subcases.
6.1. z - ( z 1 - z 2 ) and (xi--,az{) ( i - 1, 2). Hence we can apply SIH to (3.2).
If y - ( Y I - Y2) and (z i - , , Y{), we get (xi--, ~ Yi) (for i - 1, 2) and hence
D6(c~ , x, y). The remaining cases are similar.
6.2-6.3. z - 0 and x 1--,(xc, x2--,(xc , for some c, or z - O and x 1-~a0 (O),
x 2--,a O (0). Then we apply SIH to (3.2), in order to get y - g ( y O). !-1
28.3.1. R E M A R K . SENT-(c~) implies:
P W - + GID t- (a ::>b) A P T E R ( b ) ~
PROOF
PTER(a).
of the main theorem 28.1. We assume by IH that for every fl < c~,
Church-Rosser Theorem for ~
VI.28]
CR(:=~9) holds.
173
We further observe that IH and 27.11 (iii)imply:
(,)
CR(RED( < a)),
whence by 27.8"
(**)
Assume a :=~ b and a ::~a c: we must produce d such that b :=~a d, c =~a d. We
distinguish six cases according to the form of a.
1.-,PTER(a) is assumed. Then by S E N T - ( a ) , a - b - c
d - a (x=:~ax holds by EXTN and closure of RT under
choose
2. a is a t r u t h value. Apply T V - ( a ) and choose d -
3. At(a) holds. We apply A W - ( a ) . If a we see that d - g (0).
4. a - - - x
and we can
reflexivity).
a.
b, we choose d -
c; if b - 0 (O),
for some x. Apply N E G - ( a ) .
4.1. Assume that b - - - y , c - - - z , for some y, z such that x ~ y, x---,az.
Then by (**), there exists w such that y ~ a w , z ~ a w , which yields (by
27.13) b =:~ (--w), c ::~a (--w)" choose d - -~w.
4.2. Assume b - - - y , c - 0 , where
27.13 (NEG.2), we can choose d c - O and x---,s 0.
x~ay
and x ~ o~ O. By (**), T V - ( a ) ,
0; a similar argument yields d - O if
4.3. Assume b - 0, x ~ a O . If a=~ac is introduced by NEG.1 (see 27.13), the
argument is symmetric to 4.2. Else, notice that we cannot have c - O (this
would imply the contradiction O - 0 b y
(**), T V - ( a ) ) . Hence c - 0 a n d we
can choose d - 0.
4.4. b - O, x---,a 0 : argue as in 4.3.
5. Assume a - [-1 f. Apply C O N J - ( a ) . This is the most delicate step in the
proof: it makes a(n essential ?) use of the selection theorem 20.4, and hence
of EA, plus class induction on numbers.
5.1. Let b - - [7 g, c = ~ h, where f, g, h are function-classes satisfying, for
every u, the condition
f(u) ---,~g(u)
If
F((~,g,h)
= {(u,w):
and
f(u) ~ h(u).
(g(u)--,(~w) A ( h ( u ) ~ w ) } ,
w3 ((u,
g, h)).
(1)
(**) implies
(2)
Hence by selection 20.4 we can find a function class r = Sel(Fia, g,h),Y),
such that, for every u, g(u)~ar(u ) and h(u)~ar(u); hence we can choose
d = F1r by CONJ.1 of 27.13.
Levels of Implication and Intensional Logical Equivalence
174
[Ch.6
5.2. Let b = i, f(u)---,aO, and c = n h, where f ( u ) ~ a h ( u ), for every u, and
f , h are function classes. Then we can choose d = 0 b y CONJ.2 (apply (**)
and notice that T V - ( a ) i m p l i e s h(u)---,aO, for every u).
If we m a i n t a i n the same hypothesis on c, but we assume b - O
and
f(u)---'a O, for some u, we can choose d = O (by CONJ.3).
5.3. If both b and c are t r u t h values, they m u s t be identical and hence
d - b; the remaining cases are s y m m e t r i c to those already considered.
6. Let a = x 1 -- x 2. We apply D E - ( a ) .
6.1. Assume t h a t b = 0 and x 1---,aw, x 2 ~ a w . It is easily seen t h a t c ~ 0
(by (**), T V - ( a ) ) . Hence d = 0 i f c is a t r u t h value, and we can suppose
t h a t c = Ya -- Y2 with x i ---*a Yi (i = 1, 2). Then by a first application of (**)
we can find z i such that yi---,azi (i = 1,2) and w---.aZl, w--~az2; again
with (**) we get z such that z i---,az (i = 1, 2); DE.2 grants that d = 0 is the
right choice. A similar a r g u m e n t , together with DE.3, shows t h a t d - O if
b-O.
6.2. Assume that b = ( y I --Y2), c--(Zl--z2) with xi--.(~yi, xi---,azi
(i-1,2).
By (**) we can find d i such that yi-->o~di, z i - - , a d i , whence
x i - - - , ~ d i ( i - - 1,2);
an
application
of DE.1
yields
b---,,~(d 1 - d 2 ) ,
c -%~ (d 1 - d2).
The remaining cases are s y m m e t r i c to those already treated. 0
w
A m o d e l of type-free logic based on intensional equivalence
We first define a translation c of -5B into .5; the basic idea is t h a t - is
m a p p e d into the n a t u r a l equivalence relation, generated by the confluent
reduction =:~.
29.1.DEFINITION
(i) = B := { ( x , y ) " 3 z ( x ~
for (x, y ) r / - B"
(ii)
z A y ~ z)}; we agree to write x -- BY,
Let G r ( f ) "- { u ' 3 v ( u - (v, fv/)};
Conj(x,y)
"- {u" 3 v 3 w ( u - (v, w) A ((v -- -0 A w -- x) V (-~v - -0 A w -- y ) ) ) } .
A L L * "- A f . ~ G r ( f ) ;
A N D * "- AxAy. F] C o n y ( x , y).
(iii) The m a p c is the identity m a p on variables and constants K, S, D,
PAIR,
LEFT, RIGHT,
O, S U C , P R E D ; it transforms the remaining
constants of s according to the following stipulations:
A Model of Type-free Logic
VI.29]
175
( N A T ^ ) r := N A T ;
(ID^) C := ID;
(NEG^) c := N E G ;
(EQ^) e := DE;
(ALL^) C := ALL*;
(AND^) e := A N D * ;
(o^) ~ := o
(0^) e : = ,.
;
{In (ii)-(iii) above, I D , N A T , N E G are the combinators of 7.1; D E , O, O,
V1 are defined in 27.1.}
The map r is inductively extended to terms and formulas of s
(ts) c - tCsc;
( N t ) c - N(te);
(t - ~)~ - (t ~ - B ~ ) ;
(~A) ~ - ~(A~);
(t-s)
c-(t
e-se);
(t ~ ~)~ - (t~) ~ - Bo;
( A ^ B ) ~ - A ~ ^ B~;
(WA) ~ - W(A~).
The specific choices of 29.1 (ii) are motivated by the following:
29.2. L E M M A
(i)
M F - F F u n c l ( C o n j ( a , b)) A (AND*ab)(O) - a A
A Vx(--,x - 0 ~ ( A i D * a b ) ( x ) - b);
(ii)
M F - t- F u n c l ( G r ( f ) ) A V x ( ( G r ( f ) ) ( x ) - i x ) .
{Recall the definition 27.2 for the the notation (...)(x).}
29.3. T H E O R E M . If BL c F A, then P W c + GID + EA F A r
Since A r
A, whenever A is in the language of UP without T, and
P W c + GID + EA is a conservative extension of UP (by 15.5(i)), we can
conclude:
29.4. C O R O L L A R Y . BL c is a conservative extension of UP.
PROOF
assume:
of 29.3. Let us prove the translation of class N-induction; we
Vx(bx-sOVbx-sO)AVx(bx-sO~b(x+l)-sO)AbO-sO.
(1)
Then by the first conjunct of (1) and 28.3 (see TV-(c~)), we obtain, if we set
b1 - {x" b x - B 0} and b2 - {x" b x - - B O } "
Vx(xlib 1 V xllb2) A ~={x(xrjb 1 A xrib2).
(2)
Hence by 16.11 and (2), there is a class c such that c - e bl and - c - e b2;
class N-induction yields V x ( N x ~ b x - B 0).
It remains to check the provability of the C-translations of the - - a x i o m s of
176
Levels of Implication and Intensiona/ Logical Equivalence
[Ch.6
groups 2-3 (p.164; 26.2). As to (E.1) ~, -- B is trivially symmetric, reflexive
by the corresponding property of ~ (27.6(i), 27.13, EXTN), while it is
transitive by the Church Rosser property of 28.1. (E.2.1)e-(E.2.2) e are
straightforward applications of 27.13 (see DE.l, CONJ.1, NEG.1) and 29.2.
As a sample, consider (E.2.1) C and the case of A-preservation. Assume
( a - b) C and ( a ' - b ' ) ~. By 27.11 ( i i i ) a n d the closure properties of R T (see
27.6), there is an c~ such that a ~ z ,
b ~ z , a ' ~ w , b ' ~ w , for some z
and w. By definition of AND*, 29.2 and CONJ.1, we obtain:
AND*aa' ~ a A N D * z w and AND*bb' ~,~AND*zw,
whence ((a A a') = (b A b')) r
As to = - a x i o m s of group 3, first observe that by TV-(c~), a = B g (O) is
equivalent to a=V0 (O). Then we argue as in the previous case, with
introduction and inversion lemmata 28.3 and 27.13. Let us explicitly
consider three instances.
Ad (E.3.1) r If A = x E y, the translation reduces to an instance of the
identity principle. Let A = (x = y): then we must check
x - By~--~(DExy) =_BO~-+DExy:::~O.
From left to right, we just apply closure under DE.2; in the opposite
direction, we use D E - (a).
Ad (E.6) c. Assume (Va -g^)r = (ALL*a -- B g): by 29.2 (ii) and C O N J - ( a ) ,
we must have a ( x ) - B 0, for arbitrary x, i.e. (Vx. a x - 0^)r In the reverse
direction, we apply CONJ.2.
Ad (E.7) c. If ( ( a - b ) - O^) r we get (DEab)=~O: by D E - ( a ) , either a::~O
and boO, or else a::~0 and b o O . In either case (Da)r
r hold; were
( a - b) r we could derive the contradiction O = 0 by means of CR(:=~) and
T V - ( a ) . The opposite direction is a consequence of DE.3. [3
CHAPTER 7
ON THE GLOBAL STRUCTURE OF MODELS
FOR REFLECTIVE TRUTH
w
~31.
w
w
w
w
The lattice of fixed point models for the neutral minimal theory
The sublattice of intrinsic fixed point models and the cardinality
theorem
Variations on the encoding technique: non-modularity and other
oddities
A model for an impredicative extension of reflective truth
On Kripke's classification of self-referential sentences
On the consistency of coinduction principles
Appendix: a variant to the basic operator F and the restriction
axiom
According to chapter II, M F - and its variants admit, as natural
interpretations, the minimal fixed point models of a simple operator F,
which can be described in the very language L of MF. This basic feature has
been exploited at length, in order to establish the consistency of the
approximation axioms and the generalized induction schema.
The starting point of this chapter is the observation that the fixed
points of F are exactly the models of NMF-, the subsystem of MF-, which
omits number-theoretic induction and consistency. This fact permits an
extensive, natural application of lattice-theoretic techniques to the
investigation of the global structure of NMF--models. In fact, we
concentrate upon the structure FIX(.At~) of the fixed point models over an
arbitrary combinatory algebra .AI~, and we prove that this structure is highly
non-trivial and complex. An unexpected moral, which will emerge below, is
that self-referential statements and conditions stand beforehand as flexible
encoding tools for conveying information; their nonsensical character make
them available to be filled with whatever sense we like. This idea is
illustrated ad nauseam by the crucial results.
In particular, w first establishes a simple characterization theorem,
implying that the collection of sentences internally true in arbitrary models
of the theory MF-(extended with the restriction axiom of w is recursively
axiomatizable. Then we introduce a lattice-theoretic structure on FIX(.At,).
178
On the Global Structure of Models
[Ch.7
It is shown that FIX(3b) is a complete involutive lattice, which is
well-defined up to isomorphism of combinatory algebras. In a similar way,
the least inductive model O(.Ag) of M F - only depends on the isomorphism
type of dig.
w deals with the complete sublattice of the so-called intrinsic models
(see Kripke 1975). Roughly speaking, a model is intrinsic if it is obtained
from the minimal fixed point model O(vtt~), by adding information which is
compatible with that of any other model. The main result of w is an
encoding method for producing plenty of intrinsic models; the cardinality
theorem shows that there are 2 carat(M) intrinsic models (.Ab being the
underlying combinatory algebra).
The subsequent section w establishes that INT(vlg), the set of all
intrinsic fixed points (and hence FIX(dig)), is a non-distributive (indeed
non-modular) lattice, which can be further "split" into disjoint intervals (in
the sense of W h i t m a n 1944). We finally produce infinite strictly descending
sequences of models in INT(Mg).
In w we use a self-referential property, in order to encode models of
combinatory logic, embedded in second-order logic. The outcome is that we
can consistently enlarge the framework of reflective truth with extensional
objects, closed under impredicative comprehension. Again, this shows that
the non-minimal models of M F - can be quite complex.
w
adapts to the present context Kripke's classification of selfreferential sentences: the notions of grounded, intrinsic, paradoxical object
are introduced and shown to be non-trivial.
The final section w is a complement to w of Ch. III. We concentrate
on the pair (O(.Ab),B(.Ab)), where O(.Ag) and U(.Y~) are the least and the
largest fixed point of FIX(JIg), respectively. We prove that D(vtb) satisfies a
generalized coinduction principle, while O(.Ab) verifies a generalized
induction principle, which involves the complement of the largest fixed
point of any given operator (for terminology, see 10.3).
To some extent, the results of this chapter are contained in Cantini
(1989), (1993).
NB. We must warn the reader that we stick to the notations and
simplifying conventions of chapters II-III, namely:
(i) Once ~l=OP-( = OP without N - i n d u c t i o n ) i s fixed, we assume
that our basic languages are extended with distinct names for elements of
M, the domain of aft~; thus we deal with s
s
respectively. We
do not typographically distinguish between elements of M and their names.
(ii) Syntax: -~, A, V are also used for the proper terms NEG, AND,
ALL of 7.1 (hence --1t, t A s, Vt are abbreviations for NEGt, ANDts, ALLt
(in the given order)); in addition, --,--,t := ~ ( - - t ) and TA := T[A].
VII.30]
179
The Lattice of Fixed Point Models
(iii) Semantics: if att~ is any given model of OP- and b, c E M, then bc,
-,b, bAc, Vb shorten the proper atl~(Ap(b,c)), alg(NEGb), ~ ( A N D b c ) ,
.Ag(ALLb); id(b,c), tr(b), nat(b)stand for Mg([a- b]), aig([Tb]), alg([Nb])(in
the given order).
w
The lattice of fixed point models for the neutral minimal theory
For the reader's sake, we repeat a few basic facts from w First of all,
P F O R ( x ) is the s
saying that x has one of the following forms:
-~y; Vy; [Ny]; [Ty]; [y = z]; y A z. The set M - P F O R is
{a" a E M and .Ag [=PFOR(a)}.
30.1. LEMMA. There exists an operator F" ~ ( M ) ~ ( M )
such that:
((see 7.3(v))
(i) if P C_ M, r(P) c_ M - P F O R .
(ii) F 9~(M)---,~(M) is monotone: S C_ S'::~ F(S) C_F(S').
(iii) Assume a, b E M:
if a ~ M - P F O R , (~a) E F(S);
if A is an e-atom or a negated e-atom, Ml~([A])E F(S) iff A holds
in vlg ;
(a A b) EF(S) i f f a E S a n d b E S ;
(~(a A b)) E r ( s ) iff (-~a) E S or (-~b) G S;
(Vf) ~ F(S) iff ~ ( f a ) ~ S, for all a ~ M;
(-- (Vf)) E r(S) iff (-~(fa)) E S, for some a E M;
(-.tr(a)) E F(S) iff (-~a) E S;
(tr(a)) E F(S) iff a E S;
(----a) E F(S) iff a E S.
30.2. DEFINITION
(i) S c_ M is consistent (complete) iff for every a E M, either a ~ S or
(--a) ~ S (a E S or (-~a)E S).
CONS(.AI~) "- {S" S C_ M, S consistent};
COMP(.At~) "- {S" S C_ M, S complete}.
(ii) S C_ M is F-dense (r-dosed) iff S c_ F(S) (F(S) c_ S);
(iii) FIX(.AI~) . - FIX(F,.AI~) . - {S" S c_ M an'd F ( S ) - S};
180
[Ch.7
On the Global 5tructure of Models
FIXcs(~I~) "- FIX(JII~) M CONS(Jtl~);
F I X cp(J?I~) "- FIX(J~b) fq COMP(J~).
30.3. LEMMA. If S C_M is F-dense and consistent (complete and F-closed),
then F(S) is F-dense and consistent (complete and F-closed).
We recall from 7.10 that RES is the sentence:
Vx((Tx ---,PFOR(x)) A ( - ~ P F O R ( x ) ~ T-~ x));
N M F - is M F - without the consistency axiom (see 7.10, T.6). We now state
a semantical pendant of proposition 7.12"
30.4. T H E O R E M (Characterization)
(i)
S E FIX(R)
iff ( ~ , S)I = N M F - + RES.
(ii)
S E FIX~(.At~) iff (Jtl~,S) I = N M F - + RES + CONS.
(iii)
S E FIX~p(.At~) iff (.AI~,S)[= N M F - + RES + COMP.
(iv) If
-mod l
,nt p tatio of N
can omit the restriction- in (i)-(iii) above.
ta dard),
PROOF. (ii)-(iv) are straightforward by (i).
(i): it is immediate from left to right, since the axioms simply establish that
the interpretation of the truth-predicate is F-dense and F-closed. In the
opposite direction, F(S) C_ S is a consequence of the second part of RES, the
appropriate axioms of N M F - and the inversion properties of lemma
30.1 (iii); the first part of RES with the truth axioms of N M F - yields
S C_F(S). Alternatively, we could also observe that N M F - + R E S
is
axiomatized by O P - + CONS + the fixed point axiom F P T of 7.12. E!
If we regard the fixed points as possible notions of truth, we can
immediately rephrase theorem 30.4 in the form of a completeness result,
characterizing the "internal" and "external" logics of reflective truth.
30.5. DEFINITION. S E N T ( L ) : = {A: A sentence of L};
~
:= {A: A E S E N T ( Z ) and (.hl~,X)I= A, for every ~ I = O P and every X E FIXcs(Jll~)};
] ~ : - {A: A E S E N T ( L ) and ~I~([A])E X, for every ~ I=OP and every X E FIXcs(Jf6)}.
~1"(]~) might be called the external (internal) logic of consistent reflective
truth.
30.6. COROLLARY. ~1" and ~" are axiomatizable (i.e. the sets of their
GSdel numbers are recursively enumerable).
The Lattice of Fixed Point Models
VII.30]
181
PROOF. By the Ghdel completeness theorem and 30.4. Vl
30.7. DEFINITION. S d "- {a" a E M and (--a) ~ S} ( - the dual of S).
30.8. LEMMA
(i) If S E FIX(Jig), then S d E FIX(JIg); moreover, if S is consistent
(complete), S d is complete (consistent).
(ii) If S E FIX(.Jtt~), then (sd) d - S; S C_P implies pd C_S d.
PROOF. (ii) is trivial by definition of d, hypothesis, and lemma 30.1 (iii);
as to (i), observe that S d I= N M F - by theorem 8.11 and apply 30.4. V!
We shall now investigate the closure properties of FIX(Jtt~) and we
directly introduce the appropriate lattice-theoretic operations.
30.9. DEFINITION. Let S be a subset of M. We define by transfinite
recursion on ordinals (ON = collection of ordinals):
(i) v P ( s , o ) = s;
v P ( s , a + l ) = r(vP(s,a));
U P ( S , A ) - U {UP(S,a):a < A}, where A is a limit;
(ii) DOWN(S,O)= S; DOWN(S,c~+I)= F(DOWN(S,a));
DOWN(S,A) = M {DOWN(S, c~) : c~ < A}, where A is a limit;
(iii) UP(S) = U { U P ( S , a ) : a E ON};
DOWN(S) : M {DOWN(S, a):a E ON};
(iv) O ( 3 g ) = UP(O)and D(dlg)= DOWN(M).
(v) If S is of the form UP(S) (DOWN(S)), then S(c~):= UP(S, a)
(DOWN(S, a) respectively) is called the a-th approximation of S.
30.10. LEMMA
(i) If X C_Y, then UP(X) C_UP(Y) and D O W N ( X ) C DOWN(Y).
(ii) If S is F-dense, then a <_fl implies UP(S,a) <_UP(S,/3);
if S is r-closed, t h e n ~ <_ ~ impties DOWN(S,/3) <_DOWN(S, a).
(iii) If S is r-dense (and consistent), UP(S) is the C-smallest
(consistent) S E FIX(31~) extending S; in particular, the minimum
(iv) If S is r-dosed (and complete), DOWN(S) is the C-largest
(complete) X E FIX(dtt~) contained in S. In particular, O(.]tg)- the
maximum of FIX(.Jfg)- is complete and U(.]~)d - O(3g).
182
On the Global Structure of Models
[Ch.7
PROOF. Ad (i)-(ii)- by induction on a, using F-monotonicity. In (iN), the
assumption on S is essential for the case a - 0.
Ad (iii). By (iN) and Cantor's theorem, the sequence {UP(S, a ) ' a E O N } is
eventually constant. Hence, for some 6,
UP(S)-
UP(S, ~) - F ( U P ( S ,
5))
c FIX(.At~).
If X is F-closed with S C_ X, then UP(S,a)C_ X (easy ON-induction).
Hence UP(S) is the C_-least F-closed _D S. In addition, if S is consistent,
each UP(S, o t ) i s consistent by lemma 30.3 and (iN): hence U P ( S ) i s
consistent.
Ad (iv). The dual claim for D O W N ( S ) i s
left as exercise. As to
=
pply 30.s. D
30.11. D E F I N I T I O N
If E C_ FIX(Jft~), we set:
U E "- UP( U E)( - sup of the family C);
M r "- D O W N ( M E) ( - inf of the family r
We adopt the standard abbreviation "X is a poser" for "X is a partially
ordered set". As usual, X U Y "- U {X, Y} and X M Y "- M { X , Y } . If X,
Y E F I X ( . ~ ) , X _< Y iff X M Y - X (iff Y - X U Y).
(i)
(iN) If ~ is a poset with order _ , C C_ ~P is coherent iff for every pair a,
b in E, there is c E E, such that a _ c and b < c.
(iii) ~P is a complete coherent poser (in short ~ is c.c.p.o.) iff ~P is a
poset and every coherent subset of ~ has a sup in ~.
(iv) If ~ is a poset with order _< and d is a map of ~P into itself
satisfying (ad) d -- a and a < b ::~ bd (_ a d, then d is an involution. A poset
with an involution d is called involutive.
30.12. L E M M A
(i) If E C_ F I X ( . ~ ) , U E is F-dense and M C is F-closed.
(iN) /f X, Y E FIX(Jtl~), then X <_ Y iff X C_ Y.
(iii) UE is lhe sup o f t and ME is the inf of C (with respect to < ) .
P R O O F . (i): trivial with F-monotonicity and r C_ FIX(~l~).
(iN): X C_ Y implies X U Y - Y, whence
U P ( X U Y) - X U Y - U P ( Y ) - Y, i.e. X _ Y.
Conversely, assume X _ Y, i.e. X U Y - r . Since X U Y is F-dense (by (i)),
we have X U Y C_ X U Y - Y, i.e. X C_ Y.
(iii) If X E r X C_ U E, whence U P ( X ) - X C_ U E (by lemma 30.10(i)),
i.e. X < U r by (iN) above. Let Y E F I X ( . ) ~ ) with X _< Y, for every X E E;
The Lattice of Fixed Point Models
VII.30]
183
let S ( c ~ ) " - U P ( U C, c~). Then we have, by induction on c~:
S(a) C_ Y, for every a.
(+)
(If a - O, use the assumption and (ii); else, apply F-monotonicity and IH).
But (+) implies H r C_ Y and hence H r _< Y (again by (ii)). The argument
for E r is dual. D
30.13. THEOREM
(i) (FIX(J~), U, F], d, O(.)tl~),D(31~))is a complete lattice with
minimum O(J~), maximum n(~), and with an involution d, satisfying, for
C C_FIX(Mg),
( H C) d -
q { X d" X E C} and ( n C)d -
U { X d " X E C}
(,)
(ii) (FIXes(Jr6), <_) is a complete coherent poser with minimum
0 ( ~ ) , which is closed under F1 over non-empty subfamilies of consistent
fixed points.
(iii) d is a dual isomorphism between ( F I X cs(Mg), <)
and
(FIXcp(Mt~), < ). In other words: d is a bOeclion of F I X cs(a?l~) onlo
F I X c p ( ~ ) s~isfying (,) above (see 30.2 for FIXcs(J?I~), tlXcp(dtg)).
PROOF. (i): straightforward by lemma 30.8 and 30.12. For instance, as to
(,), observe that X_< U C implies (uc)d_< X d, for every X E C; hence
(HC) d C_ VI{X d" X E C}. Conversely, i f X E C , then R { X d" X E C} C_X d,
whence X C _ ( H { X d" X E C } ) d, which implies I I C C ( [ - ] { x d ' x E c } )
d
(definition of sup) and finally R { X d" X E C) C_ ( H C)d.
(ii) Let C C_ FIX(Mg) f'ICONS(Mg)" if C is non-empty, NC is
consistent; hence F1C is the consistent inf of C (by 30.12 (i), 30.10 (iv)). If C
is a coherent family of consistent fixed points, U C is F-dense and consistent
(lemma 30.10). Hence by lemma 30.3 and induction on c~, it follows that
UP( tOC,c~) is consistent; therefore H C is the consistent sup of C (lemma
30.12 (iii)).
(iii) the map d is onto because every complete fixed point X has the
form r d for r E F I X c s ( ~ ) : choose Y = X d (lemma 30.8). d is one-to-one:
trivially X d - Y d yields ( X d) d - ( Y d )d , which implies X - Y ,
again by
lemma 30.8. As to the verification of (,), apply (i). El
30.13.1. REMARK. As to the map d, (i) above establishes that d is a dual
automorphism of F I X ( ~ ) .
One may wonder to what extent the lattice structure of FIX(Jr6)
depends on the underlying combinatory algebra a~l~; we prove that F I X ( J ~ )
is only determined by the isomorphism type of .Ag. Similarly, if "2~1 and
"2~2 are isomorphic, then the expanded structures (J~l,O(.)~l)) and
184
On the Global Structure of Models
[Ch.7
(.Al~2,0(~2) ) are isomorphic.
If t is a closed term of s
~ ( t ) is the value of t in M. Below we make
use of the following facts, well-known from introductory model theory"
30.14.1. Let d be an isomorphism between ~ 1 and ~A~2. Then
r
an))) - .A~2(t(r
,r
for every term t ( x l , . . . , X n ) of .~op and arbitrary a l , . . . , a n E M 1.
30.14.2. .At~1 and .fib2 are isomorphic modulo r M1---+M2, provided
.fllbl l-- A ( a l , . . . , an) iff .A~2 I- A(r
r
for every atomic formula
A of Lop , with free variables in the list Xl,...,xn, and for every
a l , . . . , a n E M 1. {Here we do not distinguish typographically between the
a l , . . . , a n C M 1, r
r
E M 2 and their syntactical representatives
in L ( M 1 ) a n d Z(M2) }.
Let rl-@(M1)~ ~(M1) and F 2 9~P(M2)~@(M2) , where each F i is the
F-operator of 30.1 and 7.3, relativized to ~ i , i - 1 , 2. Similarly, let
UP 1" ~P(M1)~ ~(M1) (UP 2" ~
) denote the UP-operator
induced by iterating F 1 (F 2 respectively) on subsets of M 1 (M2).
30.15. LEMMA. Lel .At~1 ] = O P - and .At2I-OP-; let r be an isomorphism
between "~1 and .AI~2. Then:
(i) the map r ~176
, defined by r
- {r
a E X},
commutes with the operator F, which characterizes the models of NMF-:
X C M 1 =:~ r
Hence, if X E FIX(.At~I), r
(ii)
r
~(M1)-~(M2)
F2(r
FIX(.At~2).
commutes with the UP-operator"
if X C_ MI, then r
(iii) r preserves
d,
(,)
- UP2(r
(**)
i.e for every Y C_ M1,
r
r
PROOF. (i) Let r be the given isomorphism between .A~1 and ~ 2 "
observe that 30.14.1 implies, for a E M 1"
.YI~1 ]- P F O R ( a ) iff ~1~2 ]= PFOR(r
First
(1)
We now assume X E F I X ( ~ I ) , and we check that c E F I ( X ) implies
r
E F2(r
We freely apply 30.1 and 30.14.1-30.14.2 without explicit
mention. We have to distinguish several cases.
Case 1. If c - .A~l(-~a ) and a ~ M I - P F O R , we can infer by (1) that
r
~ M 2 - P F O R , whence . 2 ~ 2 ( - , r
r
E F2(r
VII.30]
185
The Lattice of Fixed Point Models
Case 2. c - . ~ l ( [ a - b])" then c E FI(X) implies "~1
30.14.2, .)tl~2 [ = r
r
whence again:
.Ah2([r
)- r
- .)th2(r
- b])) - r
I-(a-
b); hence, by
E F2(r
Case 3. Let c - .~l(Va): then c E F I ( X ) implies Jtt, l(ad ) E X, for every
d E M 1. Assume that u is an arbitrary element of M2; since r is onto,
u-r
for some v E M 1 and hence by hypothesis .Al,l(av ) E X, i.e.
r
J~2(r
r
which implies
J~2(V(r
.A~2(r
F2(r
Case 4. c - Jtt, l(-~-~a): then c E FI(X) implies ~l~l(a ) E X, whence
r
- J~2(r
Er
which yields
Mh2(~-~r
.At~2(r
E F2(r
The remaining cases are similar and left to the reader. The verification of
F2(r
C r
follows the s a m e p a t t e r n , by repeated use of 30.14.130.14.2 and lemma 30.1. As to the final claim, if X - F I(X),
r
r
r2(r
(ii) If X E M 1, trivially
r
- r U {UPI(X,o~ ) " o~ E O N } ) - U {r
o~ E O N } .
We check, by induction on c~:
r
- UP2(r
).
(***)
If c~ - 0, (***) holds by definition; if c~ is limit,
r
ol)) - r U { V P l ( X , t3 ) 9/3 < c~}) - U { r
-- U { U P 2 ( r
), ~) " fl < ~} - U P 2 ( r
/3 < ol} ~),
where IH is applied in the third step. If c ~ - fl-+-l, we get by definition
step (i)"
r
r
whence by IH r2(r
(iii) By definition of
) - r2(r
~))) - r 2 ( u P 2 ( r
d
and
~))) - UP2(r
~).
and essential use of all the properties of r Yl
30.16. T H E O R E M
(i) If r is an isomorphism of the combinatory algebra J~l onto the
combinatory algebra Jft~2, then r induces a complete lattice isomorphism
between FIX(Jr61) and FIX(.g&2).
(ii)
Under the same hypothesis of (i), the structures (.)tl~l, O(Ml~l) ) and
186
[Ch.7
On the Global Structure of Models
(.A~2, O(.AI~2)) are isomorphic.
PROOF. (i)" define r
{r
a E X} (X E FIX(~I~i) ).
Obviously r F I X ( ~ I ~ I ) ~ FIX(Jr
by lemma 30.15 (i) and r is bijective.
Since r
is an isomorphism between .)tt~2 and ~1~1, we have, with the
lemma above:
rl(r
r
)_ r
(for
Y E F I X ( ~ 2 ) ).
Therefore r
FIX(~I~I) and r is onto M 2. Since r is a bijection
between FIX(Jfl~l) and FIZ(~t~2) , r induces a complete lattice
isomorphism, as
r
I})-
U2{r
where El 1 - - s u p in FIX(.Zlbl); U 2 - s u p
lemma 30.15 (ii), we have:
r LJl{Xi" i E I}) -- r
E I},
(****)
in FIX(AI~2). Indeed, using the
U { X i " i E I})) -- UP2(r ( U { X i 9i E I})) --
- UP2(U {r
i E I}) -- U 2{r
i E I}.
(ii): straightforward corollary of lemma 30.15 and the recursive definition of
O(.A~I). n
w The sublattice of intrinsic fixed point models and the cardinality
theorem
Following Kripke (1975), we introduce an
consistent fixed point models. Intuitively,
of F, which "smoothly and consistently"
that it is compatible with every possible
model of OP-.
interesting subclass of the class of
an intrinsic model is a fixed point
grows from O(.Ab), in the sense
fixed point. Below, .Ah is a given
31.1. DEFINITION
(i) Let X E FIXcs(Al~): then X is intrinsic iff X is compatible with
every Y E FIXcs(Jfb ), i.e. for every Y E FIXcs(Jtb ), there exists some
Z E FIXcs(AI, ) with X, Y _< Z.
INT(.Alb) "- { X E FIXcs(J~ ) 9X is intrinsic}.
(ii) If X E CONS(.Ah), X is intrinsic iff X C_ Y, for some intrinsic
Y E FIXcs(Al~ ).
(iii) P "- P ( A h ) ' - U INT(AI~) ( = t h e maximum intrinsic fixed point,
by 31.3 (iii) below).
(iv) A consistent fixed point X is maximal iff X -
Y, whenever X _< Y
The 5ublattice of Intrinsic Fixed Point Models
VII.31]
and Y E
187
FIXcs(31~);
MAX(.AI~) "- { X E FIXcs(21b)" X is maximal}.
That the previous notions make sense, is granted by the following lemmata.
31.2. LEMMA. Every consistent fixed point can be extended to a maximal
consistent fixed point.
PROOF. Every <_-chain in FIXcs(.At,) has an upper bound by 30.13; hence
the existence of maximal elements of FIXcs(dtl~) is a consequence of Zorn's
lemma. F1
31.3. LEMMA
(i) If X C i
is consistent, intrinsic and F-dense (F-closed), then
U P ( X ) ( D O W N ( X ) ) is intrinsic.
(ii) X E INT(Jtl~) iff X < rq MAX(31~).
(iii) P(dtl~)- V1MAX(31~); hence X E INT(31~) iff X < P ( ~ ) .
PROOF. (i) By assumption and lemma 30.10.
(ii) O: by definition of maximality and M. ~ " let X _< V1MAX(31~) and
pick an arbitrary Z E FIXcs(31~). By lemma 31.2, Z can be extended to a
maximal element Z', whence X _< Z' and Z _< Z', i.e. X E INT(31~).
(iii) (U INT(Jtl~)) < M MAX(Jfb): by (ii) and definition of U, M.
V1MAX(31~) < (U INT(JII~))" MAX(31~) C_F I X c s ( ~ ) is non-empty by
lemma 31.2; hence we have n M A X C FIXcs(Jtl~) by 30.12(ii) and
V1MAX(31~) E I N T ( ~ ) by (ii) above, which implies
M MAX(.Ag) < U INT(.db).
[3
By lemma 31.3, we immediately derive:
31.4. THEOREM
I N T ( J ~ ) := (INT(.AI~), U, M, O(Jtl~), P(.kt,)) is a complete sublattice of
FIX(31~).
We now show that INT(JII~) is indeed very rich. To this aim, we
encode arbitrary subsets of the ground combinatory structure by means of
self-referential properties: as a by-product, we have a method for generating
plenty of intrinsic fixed points. Recalling that A x . ( x - x A-~(Jx A -,Jx)) is
an abbreviation for
)~x.AND(IDxx)(NEG ( A N D ( J x ) ( N E G ( J x ) ) ) ) ,
we
state:
188
On the Global Structure of Models
31.5. LEMMA. There exists a closed term J of s
then:
(i)
[Ch.7
such that, if ~ ]=OP-,
. ~ I=J = Ax.(x = x A - , ( J x A -~Jx));
.Ag l= J x = J y ---~x = y;
(ii)
if X E F I X c s , Xl=Yx(x~Tg~(x~?g Y x~g)) A Yx(-~x~g);
(iii)
O(~1~)I= Vx(-~xT/g).
PROOF.
(i): apply the paradoxical combinator to AyAx(x = x A--,(yx A-~(yx))). As
to the second claim, we apply the independence conditions of A N D and I D
(cf. 7.1.1): J x - J y implies Ix - x] - [ y -- y], whence x - y.
(ii): apply type-free comprehension 9.2 and consistency of X.
(iii): pick c~ minimal such that O(c~)]=ar/g, for some a E M, where O(c~) is
the c~th-approximation of O ( ~ ) . By definition of O(alg), c~ > 0. Then by
(i)-(ii), O ( c ~ ) [ : T ( a - a A - ~ ( ~ J a A Ja)), whence it follows, for some /3 < c~,
either O(Jtg,/Y) l : a r / J or O(J~,/3) i=a~J: but the former is impossible by
minimality of c~, while the second implies that O(J~) is inconsistent:
contradiction! [-1
The reason for introducing J is that J plays the role of a generic
property: if P C M, then there is an intrinsic fixed point, where J
represents P. The redundancy I x - x] in the choice of J is only needed to
make J injective.
31.6. DEFINITION
(i) Let P C_ M: then
J ( P ) "- {.J~(Ja)" a E P} U {.Jfg(--,--,Ja)" a E P} U
U { ~ ( - ~ ( J a A ~ga))" a E P} U {~l~(a - a)" a E P}.
(ii) G ( P ) "- U P ( J ( P ) ) .
If h i , . . . , a n E M, G ( a l , . . . , an) stands for G ( { a l , . . . , an}).
We collect a few simple facts about G.
31.7. LEMMA
(i) J ( P ) is F-dense, consistent and G(P) E INT(.AI~).
(ii) G ( P ) I-arlJ iff a E P;
(iii)
e ( P ) [--~Cl(J), whenever P C M .
G(0) - O(~1,) and P(.~)]= Vx(xrlJ).
VII.31]
(iv)
The 5ublattice of Intrinsic Fixed Point Models
189
If P C_ M is non-empty,
O(J~) < G(P) - G(P) M (G(P)) d < G(P) " (G(P)) d - G(P) d < D(.Ag).
P R O O F . (i) As to F-density, if .Al~(Ja) E J(P), we have by definition:
.Al~(--,--,Ja) E g(P) and .At~([a = h i ) E g ( P ) ; hence by l e m m a 30.1 and 31.5
Jfg(a = a A-,(--,ga A ga)) = .Ag(Ja) E r ( J ( P ) ) . If .]fi~(-~-,Ja) E J(P), also
alg(Ja) E J ( P ) , whence MI~(-~Ja) EF(J(P)). On the other hand, if
~ ( [ a = a]) E J(P), .Ag([a = a]) E F ( J ( P ) ) holds by definition of F.
If .]~(-~(Ja A-~Ja)) E J(P), then J~(--,--,Ja)E J(P), which implies, with
30.1, .Al~(--,(Ja A-~Ja)) E F ( J ( P ) ) .
As to the consistency of J(P), assume that there is an a E J(P), such that
(--,a) E g(P). Then we derive a contradiction, using the independence
properties of ID, A N D , N E G (see 7.1.1). There are sixteen cases to check,
but we only consider two typical instances, because the verification is
routine. Let a = .At,(--,--,Jb), ( ~ a ) = .At,(--,--,gc), for some b,c E P. Then we
have .]ft,(--,---,gb)= .At,(--,--,Jc), which implies .Al~(--,Jb)= .At~(Jc), whence, by
31.5:
.At~(--,Jb) - .Ab(c - c A--,(Jc A --,Jc)).
(1)
But (1) yields the identity .AI~(NEG) - .At~(AND), against 7.1.1 !
Let a - .Al~(Jb), ( - , a ) - .At,(--,(Jc A--,gc)), for some b,c E P. Then we have
.2~(Jb) - J~(Jc A ~gc), which implies by 31.5:
.Ab(b - b A--,(Jb A ~Jb)) - .Ab(Jc A ~Jc),
whence, again with 31.5 and the properties of AND:
.At~(b - b) - .Ab(c - c A--,(Jc A --,Jc)).
(2)
But (2) implies .At~(ID) - .At,(AND), against 7.1.1.
We now prove that G(P) is intrinsic. If Y E FIXcs(.At~), then
Yo "- y U g(P) is F-dense, as both sets are F-dense (use F-monotonicity).
Assume that Y0 is inconsistent; since Y is consistent, as well as J(P), there
is an a E Y with (--a) E J(P) (or vice versa). If (--a) - .At~(b - b), then
31~(-,b- b) E Y 9 contradiction! Let (--a) - Jtl~(Jb)" as (--,--,a) E Y, Yi=b-~ J,
against 31.5 (ii). If .Ale(--a) - .At,(--,--,Jb), we argue as in the preceding case.
If (--a) - .AI~(~(Jc A --,Jc)), a - ..~( gc A - , g c ) E Y, against the consistency
of Y! If a E J(P) and (--a) E Y, the argument is similar. It follows that Y0
is consistent and V P ( Y o ) E FIXcs(.flb ) (by lemma 30.10(iii)); moreover,
J(P) C_ G(P) <_ UP(Yo) and Y <_ UP(Yo) , and hence G(P) E INT(.Ag) by
definition 31.1.
(ii) If a E P, we have G(P)i=arIJ by definition of G. Conversely, assume
that G(P)l=arlg. By consistency and definition of G(P), we must have
190
[Ch.7
On the Global Structure of Models
.3[t~(Ja) E J(P). As AND, ID, N E G denote pairwise distinct elements of
.Al~, .~(Ja) - 31~(Jb), for some b E P; but 31.5 (i) implies a - b and a E P.
The second part is immediate by consistency of G(P) and the preceding
fact.
(iii) Trivially J ( 0 ) - q)C_ O ( ~ ) ; but G(q})is the least fixed point extending
J(q)), hence G(q})C_ O(Jtt~). As to the second part, note that by (ii)
G(M) ]= Vx(x~J) and G(M) C_P(.31~).
(iv) By consistency of G(P), G ( P ) i s strictly contained in G(P) d and
G(P) < G(P) d by lemma 30.12 (ii). Hence G(P)U (G(P)) d = G(P) d, which
also implies G(P) n G(P) d - G(P) (use 30.13 (i)).
P r q) implies O(dtl~) < G(P), whence G(P) d < B(~), again by 30.13. D
31.7.1. REMARK. (iv) shows that FIX(31~) is not even an ortholattice
with respect to d (see Birkhoff 1967, p. 52).
31.8. T H E O R E M (Embedding)
(i) Th~ map C" V ( M ) - . I N T ( J ~ ) i~ i~j~cti~ ~nd
G(U {Pi" i E I } ) -
U { e ( P i ) " i E I)}.
(ii) If X E INT(31~), let J ~ ( X ) " - {a E M" X ]=a~J}.
Th~n J~" INT(2~)--~ V ( i ) ~ a co.~pl~t~ taU,c~ ~pi.~o~ph,~.~, ,.~.
J~ is onto and
J~(UC)-
U{J^(X)'X~C};
J~(nC)-
n{J^(X)-X~C}.
PROOF. (i) If P r Q are distinct subsets of M, say a E P and a ~ Q, we
have, by 31.7 (ii), e(P)]=arlJ and G(Q)]=-,arlJ, whence e ( P ) ~ G(Q). On
the other hand, P C_ Q implies g(P) c_ J(Q), whence G(P) <_G(Q) (by
30.10 (i), 30.12 (ii)), and U {G(Pi) " i E I} C_G( U {Pi" i E I}). Conversely,
we show by induction on a, with Z : - U {P i" i E I}"
u n ( g ( z ) , ~ ) c u {G(P~). i c I}.
If a -- O, UP(J(Z), O) - J(Z) and
J(Z) C_ U {J(Pi)" i E I}
C_ U {G(Pi)" i E I}
C_UP( U {G(Pi)" i E I}).
If a is a limit, we apply IH; if a - fl+l and a E F(UP(J(Z),fl)), we easily
obtain a E Y "- U {G(Pi)" i E I} by IH and F-closure of Y.
(ii) If P C_M, J ^ ( G ( P ) ) - P and hence J^ is onto. Trivially, J^ preserves <__
and hence:
J^( V1r C_ M {J^(X)" X E r
U {J^(X)" X E r
C_ J^( U r
(,)
VII.31]
The 5ublattice of Intrinsic Fixed Point Models
Assume a E J^(X), for every X E r
191
then
J(a) :- {Jfl~(Ja), Jl~(-~-~Ja), .)l~([a- a]), Jl~(-~(Ja A-~Ja))} C X,
for every X E C ; thus G ( a ) _ [-1C, whence MCI-a71J , i.e. a E J ^ ( M C ) . If
a E g^( U C), we have UP( U C)]-a~lg, which implies JIl,(Ja) E U C, i.e.
JfI~(Ja) E X , for some X E C , and finally a E U { J ^ ( X ) : X E C } . This
completes the verification of the converse to (,). F1
The embedding theorem, together with 30.13 (iii), immediately yields:
31.9. COROLLARY
card(FIXcs(JIl~)) - card(FIXcp(.At,)) - card(FIX(Ml~)) - 2 card(~),
(where card(P)= the cardinal number of P).
The "thickness" of INT(.AI~) is dramatically exemplified by a "local"
version of the cardinality theorem: even if we consider an intrinsic fixed
point "quite close" to O(3t~) (see F below), we still get an interval of the
highest possible cardinality.
31.10. PROPOSITION. Let ~-(31~)'- M {G(P)" P C_M and P :/: 0). Then
card([O(Jfl~), F(.)I~)])- 2 card(M).
([X,Y] "- {V E F I X ( . ~ ) " X <_V <_Y} -closed interval determined by X,
Y).
PROOF.
Choose, by fixed point for operations, a term E such that
E - )~x.(x -- x A - , ( ~ E x A Vy-,(Jy))); then, if X E FIX(.At~),
x I= V ( uE + (x lE V 3y(y lJ))).
If P C_ M, the set
Ho(P ) :- {Jtl~(Ea)" a E P} U {Jfl~(-~-~Ea)" a E P} U
U {.Al~(a - a)'a E P} U { ~ ( - - (-1Ea A 'r
E P}
is consistent (by 7.1.1), F-dense, and intrinsic" if Q r 0, G(Q)i=3y(yr/J),
hence G(Q) i=ar/E, for every a E P, yielding Ho(P ) C_G(Q). Thus there
exists the _<-least H ( P ) E INT(.At~) with Ho(P ) C_H(P) by lemma 31.3,
and clearly H(P) C_G(Q). Since Q is an arbitrary non-empty subset of M,
we actually obtain H(P) _< F(Mt,). But if P r 0, then O(d~) r H(P) (recall
the inductive definition of UP and lemma 31.7). In addition, a ~ Q and
a E P imply H(Q)I=~arlE and H(P)[--ayE; in this step, we implicitly rely
on the fact that, by choice of E, Jft~I:Ea- E b - - , a - b. Hence the map
P H H(P) is injective. 17
192
On the Global Structure of Models
w
Variations on the encoding technique:
oddities
[Ch.7
non-modularity and other
A lattice L - (L, F1 U) is modular (see Birkhoff 1967, p . 1 3 ) i f f it satisfies
the condition:
xu(ynz)- (xuy)nz,
whenever x < z and x, y, z E L.
32.1. THEOREM. INT(J~) is non-modular.
PROOF. Fix four distinct elements d, a, b, c of M; this can be done since
Jtt~]=OP-. Then there exists a term V = Y(a,b,c) such that, for every
X E F I X cs(.AI~):
J ~ ] = V = )tu.(-~(-~Vu A (-~(Ja A Jb) A --,(Jb A Jc))));
(.)
Xi=Vu(u~IY~--~(u~lY V (a~lJ A b~lJ) Y (b~lJ A cT/J)));
(**)
x
(***)
(.)-(**) are clear; as to (***), X l=e-ffY (e arbitrary) would imply either
X I=a~g, or X ]=b~J, or else X I = @ J , contradicting lemma 31.5 (ii). Let
us consider the consistent intrinsic fixed points N O and N1, where
No=G(b,c) MG(a,b);
N 1 = G(b) U (e(c) [7 G(a, b)).
We show"
not N O C_ N 1.
(1)
G(b) < G(a,b) and G(b,c) - G(b)H G(c),
(2)
Since we know by 31.8 that
(1) will refute modularity of INT(JfI~). In order to check (1), we show
N OI= d~lY ( = V(a,b,c));
(3.1)
N1]= ~dTIV.
(3.2)
Ad (3.1): if No(a ) = the ath approximation of G(b,c) FlG(a,b)(see the
inductive definition of D O W N in 30.9), we check by transfinite induction:
Jfl~(Vd) E N0(a);
(3.1.1)
JII~(-~-~Vd) C No(~ ).
(3.1.2)
c~ = 0. Then No(0 ) = G(b,c) M G(a,b); by 31.7(ii), .Al~(Jb)E G(b)C_ G(b,c)
and .At~(Jc) E G(c) C_G(b,c), whence G(b,c) I=bTIJ A cTIJ, which implies by
(**) .At(Vd) E G(b,c). A similar argument yields .At~(Vd) E G(a,b).
Variations on the Encoding Technique
VII.32]
193
Moreover, alg(----1Vd) E G(b, c), since G(b, c) E FIX(.At).
If a is a limit, the conclusion is trivial by IH.
a = / 3 + 1 . By I n we assume .Ag(Vd)E No(/3 ) and .Ag(--,--,Vd)rlNo(t3). Then
by definition of F, we obtain (in the given order): .)~(--,--,Vd)E No(a ),
.Al,(Vd) E No(a ).
Ad (3.2). We check by transfinite induction on a:
.Al~(Vd) ~ N l ( a ) ,
where N l ( a ) - t h e a t h approximation of N 1 (see 30.9). The limit case is
trivial by IH. Let a - 0 .
Since NI(O)=G(b)U(G(c)MG(a,b)) , we have to
prove:
J (Vd) r a(b);
(3.2.1)
.A~(Vd) ~ G(c) [1G(a, b).
(3.2:2)
Ad (3.2.1). By contradiction, let .At~(Yd)E G(b) and choose a minimal such
that .At~(Yd)E G(b, a); then a = t3+1, for some Z, because .At(Yd) ~_G(b, O)
(we apply 7.1.1 to check that Yd =/=Je, for all e). {Here G(P,a)is a
temporary abbreviation for vP(g(P),o~), the a-th approximation of G ( P ) } .
Then by (,)
.Al~(~(~Vd A (--,(Ja A Jb) A -,(Jb A Jc))))
E G(b, ~+1).
either .AI~(--,--,Vd)EG(b,~) or
In the first case, we get
the minimality of c~ = ~+1. In
the second case, there exist ~, ~ < 13 such that:
By
definition
of
G(b,Z+I)
and
30.1,
.At~(~(--,(JaAJb) A~(JbAJc))) E G(b,~).
.At(Yd) E G(b, 7), for some 7 </~, against
either
.A6(ga), .At~(gb) E G(b,~)
Thus either G(b)]=arlg or
31.7): contradiction!
G(b)I=crlJ,
or
.At~(Jc), .At,(Jb) E G(b, ;).
which imply either a = b or b = c (by
Ad (3.2.2): assume .At~(Yd) E G(c) I-1G(a,b).
Then from G(c) MG(a, b) C_G(c) M G(a, b) we have .At(Yd) E G(c). If we
repeat the argument of (3.2.1) by replacing in it b with c, we still reach a
contradiction.
We conclude the verification of (3.2). Let a = 7 + l .
Assume by IH
.~(Yd) ~ N1(7) , and by contradiction:
.2s
E N1(7+1 ) =
r(N~(-r)).
(+)
Then we see that, for some 5 < 7, either .At~(Ja)E N1(5 ) or .At~(gc)E N l ( 6 ).
But .Al~(Ja)E N1(5 ) implies ~ = 0, i.e..At,(ga) E G(b) U(G(c)MG(a,b)).
Since .]fl~(ga) ~. G(b) (by 31.7 with a ~= b), we must have .Ab(Ja) E G(c), i.e.
a = c: contradiction. The case .]~(gc) E Nl(~i ) is similar. [-1
194
On the Global 5tructure of Models
[Ch.7
32.1.1. REMARK. Theorem 32.1 was stated by the author with a fake
argument in a preliminary version of this work; fortunately, P.Minari
(1987) found a nice proof of an analogous theorem for an infinitary
propositional logic of truth and a trick of his could be adapted to the
present framework; see also Visser (1984).
As a corollary, FIX(Jfi~) is non-modular, hence non-distributive. We
further indulge in self-referential constructions, in order to see that
I N T ( J ~ ) admits infinite strictly descending sequences (with respect to the
lattice ordering).
32.2. LEMMA. Let ~
be an w-standard model of OP and let
A D D ( f ) - FP()~y)~x [yx V f x]).
Then we can find an operation )~x.J(x) such that, for every n, a, c:
(i) ~ I= J ( 0 ) = J A Vn(J(n+l) = ADD(J(n)));
I= ~(J(n)a = J(n+l)c) A---,(J(n)a = ~ J ( n + l ) c ) .
(ii) O(Jlt~)I= Vx(~x~?J(n));
(iii)
if X E FIX~(JfG), X I= Vx(~x-~g(n)).
PROOF. (i) The first line is immediate by fixed point theorem
operations; the second part follows by induction on n~ using
independence properties of A N D , N E G (see 7.1.1).
(ii) By induction on n. If n = 0, J ( 0 ) = J and we apply 31.5. Let n and choose fl minimal such that O(.Al~,fl+l) l=a~?g(k+l ) ( O ( ~ , c ~ ) =
ath-approximation of O ( ~ ) ) . Since (i) implies:
for
the
k+l
the
.]fb I- g(k+l)a - (ADD(J(k)))a - [(ADD(J(k)))a Y (J(k))a],
then either for some 5 < fl, O(.)[b, 5)l=ariJ(k+l ) or O(.At~,5) l=a~J(k ). By
minimality of fl, O ( ~ , 5) I= m?J(k), whence O(~t~, fl)I=m?J(k), against IH.
(iii) Easy induction on n, with lemma 31.5 in the case n - 0. n
32.3. DEFINITION (i) If P C_ M, let J o ( P ) - J(P) (see 31.6);
Jn+l(P)-
{Jfb(J(n+l)a)" a C P ) U { ~ ( - , ~ ( J ( n + l ) a ) ) " a C P);
(ii) Go(P ) - G(P) and G n + I ( P ) - UP(Jn+I(P)).
32.4. THEOREM. Let ~1~ be an w-standard model of OP. Then, if P C_ M
is non-empty, we have, for every n C N"
(i) Gn(P ) C INT(Jft~) and O(~t~) < Gn(P);
(ii) Gn+I(P ) < Gn(P ) (here X < Y "- X ~_ Y and Y ~ X).
Variations on the Encoding Technique
VII.32]
195
PROOF. (i) Jn(P) is consistent by 7.1.1, and F-dense; hence Gn(P ) is a
consistent fixed point > O(MB), as P r 0 and Gn(P)[=aT1J(n) for a E P.
(ii) By construction, we have, for every a E P, Gn(P)]=a~g(n); hence, by
32.2, Gn(P)i=arlg(n+l), which implies Jn+l(P)C_ Gn(P ). Since G n + l ( P )
is the least fixed point _DJn+l(P), then Gn+l(P) <_Gn(P ). On the other
hand, if a E P,
Gn+I(P) I- - 7 a~TJ(n);
for Gn+l(P)]= a~g(n)implies that Jfg(J(n)a)is in Gn+l(P) at stage 0:
hence either Jfg(J(n)a) - .]f6(g(n+l)c) or .J~(g(n)a) - .]fg(~J(n+l)c), for
some c E P. This contradicts 32.2 (i).
Since Go(P ) is intrinsic by 31.7 and Go(P ) > Gn(P), Gn(P ) E INT(Jf[~). [7
A closer look into the structure of FIX(.Ag) can be gained through
Whitman's notion of splitting of a lattice (see Whitman 1944).
32.5. DEFINITION. Let X, Y be elements of F I X ( ~ ) :
(X, Y) is a
splitting
pair for FIX(atg) iff
(i)
[O(J~),X] U [Y,~(.AI~)] -
FIX(.lfg);
(ii)
[ 0 ( ~ ) , X] M [Y, D(~)] -
0;
(iii) X, Y are _<-incomparable.
32.6. THEOREM.
There exist at least card(M)-many splitting pairs for
FIX(..31~).
PROOF. Choose a E M and define
L(a)- U {H E F I X ( R ) " H l=-~ayJ ).
Then L(a)E FIX(alg) by 31.4; we claim that
pair. First, observe that
(L(a),G(a)l is a splitting
L(a) ]--,arlg.
(1)
If there were some H E FIX(.Yfg) such that G(a) < H < L(a), we should
have L(a) i=aTIg, against (1). Hence condition ( i i ) o f 32.5 holds for
(L(a), G ( a ) ) a n d also
G(a) ~ L(a).
Moreover
(2)
L(a)~ G(a)" indeed, assume by contradiction
L(a) < G(a).
(3)
Choose c :/= a and a closed term S such that S - $x.-~(-~Sx A -~Jc) holds in
.~. The set W ( c ) - {.]~(Sc)}U{.]~(-~-~Sc)} is r-dense, consistent (with
196
On the Global Structure of Models
[Ch.7
7.1.1), and hence there exists a consistent fixed point W " ( c ) - UP(W(c)).
Since Jtt~(Ja) ~ Jft~(Sc) and Jft~(Ja) ~ .AI~(-,--,Sc), we have:
W^(c) l--~aTlJ.
Hence W ^ ( c ) ~ L(a) and by (3), W^(c)~ G(a), i.e. G(a) i-crlS , which
implies either .At~(Jc) E G(a, 0) or .AI:(Sc) E G(a, 0) (as usual G(a, o~) - c~th
approximation of G(a)). The first possibility is excluded by the choice of c
and 31.7, while the second is impossible, as .At~(Sc) ~ J(a). Hence (3) is false
and condition (iii) of 32.5 holds. Condition (i) of definition 32.5 is fulfilled,
too" indeed, given any X E FIX(.Ag), either X J=arlJ and hence G ( a ) < X,
or X I=--,artJ and by definition X < L(a).
As to the verification of the cardinality condition, it is enough to observe
that, if [ O ( ~ ) , L(a)] - [O(Jl~), L(b)] and [G(a), g(.Al~)] - [G(b), n(.Al~)], then
G(a) - G(b) and hence a - b (by 31.7). [3
32.6.1. REMARK. (i) The argument is readily extended to INT(.A[~).
(ii) X stronger form of 32.6 is proven by n i n a r i (1987) for an infinitary
propositional logic of self-referential truth.
We conclude with a useful modification of G. An essential feature of the
map P ~ G ( P ) , is that it positively generates G(P); but it also follows from
31.5 (ii) that J is in general never total, and hence P cannot be handled as
a standard total object. This suggests a new map P ~ D(P), such that P is
represented by a class in D(P). Moreover, the operation D yields additional
cardinality information.
32.7. DEFINITION
(i)
First of all, let E X T be the closed term satisfying
I= E X T - )~x. (x - x A E X T x ) .
Clearly ~ [= E X T a - EXTb--+ a - b, for every a, b E M.
(ii) I f P C M ,
weput"
E X T ( P ) "- {.At~(EXTa)" a E P} U {.At~(~EXTa)" a ~ P} U
U {.Al~(a- a)" a E P}.
D(P) "- U P ( E X T ( P ) ) .
32.8. LEMMA
(i)
(ii)
O(Jtl~)I= Vx(~x~TEXT A --,x-ffE X T ) .
If P C_ M, D(P) is the least consistent non-intrinsic fixed
point D_E X T ( P ) , such that:
VII.32]
Variations on the Encoding Technique
D(P) I-CI(EXT);
D(P) I-a~EXT iff a E P, for every a E M.
(iii)
197
(*)
(**)
The map P ~ D ( P ) is injective.
P R O O F . (i) is a consequence of the inductive generation of O(~l~).
(ii) Clearly EXT(P)is F-dense. As to consistency, assume by contradiction
that a E EXT(P), (-~a)E EXT(P) with a = JIt~(EXTb)for some b E P,
(-~a)- .~(-~EXTc), for some c ~ P. Then ~ ( E X T b ) - - J ~ ( E X T c ) and
hence b - c by injectivity of EXT: contradiction! The remaining cases are
disposed of with the independence properties of AND, ID, NEG. It follows
that D(P)E FIXcs(P ). Moreover, D(P) is non-intrinsic, since D(P) is
incompatible with D ( M - P).
(,)-(**) are immediate by construction (injectivity of EXT is required for
checking D(P) ]=a~EXT ~ a E P).
(iii) If a E P and a~Q, D(Q) I=a-~EXT and D(P) I=mlEXT , hence
D(P) :/: D(Q). [3
32.9. T H E O R E M
card(iAX(~l~))- card(FIXcs(.llt~)- I N T ( ~ ) ) - 2card(M).
(ii) There are 2card(M) <-incomparable elements of F I X ( R ) , which
are neither complete nor consistent.
(i)
P R O O F . (i) Since every maximal consistent fixed point is non-intrinsic (by
31.3(iii)), it suffices to check that there are 2card(M)-many elements in
MAX(.)~). Now by lemma 31.2, for each D(P), there exists a consistent
fixed point D*(P)E MAX(J~), which extends D(P); furthermore, P :/: Q
implies D*(P) :/: D*(Q).
(ii) Let P be a subset of M with at least two elements and define I(P)"= U {D(a)" a E P}. Then I(P) is inconsistent: if a r b and a, b E P, then:
D(b) I=a~EXT and D(a) [=a~EXT,
which implies I(P) I=a-~EXT A a~EXT.
On the other hand, let L be such that Jfbi=L = [-~TL]. Assume by
contradiction that I(P) is complete, and consider the case . ~ ( L ) E I(P). If
I(P,a) = a-th approximation of I(P), we should have Jtl~(L)E I(P,a+l),
where a is minimal with respect to this property, and hence
~4t~(-~L)E I(P, a), i.e. dit~(L)E I(P, 7), for some 7 < a: contradiction !
A similar argument works for the extant case. It is also easy to check that
I(P) ~ I(Q) whenever P ~ Q. [1
198
w
On the Global Structure of Models
[Ch.7
A model for an impredicative extension of reflective truth
The systems we have been analysing so far, are basically intensional; also,
the abstraction process is more or less reduced to an inductive valuation
process, at least if we stick to the minimal model O(dil~). But it would be
conceivable to have, besides properties, sets(extensions), which can be
circumscribed by means of impredicative definitions.
We here assume that sets and properties coexist in the same universe; in
particular, we identify sets with certain canonical total properties, whose
specification is, so to speak, kept at a minimum of logical complexity.
According to these desiderata, operations and truth predicates ought to
apply to sets as well; moreover, extensional equality and the basic equality
of the given universe (which is a model of combinatory logic) should
coincide on sets.
33.1. D E F I N I T I O N
(i) We expand the language s of M F - to the langua.ge s
in which
there are new primitive individual constants ID, TR, N A T , NEG, AND,
ALL, V2 and a denumerable list of set variables X1, X 2 , . . . ; X, Y, Z will
be the corresponding syntactical variables.
(ii) We enlarge the notion of term by requiring that set variables are
individual terms.
(iii) We admit quantification on set variables: if A is a formula of s
V X A is a formula of .La.
(iv) We define the map A ~-~ [A] for arbitrary La-formulas by means of
the standard clauses and the new constants of (i) (cf. Ch. II, w appendix
III); in particular, we set [VXA] = V2(Ax.[A]) ).
Now the crucial question concerns set existence principles. The solution
we choose comes from a nominalistic interpretation of sets, suggested, for
instance, by Gilmore (1980, 1986). Under this interpretation, we must be
careful in mixing up too freely occurrences, in which set variables a r e - so to
s p e a k - m e n t i o n e d (as in the contexts XrlZ or X = x), with occurrences, in
which X is properly used as an extension (see ZrIX ).
This point of view leads to conjecture the consistency of the basic
theory of reflective truth with the principle stating the existence of a set
corresponding to any given condition A, provided A does not contain mixed
occurrences of set variables (mixed roughly means that no X can be used
and mentioned at the same time). In order to give a precise version of this
idea, we introduce analytical s
a sort of liberalized second order
formulas.
A Model for an Impredicative Extension
VII.33]
199
33.2. DEFINITION
(i) The class EF of rl-free formulas is the smallest class of formulas,
which is closed under classical sentential connectives, quantification on
either variable sort, and is generated from atoms of the form Nt, t - s,
where t, s are arbitrary terms.
(ii) The class ST of stratified formulas is the smallest class, which is
closed under classical sentential connectives, quantification on either
variable sort, and is generated from atoms of the form Nt, t - s , trlX ,
provided no set variable occurs free in t, s (for related notions, see
Feferman 1975, 1979; Cantini 1988).
(iii) A formula A of s
is analytical iff A is r/-free or stratified.
(iv) MFS- is the theory, which includes:
1. o e - and (finitely many), axioms, estab!ishing the independence of the
primitive constants ID, TR, NAT, NEG, AND, ALL, V2 (cf. Ch. II;
axiom LOG of Appendix III);
2. the truth axioms T.1-T.6 of 7.10 with ID, TR, NAT, NEG, AND, and
ALL replaced by ID, TR, NAT, NEG, AND, ALL (respectively);
3. the natural axiom relating T and set quantifiers:
TV 2
(TV2(f) ~-~V X T ( f X ) ) A (T-~V2(f)~-~ 3XT--,(fX));
4. the set principles SET.I-SET.3:
SET.1.
Cl(X);
SET.2.
X -- eY--~ X - Y
(extensionality);
SET.3. 3 Y ( Y - e { x " A}) ,
provided A is analytical (as usual, a - e b stands for Vu(urla ~ urlb)).
The logic underlying MFS- is simply the two-sorted version of classical
predicate logic. In particular, we include the generalization rule for
introducing VX and the schema V X A ~ A [ X "-Y] (Y free for X in A).
MFS is MFS- with the schema of N-induction for arbitrary s
(v) R - - 3 f ( f " V ~ S E T ) "there exists an operation establishing a
bijection between the universe V and the collection of sets".
The operation f of R cannot be the identity operation (otherwise everything
would be a class); in any case, there is at least a set X, which is left fixed
by f. We show that MFS + R is interpretable in the theory of suitable
models of the form D(S) over arbitrary combinatory algebras, D being the
operation of 32.8.
200
[Ch.7
On the Global Structure ot: Models
33.3. DEFINITION. E X T ( a ) : = A x . E X T ( a , x ) ,
term of the previous section.
where E X T
is the closed
33.4. LEMMA. Let J~ I=OP-. Then:
aig I = E X T ( a ) = EXT(b)~-~a = b (a, b arbitrary elements of M ) .
PROOF. Apply 32.7 and the pairing axioms. [3
Let S = {Sa: a E M } be a M-indexed family of subsets of M. S can
be encoded by a subset of M, also named S: define S := {afb((a,b)): b E Sa}
(where ( - , - ) is the term defining the pairing function).
33.5. LEMMA
(i)
I f S is a subset of M, which encodes the family {S a : a E M } and
.Ab [= OP-, then
D ( S ) I= V x C I ( E X T ( x ) ) and S a = {b E M : D ( S ) I=b~IEXT(a)}.
(ii) If S = {S a : a E M }
has no repetition (i.e. S a -- S b implies a - b),
then
D ( S ) I= E X T ( a ) = e EXT(b)---, E X T ( a ) = E X T ( b ) .
PROOF. By 32.8, definition of E X T ( a ) , assumption and 33.4. l-I
33.6. DEFINITION
(i) The language s + is obtained from s by omitting the predicate T.
s
are just the usual terms of OP (thus no set variable occurs in
s
s
have the form t = s, Nt, X t , where X is a set
variable and t, s, are s
s
are inductively generated
from atoms by means of -1, A and quantification over individual variables
and set variables.
(ii) OP 2 is the second-order extension of OP-, which is based on
classical predicate logic for the two-sorted language s + , and it includes the
full second-order comprehension schema CA:
3 X V u ( X u ~ A(u)) (A C s +, X not in A).
(iii) If ~ ]=OP-, .At~2 := (.Alo,~(M)), where 9~(M)is the power set of M;
s
is s +, expanded with constants a, b, c,... for elements of M and
with constants R, P, S, ... for elements of 9~(M). Thus Ra is an atom of
.L2+(M), provided a E M, R C_ M.
(iv) If S C_ ~(M), A is a sentence of s
(.Ate,S)I=A is inductively
defined according to the standard Tarskian clauses; in particular:
(NI~,S) I= V X A
iff (.Al~,S)]= A [ X := P], for P E S.
VII.33]
A Model For an Impredicative Extension
201
33.7. LEMMA
(i) There exists a family without repetitions S = {Sa:a E M}
subsets of M, such that, if .At~+ = ( ~ , S ) , then for every a, b E M
i= A(.. c. Sb) .ff
for every s
I=
of
A(.. c. Sb).
A(u, v, X) with the free variable shown.
(ii) ~ + ]=CA.
PROOF. This is a standard application of Skolem functions. Well-order
~P(M); then, if A ( u , X , Y ) E s +, there exists a function f A such that, if
a E M , R C_ M,
~ 2 1= 3YA(a, R, Y ) ~ A(a, R, FA(a , n)),
where F A is a new function symbol whose denotation is f A" Let H be the
Skolem hull of the empty set under the f A' s, i.e.
H o - ~, Hn+ 1 - {fA(a,R)" R E H n, A E s +, a E M};
H = U {Hn: n E w } .
Since card(H)=card(M), we pick a bijection F : M---~H and we set
S a -- F(a), S = { S a : a E M} and Ml~+ = (dI~,S). By construction, if R E S
and .AI~2 1= 3XA(a, R, X), then 31~2 I= A(a, R, Q), for some Q E S; by basic
model theory, we derive (i) and hence (ii). !-1
We now embed s
into s essentially by interpreting sets as the
denotations of terms having the form E X T ( x ) .
33.8. Inductive definition of the translation- of .fi`, into s
(i) Translation of s
(Xi)--- X2i and ( X i ) - - EXT(x2i+l ) ( for i E w);
if c -- ID (TR, N A T , NEG, AND, ALL), then ( c ) - - ID (TR,
N A T , NEG, AND, ALL in the given order; ID, TR, N A T ,
NEG, AND, ALL being the terms of 7.11); else, (c)- = c;
(V2)- = )~f.ALL()~xf(EXT(x)));
t
;
(ii) Translation of s
(Tt)- = T(t-);
(t = s ) - = ( t - = s-);
( N t ) - = N(t-);
(~A)-=--
(A A B ) - = ( A ) - A (B)-;
(A-);
(Vx~A)- = Vx2~A-;
202
On the Global Structure of Models
[Ch.7
(VXiA)- = Vz2i+l A-.
In practice, A - and t- are obtained 1) by relativizing set variables to the
range of Ax. EXT(x); 2) by replacing distinct variables of different sort by
distinct general variables; 3) by renaming bound variables in order to avoid
clash of free and bound variables. For instance, the term {x: 3Y(Y = x)}
becomes {x: 3y(EXT(y) = x)}.
33.9. T H E O R E M (Interpretation). We can find S C M such thai,
MFS F A(u,X), then D(S) I--A-(a, EXT(b)), for every a, b E M.
if
P R O O F . Choose S by lemma 33.7 and define D(S) as in w
First of all,
note that (R)-, ( S E T . l ) - , ( S E T . 2 ) - h o l d in D(S) by 33.4-33.5. As to
(TV2)- , we have, by definition o f - - t r a n s l a t i o n and renaming of bound
variables:
( T V 2 ( f ) ) - ~-+TV(Axf(EXT(x)))
VxTf(EXT(x))
(by T.5.1)
~-~(VXT(fX))The second part of (TV2)- is similar. Thus it remains to check:
D(S) I= 3yVx(xTIEXT(y) ~ A-(x, c, EXT(b))),
A(x, u,X) being analytical and c, b E M.
Case 1. A is r/-free. Then {a E M" D(S) [=A-(a, c, EXT(b))} - Sa, for some
a E M, as A-(u,c, E X T ( b ) ) i s a formula of s
and S is closed under
CA. Hence S a - {d E M" D(S) [=drlEXT(a)} and we are done.
Case 2. If A(x, Y) is stratified, we inductively define the transform A2(x , Y)
of A, as follows:
( t - s)2 - t - - s-; (Nt)2 - N ( t - ) ; (tT/Y)2 -- Y(t-);
the 2-transform commutes with -~, A, Vx and VX.
Then A2(x , Y) is a formula of s
A, that for every a, d E M:
and we can check, by induction on
.AI~+ I=A2(a, Sd) iff D(S) ]=A-(a, EXT(d)) (we use the notations of 33.8).
But .At + ]= CA and hence, for some c E M,
S c - {a E M" D(S)I=a~EXT(c)} - {a E M" .AI,+I=A2(a, Sd)},
which implies D ( S ) I = 3 y ( E X T ( y ) - e{x" A-(a, Sd)}), which is (SET.3)-. 11
33.10. COROLLARY. MFS + R is consistent.
We stress that MFS is a very experimental system, which proposes a blend
Kripke's Classification of 5elf-Referential Sentences
VII.34]
203
of extensional and intensional elements, and we do not know whether MFS
is stronger than full second order arithmetic. Of course, MFS has a few
closure properties, which are typical of impredicative systems. They are
quickly summarized below.
We say that S E T = {u: 3 Y ( Y = x)} is closed under a given n-ary
operation f, if MFS ~- 3 Y ( Y = e f X l ' " Xn)" We then define:
(i)
p w X := {u: 3Y(u = Y N X)} ( = weak power set of X);
(ii)
i g X R := {u: VZ(Progr(X, R, Z ) ~ urlZ)} ( = inductive generation).
Of course:
Y N X := {u: urlX A urlY},
Progr(X, R, Z) := V x ( x y X A Vy(yRx --, y~Z) --~ xyZ),
where yRx is a shorthand for (y,x)~iR; note also that
conditions of pw, S E T , ig are analytical.
the defining
33.11. P R O P O S I T I O N
(i) S E T contains the empty set and the universal set, is closed under
boolean operations and analytical comprehension:
MFS F 3 Y ( Y - e{u" A(x, u, X)} A Vv(v~Y ~ A(v, u, X))),
where A(x, u, Y) is analytical.
(ii) S E T is closed under weak power and inductive generation; in fact,
p w X and i g X R are classes, provably in MFS. Moreover:.
3Y(Y-
e S E T ) ( - there is the set of all sets);
Vx(xrlpwX - . 3 Y ( Y C_ X A Y - e x) A
A V Y ( Y C_X - ~ 3b(Cl(b) A b - eY A b~pwX)));
Progr(X, R, Z)-~ i g X R C Z A Progr(X, R, igXR).
Related principles have been investigated in the context of explicit
mathematics by Feferman (1979). As to the relation between classes and
sets, we remark that there are classes, which are provably not sets (e.g.
{u" 3 Y ( Y - u A-, u~Y)}).
w34. On Kripke's classification of serf-referential sentences
In the seminal Outline of a theory of truth of 1975, Kripke shows how to
use the fixed point semantics for classifying self-referential statements. We
extend K r i p k e ' s notions to the present framework.
On the Global Structure of Models
204
[Ch.7
34.1. DEFINITION. We fix ~ I=OP-; the definitions are uniform in .A~.
(i) If XEFIX(.AI~), X~a :-- X I--Ta V Fa; XTa :- not (X~a). If X~a
(XTa), we also say that X converges (diverges)on a, or X is (un-)defined
on a. If X E FIXcs(.AI~), Jf is the partial function defined by the clauses:
Jr(a) - 0 if X I= Ta and Jr(a) = 1 if X I= Fa.
(ii) An element a E M is grounded (it is a proposition, according to
chapter II), if O(.Al~)~a; a E M is ungrounded, otherwise.
(iii) An element a E M is paradoxical iff XTa, for every X c F I X cs(.AI~);
a is unparadoxical, otherwise.
(iv) An element a E M is intrinsic (or has an intrinsic truth value) iff
X~a, for some X C INT(.AI~).
(v) An element a E M is coherent iff for every X, Y E FIXcs(.]~ )
such that X~a and Y~a, J r ( a ) - Y(a) (i.e. wherever a is defined, a always
receives the same truth value), a C M is incoherent iff a is not coherent.
Gr(.A~) := {a E M : a grounded};
Intr(At~) : - {a E M : a intrinsic};
C o h e r ( ~ ) := {a E M : a coherent};
Paradox(.Al~) := {a E M : a paradoxical}.
34.2. REMARK. It is immediate to see that:
1)
2)
3)
if a ~ M - P F O R (cf. 30.1), then ( ~ a ) is grounded;
Gr(Jll~) C Intr(.Al~);
Intr(.A1~) U Paradox(.A1,) C Coher(Jtl~).
34.3. THEOREM
(Classification)
There exist paradoxical elements;
there exist elements of M which are ungrounded, incoherent,
unparadoxical;
(iii) there exist elements of M which are ungrounded, unparadoxical,
coherent, non-intrinsic;
(iv) there exists ungrounded intrinsic elements of M.
(i)
(ii)
PROOF. (i): pick the "Liar" term L with ~ I - L -[-~ TL].
(ii) Pick the term s such that ~ ] - s - [Ts], the so-called "truth-teller". s is
ungrounded. Indeed, if O(~1~)I=Fs, there would be a minimal such that
Ah(-~s) E O(Ah, a + l ) ; but Ah(-~s)= Ah(--Ts), whence At~(-,s)E O(Al~,a):
against minimality of a. The case O(Al~)I=Ts is similar. Consider
s + = {Jl~(s)} and s - = {Al~(--s)}; clearly either set is consistent and
Kripke's Classification of Self-Referential Sentences
VII.34]
205
F-dense. So there exist S +, S-E FIXcs(Al, ) such that S+I=Ts and
S-I = Fs: hence s is incoherent (and also non-intrinsic) and unparadoxical.
(iii) Let r - s V L, where s, L are the closed terms of (i)-(ii) above. Then r
is clearly ungrounded; also, using the above notation, S+]=Ts implies
S+]=T(s V L), hence r is unparadoxical, r is coherent" if X I=F(s Y L),
then X I=Fs A FL, and therefore X I=FL: absurd! Hence, if X~r,
X ( r ) - 0. r is non-intrinsic: if X I= Tr, then X I= Ts, hence S + < X. So a
consistent fixed point extending both X and S - would also extend S + and
S-, which is absurd.
(iv) We choose a closed term q such that .At~l=q--(--,(qA--,q)). q is
obviously ungrounded (usual reasoning on the inductive definition of
O(J~)). But q is unparadoxical: the set {Jig(q), .A~(---~q)} is trivially Fdense, consistent (once more apply 7.1.1) and there exists Z(q)E FIXcs(.At)
such that Z(q) I=T q. On the other hand, if Z E FIXcs(.A~), the set
X 1 - X U { J ~ ( q ) , J~(-~-~q)} is likewise F-dense and consistent (were X 1
inconsistent, J~(-~q)E X, whence X I=F q A Tq" against consistency of X).
Thus UP(X1)E FIXcs(.Ag), UP(X1)> X, Z(q), i.e. Z(q) E INT(J~), and
hence q is intrinsic. [3
34.3.1. REMARK. If X E MAX(Jtg) and a is intrinsic, then XJ.a. In fact, if
a is intrinsic, P ( ~ ) + a , but P(J~) < FI MAX(./[g) by 31.3 (iii). On the other
hand, if a is a coherent, non-intrinsic element of M, there may be maximal
consistent fixed points diverging on a: consider the fixed point S - in the
proof above. Then S - has maximal consistent extension S 1- (by 31.2) and
clearly S 1- [= ~ T(s Y L).
34.2 and 34.3 induce a corresponding classification on s
uniformly in any given Al, [=OP-. For instance, a sentence A E s is intrinsic
(paradoxical, etc.) iff ,AI,([A]) is intrinsic (paradoxical, etc.).
The classification of self-referential notions produces highly complex sets
(from the recursion-theoretic point of view). The moral is that "the truth is
never simple" (Burgess 1986). Let us recall a couple of definitions from
standard recursion theory.
A set X C_ M is E~(M) (resp. E12(M)) iff there exists a formula B(x) of the
language s
(see 33.6)such that B ( x ) - 3YA(Y,x) (3YVZA(Y, Z, x))
and 1) no set quantifier occurs in A; 2) X - {a E M" (.Ag,EP(M))I= B(a)}.
A set X C_ M is A](M) iff both X and its complement are El(M); X C_ M
is I I ~ ( M ) i f f the complement of X is El(M). Of course, similar
classifications are readily extended to families ~ C_ EP(M) and it makes sense
to speak of E](M)-families, etc.
34.4. LEMMA
(i)
FIXcs(A1,), FIXcp(atl,), F I X ( ~ ) are Al(atg ) families c_ ~(M);
On the Global Structure of Models
206
[Ch.7
(ii) MAX(J~) and INT(.AI~) are IIl(.At~).
PROOF. (i) By the characterization theorem 30.4, it suffices to show that
the relation (At, X)I = A is A], uniformly in X over At, which is well-known
(exercise or Moschovakis 1974).
(ii) Y E MAX(AI~) ~=~VX(X E FIXcs(./ft~) A Y C_X ~ Y = X), which is
IIl(.Ah) in a Al(.Al~)-notion by (i), hence II](.A~)tout court. Define
Compat(X,Y) iff X tO Y is consistent and F-dense. Clearly Compat(X, Y) is
an elementary relation and hence INT(.Jf~)is II](Al~), since we have:
Y E INT(.AI~)r
VX(X E FIXcs(.AI~)~ Compat(X, Y)). !"1
34.4 easily implies (with the notations of 34.1 and 13.3):
34.5. PROPOSITION
(i) Gr(Ah)E IND(.J~)-HYP(.]~);
(ii) Paradox(Al~)E II~(At);
(iii) Intrinsic(.Al~)E ~l(At~).
PROOF. (i): Gr(Al~)E IgD(.As since O(.AI~)E I g D ( ~ ) (apply 13.4 and
the trivial representability of O(Al~) by Tx). Were the complement -Gr(Al~)
of Gr(.At~) inductive, there should be a closed term t (by 15.2) such that:
a ~ Gr(d~) iff O(.At~)[=arit.
It follows with consistency of O ( ~ ) that:
-O(.A~) = {a E M: O(.At,) [-a~{x : x~t v T~x}},
whence, by 15.2, -O(Al~)E EgY(O(.Alt~)) C IND(AI~), which contradicts
theorem 13.4.
(ii): a E Paradox(.Al~)r VX(X E FIXcs(.At ) =:~ XTa).
(iii): a E Intrinsic(Ate)~=~ 3Y(Y E INT(31~) A Y~a). Fl
In the literature, sharp results are known about the logical complexity
of the collections of paradoxical, intrinsic, classes, in the special case of
models of reflective truth above the standard model of PA. The interested
reader is sent to the paper of Burgess (1986).
34.6. REMARK (Cardinality theorems again). Consider the formula
C(.Ah) " - " i f At I- OP-, card(FIXcs(.All~))- 2card(~)''.
Then C(.At) is lll(ZFC ) and hence, by Levy's absoluteness lemma, in order
to check that C(.~t) is true for every Ate, it is enough to verify C(At) for
arbitrary countable Ah. Let 3 t be countable: then by 34.4 and 13.4,
F I X cs(~ ) is a A~(d~)-family of subsets of M, containing an element
On coinduction principles
VII.35]
207
which is not A~(Mt~). Thus we can apply the "Perfect Set Theorem" (see
noschovakis 1974), and FIX~,(J~)is uncountable. {For the unexplained
notions, the reader is sent to Barwise 1975}.
w35. On the consistency of coinduction principles
Assume that A(x, v) is an operator of L, in the sense of 10.3. As we know
from chapter II, if we replace each occurrence of tTIv in A(x,v) by means of
the atom Pt (P fresh predicate symbol), we obtain a formula A(x,P) of
L ( ~ , P ) "-s
{P}, which is positive in P. In chapter III we proved
that there is a closed term I ( A ) - FP(Av. {x" A(x, v)}), which represents in
O(.A~) the least fixed point of the monotone operator:
FA(S ) - {a E M" (.Ate,O(.AI~), S ) l = A(a,P)} (here P is interpreted by S).
We now verify that I(A) defines in U(.At~) (see 30.13) the largest fixed point
of F A. Formally, we are led to introduce a generalized coinduction schema,
together with a generalized induction schema for the internal complement of
I(A).
35.1. D E F I N I T I O N
(i) Let B(x) be a formula of s
L(.AI~); we put:
and let a(x, v) be an operator of
DenseA(B ) "-- Yx(B(x)--, A(x,B));
GID ^ "-- nenseA(B ) --, Vx(B(x)--, xyI(A));
*GID ^ "- Vx(B(x)-, FA(x, B))--, Vx(B(x)-~ x~I(A));
*GID "- Vx(FA(x, B)-~ B(x)) ~ Vx(x-ff I(A) -~ B(x)).
GID ^ is the generalized coinduclion schema; for comparison, remind that the
generalized induction schema has the form:
GID "- ClosA(B ) -~ Vx(xrlI(A)-~ S(x)),
where ClosA(B ) .-- V x ( A ( x , B ) ~ B(x)).
(ii) In order to give a dual version of 12.4, let MFcs be the theory
N M F - + COMP, plus N-induction axiom for consistent properties:
Cs-N-IND "- Cs(a) A OrlaA Vx(xrla -~ (x+ 1)ya) -~ Vx(Nx -~ xrla),
where Cs(a) "- Vx(-- xrla V --,x-~a).
(iii) We simultaneously introduce the transformations
formulas (where 3, V are defined via V, A , --):
+, -
on L-
On the Global Structure of Models
208
[Ch.7
(A) + = A, if A is an atom;
(Tt)- = T-~t;
( A ) - - --A, if A - ( t -
(-~ B)+ = (B)-;
(-~ B)- - (B)+;
(VxB) + - - Vx(B)+;
(VxB)--3x(B)-;
(B A C) + --(B) + A (C)+;
(B A C)- - (B)-V (C)-.
s), Nt;
As a preparatory step, we need two simple lemmata:
35.2. LEMMA
(i) If X C M and X is consistent,
(Jt~, S)I = (a + ~ A) A (a ---,-1 a-).
(ii) If X C_M and X is complete,
(Jtl~, S)I= ( - ~ a - ~ A) A (A ~ A+).
(iii) If X C FIX(JII~), X I= (A + ~ TA) A ( A - ~ FA) (A arbitrary
PROOF. (i)-(iii): induction on A, using consistency and completeness of X
for A - T t . [-1
35.3. LEMMA. If A is an arbitrary sentence of L(.~I~),
O(.At~, c~+l) I=TA implies O(.Ai~,c~) I=A+;
O(~l~,a+l) I=FA implies O(Jtl~,a) I=A-.
g(Jlt~,a) [=A + implies 0(Jfl~,a+l)I=TA;
0(gll~,~)l=A-implies 0 ( ~ , ~ + I ) I = F A .
PROOF. This is a refinement of 12.1, which uses 30.1 and the fact that
{O(.At~,a)" a E ON} is a C_-increasing sequence of consistent sets, while
{O(.At~,c~).a C ON} is a C_-decreasing sequence of complete sets. V1
35.4. THEOREM
(i) I(Jll~)I=GID ^ and l(Jll~)I=*GID^;
(ii) O(.&)]=GID and O(.&) ]= *GID.
PROOF. (i) If i(.At~,a) -- the ath-approximation of i(.Ai~), we must check by
induction on a, assuming that I(.AI~)1=Densea(B):
{a E M- 0(Jtl~)I= B(a)} C_ 0(~1~,a).
Appendix
VII.A]
209
We only consider the successor case. If i(~l~,a+l)l--~arlI(A), we have
l(Al~,a) l=-~A(a,I(A)) by 3 5 . 2 ( i i ) a n d 35.3. But -~A(a,I(A)) depends
negatively on T and I(A): since l(dtl~) C_ l(Jl~, c~) we have I(~1~)1=7 A(a,B),
whence ! ( ~ ) I = - ~ B ( a ) . The verification of *GID ^ is similar.
(ii) First apply the induction theorem 12.4 for GID. As to *GID, let
-I(A) := {x: x~I(A)} and assume:
o(~)
I= 'r
B) ---,B(x));
( - I ( A ) ) ( a ) = {a E M : O(~1~, a) l= a~I(A)} C_ {a E M : O ( ~ ) I=B(a)}.
Then O(~l~,c~+l)l=a~I(A ) and 35.3 imply O(.AI~,a)I=A-(a,I(A)); since
A-(a,I(A)) depends positively on T and (-I(A)), we get with 35.2 (iii),
O(db)l=A-(a,B), which implies O(~1~)I= FA(a,B), i.e. O ( ~ I= B(a). El
The theorem can be easily transformed into a strengthened conservation
result:
35.5. COROLLARY
(i) P W c + GID + * G I D is conservative over OP;
(ii) MFcs + GID ^ + *GID ^ is a conservative extension of OP.
Appendix: a variant to the basic operator F and the restriction axiom
Up till now, we avoided a critical evaluation of a strongly conventional
assumption concerning the predicate T; indeed, the definition of F contains
the clause "infer T~x from --,PFOR(x)", with the consequent truth of the
restriction axiom RES:
(Tx----~PFOR(x)) A (~PFOR----~T~x).
We briefly explore a natural alternative to RES.
First of all, why restriction axioms at all? The basic reason, already
mentioned in Ch.II, 7.12, is semantically embodied in theorem 30.4: RES
makes M F - into a fixed point theory, and models of N M F - + RES form a
complete lattice with nice properties. A second point is that RES is self-dual
with respect to the transformation of T into -~ T-~ (see 8.11-8.12).
Can we imagine any reasonable variant to the operator F, with respect
to the restriction axiom RES ? In particular, the condition RES.2
Vx(-~PFOR(x)~T-~x) may appear unnatural. Therefore, we are going to
consider a new operator F- defined by the formula F - ( x , P ) which is
obtained from F ( x , P ) of 17 by omitting the clause corresponding to RES.2.
On the Global Structure of Models
210
[Ch.7
It turns out that the fixed points of F - over a given model of O P - are
axiomatized by M F - p l u s the restriction principle RES-:
Vx((Tx ~ P F O R ( x ) ) A ( T ~ x ~ PFOR(x))).
More precisely, as in 7.12, we can derive"
1. P R O P O S I T I O N . Let F P T - be the sentence Yx( T x ~ r - ( x , T ) ) ,
where
is obtained from the formula r-(z,P) by replacing every
subformula of Pt with Tt. Then M F - + R E S - a n d O P - + F P T - have the
same theorems.
F-(x,T)
Of course, M F - + R E S - and M F - + RES are incompatible and we are
left with a choice. In this respect, while we stress that both RES and R E S are not required for relevant theoretic developments, we maintain a
preference for the RES-axioms for the following reasons.
First, the alternative candidate theory M F - + R E S - does not enjoy the
simple duality between T and -1T-~. On the semantic side, it turns out that
the lattice-theoretic investigation of MF--models is slightly simpler in
presence of RES: with R E S - a n d F - o n e has to relativize the latticetheoretic operations to subsets of (the set defined) by P F O R .
Be that as it may, if we assume a neutral attitude, we can establish a
few conservation relations between MF-, M F - + RES, M F - + RES-.
2. P R O P O S I T I O N
(i)
M F - + RES b A ~ M F - + R E S - F A, for any T-negative A.
(ii)
M F - + R E S - F A =V M F - F A, for any T-positive A.
(iii)
M F - , M F - + RES, M F - + R E S - have the same T-free theorems.
P R O O F . (iii)is a consequence of (i)-(ii).
(i): let A be T-negative and assume that A is not provable in M F - + RES-.
Then for some structure ( ~ , X), (Ml~,X ) I = M F - + R E S - + ~ A. Hence by the
previous proposition, X is a fixed point of F - and hence, since
F - ( X ) C_ F(X), X is F-dense. Thus by lemma 30.10, there is a set X ^ such
that X ^ - F ( X ^) and X C_ X ^. By the characterization theorem 30.4, since
-1A is T-positive and hence upward persistent, we have
(.AI~,X^)I= M F - + RES + ~ A,
i.e. A is unprovable in M F - + RES.
(ii): observe that if X is a model of MF-, then X is F--closed. Then argue
as in (i), using downward persistence of T-negative formulas. D
VII.A]
Appendix
211
Of course, there is room for exercises: we can change M F - to systems
including forms of N-induction, the approximation axioms, generalized
induction. The point is that (iii) above remains true and it can be
established by proof theoretic means; hence we conclude:
RES, RES- do not affect the proof theoretic strength of the theory involved.
A possible addition to the basic M F - is suggested by considering an
essential feature of the usual truth predicates for formal languages: they
deal with inductively defined syntactical entities, i.e. sentences. So we
might replace the notion of pseudo-form P F O R by an inductively defined
subset, whose elements would play the role of traditional sentences. Then
we could try to define a modification of T, which only acts on sentence-like
objects (see Ch. II, for a related step).
This Page Intentionally Left Blank
PART D
LEVELS OF TRUTH AND PROOF THEORY
"F'~r jede mathematische Disziplin ist es charakteristisch, dass 1 ) f f i r sie
ein derartiger Operationsbereich zugrunde liegt, wie wit ihn bier von
A nfang an vorausgesetzt haben, dass diesem 2) stets die nat~rlirchen Zahlen
saint der sie verkn~pfenden Beziehung F assoziiert werden, und dass 3) ~ber
diesem kombinierten Operationsbereich dutch den ev.sogar beliebig oft
iterierten mathematischen Prozess ein Reich neuer idealer Gegenstande,
yon Mengen und funktionalen Zusammenhangen, aufgebaut wird."
(H.Weyl, 1918)
This Page Intentionally Left Blank
CHAPTER 8
LEVELS OF REFLECTIVE TRUTH
w
w
~38.
w
w
w
w
A language and axioms for reflective truth with levels
Simple consequences
Universes and the Weyl extended iteration principle
A recursion-theoretic model
Levels of truth and predicatively reducible subsystems
of second-order arithmetic
Consistency of a reducibility principle for classes
Levels of truth and impredicative subsystems of second-order
arithmetic
Appendix: on projectibility and stronger reflection
We present a new formal framework TLR ( = truth with levels and
reflection), in which the theory of reflective truth is enriched by the notion
of level. We will prove that TLR is able to internalize, to a certain extent,
negation and quantification over classes. We will also verify that the
resulting system is mathematically non-trivial and it yields a new
characterization of predicative mathematics (this will follow from chapters
IX and XI).
In the previous chapters, we have been pursuing a logical approach to
abstraction, which is based on a self-referential truth predicate T. However,
there appear severe limitations to the reflective power of T, even in presence
of the approximation structure, investigated in chapters III-V. For instance,
T cannot seriously think of itself to be consistent, without implying its
completeness, and hence its very inconsistency by the Tarski-Russell
arguments of 8.5-9.3. Nevertheless, it makes perfectly good sense to state
that T is consistent, once, say, the inductive model O ( ~ ) of Ch. II (w is
grasped as a whole. Therefore, we wonder whether we cannot design a new
formal framework, which can better adjust negative semantic information.
A starting point towards a reasonable solution is the remark that the
truth T is, after all, a parametric notion: it always depends on a set %0,
involving complete information about given primitive predicates, which can
also be regarded as the context T is about. As a consequence, T is really
T(%o) , for some 2;0; and if we regard T(%0) as grasped, we are actually
shifting f r o m the context %o to a new one %1, which also includes a
216
tevels of Truth
[Ch.8
complete description of T(%0) as primitive! This means that, if A is any
sentence and A ~ T(%0) (A E T(%0) respectively) holds, we must have
(-,ToA) E %a ((ToA) E %1, To being the formal counterpart of T(%o) ). Of
course, we mu~st add (-,ToA) to %1 and not simply (-~A): (--A) would
dangerously conceal its context dependence and this would drive us
immediately to contradiction.
These considerations naturally advice to make the parametric
dependence of truth explicit by means of levels: the shift from T(%0) to
T(%1) is regarded as a step to a higher stage of reflection, and, formally,
from truth of ground level T O to truth of higher level T 1. Moreover, since
the step from level 0 to level 1 can be understood as a general method to
produce new truth predicates from given ones, we may identify levels simply
with ordinals and devise a new formalism TLR, where T is accompanied by
level dependent truth predicates T i. Informally, we can summarize the basic
tenets behind T L R as follows.
1) If i, j are levels and i -< j (where -< is the precedence order on the
set of levels), T i and Tj will be related in such a way that: (a) whatever is
declared true by Ti, is declared true by Tj, i.e. Vz(Tix ~ Tjz); (b) T i is
decidable with respect to Tj, i.e. T j T i A or Tj--,TiA (A arbitrary; we
neglect formalization details).
2) Each local truth predicate T i satisfies the principles of the general
theory M F - o f reflective truth (Ch. II, 7.11).
3) We still maintain a level-free truth predicate T with us, and we
conceive it as the "limit" of the local truth predicates; in addition, we still
assume that T itself has the same self-referential abilities of any T i. To sum
up, T is a model of MF-, as well as each of its local approximations.
On the surface, we have restored a hierarchy of truth predicates, which
is strictly reminiscent of the Tarskian language/metalanguage hierarchy; it
might seem that we have destroyed the type-free style of the previous
systems. As a matter of fact, the new framework is quite distant from the
Tarskian one; in particular by 2) each T i already encompasses the standard
Tarskian predicates, as to closure properties and self-referential ability.
Furthermore, the level structure greatly strengthens the deductive force, as
it should appear from the summary of the results below.
In w we outline the new formal theory TLR. We mention that, in the
axiomatic approach, the level ordering is not assumed to be total
nor well-founded; it is also necessary to postulate an injection of levels into
objects, in order to codify sentences involving levels. This is an important
restriction for building models of TLR; it also requires non-trivial properties
of admissible ordinals (projectibility). w states a few basic facts about
VIII]
Introduction
217
predicate abstraction relative to any given level i; in particular we can
distinguish /-classes, i.e. predicates which are total relative to truth of level
i. w investigates the influence of the local structure on the closure
properties of level-free statements. It turns out that CL: = {x: x class}
splits into a directed family { C L i :i level}, where every C L i := {x: x class
of level i} is itself a class at any higher level j ~-i. As a consequence,
classes are closed under an analogue of Weyl's Iterationsprinzip (see Weyl
1918), a transfinite recursion principle along CL-wellfounded linear
orderings. We can also introduce, following Feferman (1982) and J/iger
(1984), a satisfactory notion of universe. w describes a model Ct for TLR,
which is built-up by means of a suitable iterated inductive definition along
the first recursively inaccessible ordinal. A byproduct of the effective nature
of Ct is that it validates a remarkable reducibility principle for classes RPC
(w
if an elementary condition with parameters in C L i is satisfied by a
class, then it is already satisfied by a j-class for some j ~ i. w and w
establish a link between second-order arithmetic Z2 and theories of reflective
truth with levels. After a brief survey of Z 2 and its subsystems, we prove
that TLR yields a model to Friedman's subsystem ATR0, while MF c and
MFp (see Ch.II) interpret suitable versions of hyperarithmetical analysis
(namely A1-CA0 and E]-DC 0 respectively). We shall later verify that TLR
and ATR 0 have exactly the same arithmetical content (ATR 0 is a strong
version of predicative analysis).
The final section shows that TLR plus RPC interprets impredicative
subsystems of Z 2 (namely II~-CA 0 and A1-CA0 ; cf. w
It is interesting to
mention that RPC dispenses TLR with the primitive notion of natural
number. We also stress that the interpretation r e s u l t s - coupled with wellknown theorems of Friedman-Simpson's reverse m a t h e m a t i c s - grant that
the mathematical content of TLR and related systems is significant for
mathematical practice. In the final appendix, we suggest the consistency of
TLR with reflection principles, stemming from higher recursion theory and
the related study of recursively large cardinals; this, however, is only a
research to come.
As to the connection with the literature, the idea of iterating the
abstraction procedure is implicitly involved in Ackermann's approach to
type-free logic, in Lorenzen and Myhill (1959, pp.47-49), Schiitte (1960) and
Fitch (1964); it is then made explicit by Scott (1975), though in a very
different context. More recently, we ought to mention Martin-LSf's idea of
adding "universes" (Martin-LSf 1984) and, in a classical set-theoretic
context, the theories of iterated admissibility (J/iger-Pohlers 1982). For
related ideas in the investigation of the so-called logical frameworks in
Theoretical Computer Science, we send the reader to Aczel-CarlisleMendler (1991) and to the final chapter. Finally, ideas connected with truth
218
Levels of Truth
[Ch.8
and levels can be found in the philosophical papers of Burge (1979) and
Gaifman (1983). A direct ancestor of TLR is outlined in Cantini (1987).
w36. A language and axioms for reflective truth with levels
The language s is the extension of the basic language 2. for reflective truth
(w which includes"
(i) a new sort of variables for levels i0, i l , . . . (in short L-variables);
(ii) a new unary function symbol LT;
(iii) three new binary predicates ~ , - t and Y (for level ordering,
level identity and local truth respectively).
The syntax of s requires the introduction of L(evel)-terms, besides terms
in the usual sense.
36.1. DEFINITION. L(evel)-terms, terms and formulas of s
(i) L-terms are exactly the L-variables (i,j,k metavariables for Lterms);
(ii) the set of s
is the least collection which is closed under the
following clauses: individual variables and constants are terms; if j is an
L-term, L T ( j ) i s a term; if t, s are terms, A p ( t , s ) i s a term.
(iii) the set of s
is the least collection closed under the
following clauses: if j and i are L-terms, i _ j i = lJ are atoms (and hence
formulas); if t, s are terms and i is an L-term, Nt, t = s, Tt and V(i, t) are
atoms (and hence formulas); if A, B are formulas, --A, A A B are formulas;
if A is a formula, x an individual variable and j is an L-variable, then VxA
and V j A are formulas (where x, j occur bound).
36.2. NOTATION. We stick to conventions and notations of Ch.I, w In
addition, we write Tit , trlis , t-~is , trls and Cli(t ) as abbreviations for V(i,t),
V(i,(st)), V(i, N E e ( s t ) ) , T ( s t ) a n d Vx(xrht V x-~it ) (in the given order). If
Cli(t ) is assumed, we say that t is a class of level i, or, simply, an/-class. If
i, j are L-terms, i = j
stands for i = l j
; we also write i - ~ j for
(~i - j ) A (i ~ j).
Before stating the axioms for the theory of truth,
abstraction operation of chapter II to the present context.
we adapt the
36.3. DEFINITION
(i) The set A + of acceptable formulas of s is the smallest collection,
which includes the atoms Tt, Tit , Nt, t - s
and is closed under
Reflective Truth with Levels
VIII.36]
219
negation, conjunction and universal quantification on object variables. Note
that Z-formulas are acceptable.
(ii) We assume the combinators ID, T R , N A T , N E G , A L L of w
Then we inductively extend the map A ~ [ A ] to arbitrary acceptable
formulas: we only mention the new clause (which clarifies the use of the
function symbol LT):
m
[Tit ] "- (7, (LT(i), t)).
If A is acceptable, T i A "- Ti[A ] and T A "- T[A].
(iii) The standard definition of ~-abstraction (see Ch. I, 1.1) is extended
by the new clause )~x.LT(i):= K ( L T ( i ) ) (where i is an arbitrary L-term).
As a consequence, it makes sense to introduce the abstraction operator for
arbitrary acceptable formulas by stipulating {x: A} : - )~x. [A].
36.4. The theory TL ( - t r u t h with levels) is based on two sorted classical
predicate calculus (indeed, s
has L-variables and individual variables).
The principles of TL are grouped into operational and number theoretic
axioms, local truth axioms, level and connection axioms.
36.4.1. Operational and number theoretic axioms. They include the standard
principles for basic combinators K and S, pairing, projections, zero,
successor, predecessor, definition by cases on natural numbers (cf. COMB,
PAIR, NAT.I-NAT.2 of Ch.I, 2.1), plus local number theoretic induction
LIND, i.e. N-induction for/-classes:
LIND
Cli(x ) A Closi(x ) ~ Vu(Nu --~ sT}ix),
where Closi(x ) "-Oyix A Vv(v~iix---. (v-t-1)yix). In addition, we require the
projectibility axiom
PROJ
ViVj(LT(i) - LT(j)~
i - - j).
36.4.2. Local truth axioms:
4.2.1
T i A ~ A , if A - (x - y), Nx, -~x - y, -~Nx;
4.2.2
T i x - ~ TiTix;
4.2.3.
T i-~-~x ~ T ix;
4.2.4
T i ( x A y)~-~ Tix A TiY;
4.2.5
T (Vy)
4.2.6.
-~(Tix A Ti-~x);
Ti-~x --, Ti~Tix;
Ti-~(x A y) ~ Ti-~x V Ti-~y;
(Local consistency).
Levels of Truth
220
[Ch.8
36.4.3. Level axioms: they include the standard equality axioms for level
equality - l, and state that ~ is a directed unbounded partial order:
4.3.1
ViVjVk((i ~_ i) A (i ~_ j A j ~_ k -~ i ~_ k) A
A (i __ j A j __ i - ~ i - - j));
4.3.2
ViVjSk(i -~ k A j -~ k).
36.4.4. Connection axioms. They are the crucial principles of the theory,
relating truth predicates of different level.
4.4.1
T x ~-~ 3iTix;
4.4.2
i ~ j A T ix ---, Tjx;
4.4.3
T i T x ~ Tix; Ti-~Tx ~ Ti-~x;
4.4.4
i -~ j ---, ( T j T i x V Tj-~Tix);
4.4.5
T j T ix ---, i ~ j A T ix;
4.4.6
Tj-~Tix ---, (i - j A Ti-~x ) V (i ~ j A-~Tix )
Limit
Persistence
Localization
Potential Completeness
Positive Soundness
Negative Soundness
As usual, T L - denotes TL without local N-induction. A word of comment.
By the principles of groups 36.4.2, and 36.4.4.5, 36.4.4.6, local N-induction
and soundness, it is obvious that the axioms of MFc( = the system of
reflective truth with class N-induction; see 10.7) are satisfied at every level.
On the other side, potential completeness ensures that negative information
about any level i becomes internal at higher levels, while limit and
localization axioms grant that global truth statements always reduce to
local truth statement (of sufficiently high level). Finally, by persistence and
soundness, no information is lost at later levels, and later levels do not
conflict with the earlier ones, even on negative information.
w
Simple consequences
We begin with a few elementary consequences of TL.
37.1. DEFINITION
(i) Let i be any L-variable: the i-transform of A C s is the Zv-formula
Ai, which results from A by substituting (each occurrence of atoms of the
form) T t by T i t (e.g. (VxT(ax)) i - VxTi(ax); ( T T t ) i - (T[Tt]) i - T i T t ).
(ii) An s
A is T-positive iff A belongs to the least collection
which contains expressions of the form t - s, - ~ t - s, Nt, -~Nt, T i t , -~Tit ,
VIII.37]
Simple Consequences
221
T t and is closed under conjunction, disjunction, and quantifiers (of either
sort).
(iii) A is an /-formula iff A belongs to the least collection of formulas
which is closed under A, -1, universal object quantification and contains
only atoms of the form t = s, Nt, Tit.
The first result allows to freely use/-transforms of MFc-theorems within the
system TL; it follows from local truth axioms and localization.
37.2. LEMMA. If MF c F- A, then TL F- A i.
37.2.1. APPLICATION. By the previous lemma and 9.6, we immmediately
have the following useful facts:
(i) If A ( u , x ) i s an E-formula, quasi-elementary in x (cf. II, 9.5),
T L - t- Cli(x ) ~ (Ai(u , x) ~ TiA(u , x)).
(ii) If A is a T-positive E-formula,
T L - t-- A i ~ T iA.
(iii) If A(u,x) is an Z-formula, elementary in x,
T L - ~ Cli(x ) ~ T i A ( u , x ) V FiA(u,x ).
37.3. LEMMA. (i) TL-proves:
T.1
TArA
, if A = (x - y), Nx, -~x = y, -~Nx;
T.2
T T x ~ Tx;
T.3
T-~-~x ~ Tx;
T.4
T(. ^
T.5
T(Vf)--, V x T ( f x ) ;
T.6
-,(TxAT~x).
T ~ T x ~ T~x;
T . ^ Ty;
(ii) A+-soundness:
^
T-. V T-y;
3xT-~(fx) ~ T~(Vf);
T L - ~ T i A - , A (A E A+).
(iii) TL-F- T i A - ~ Ai (A E A+).
(iv) T L - t- i _~ j A T i A -~ T j A (A E A+).
(v) If A is a k-formula,
T L - F- k ~ j ~ ( T j A V Tj-~A) A ( T j A ~ A);
T L - ~ ( T A ~ A) A ( T A V T-~A).
Levels of Truth
222
[Ch.8
PROOF. (i): by limit, persistence, localization and local truth axioms,
together with directedness of the level ordering. As a sample, let us check
global consistency T.6. If Tx and T~x are assumed, then by limit axiom
T ix and T k-~X, for some i, k; hence, there exists some j ~- i, k, such that by
persistence T jx and T j~x, against local consistency.
(ii): induction on A. If A is an e-atom or has the form Tit, -~Tjt, we apply
36.4.2.1, positive and negative soundness and local consistency. Let
A := -~Tt and assume Ti-~Tt; then Ti~t by localization, hence T-~t by limit
and -~Tt by T.6 above. The remaining cases are straightforward by IH and
local truth axioms.
(iii): exercise (induction on A).
(iv): apply persistence axiom.
(v): potential completeness and A+-soundness imply the first statement,
which yields the second by limit, persistence and unboundedness of the level
ordering. D
We proceed by considering simple facts about abstraction, classes and
/-classes. In particular, we see that the notion of/-class determines a class at
any level j ~- i. Moreover, the Russell sentence relativized to level i becomes
true at strictly higher levels. As usual, we let:
37.4. DEFINITION.
CL := {x: Cl(x)};
CL i := {x: Cli(x)};
R := {x: ~xrix};
R(i) := {x: ~xriix};
x r i y := (Tix ~ TiY ) A (Ti-~x ~ Ti~Y);
x r y := (Tx +-~Ty) A (Fx ~ Fy).
37.5. LEMMA
(i)
The extended abstraction schema for acceptable formulas: for every
A E h +,
T L - F Vu(u~{x: A(x)} r A[x := u]).
(ii)
The local abstraction schema for acceptable formulas: /f A E A +,
T L - F ViVu(uy{x : A} r
(iii)
A[x := u]).
/f A(x) is a j-formula, TL-proves:
j -< i ~ Vu(u~i{x: A} ~ A[x := u]),
(where u free for x in A in (i)-(iii) above).
(iv) T L - F Vi~Cli(R ).
(v) T L - F Vi(i ~- j--,Cli(R(j)) A-~Clj(R(j)) A R(j)~iR(j)). Hence:
Simple Consequences
VIII.37]
223
T L - F ViCl(R(i)).
(vi) T L - t - i - ~ k ~ C L
i rlk CL k A CL i C CL k.
PROOF. (i): trivial by fl-conversion and T.2 of 37.3 (i).
(it): immediate by/~-conversion and localization axioms.
(iii): by (it) and lemma 37.3 (v).
(iv): by lemma 37.2 and 9.3.
(v). Let i ~- j. As to the first conjunct, R ( j ) is defined by a j-formula and
hence we apply 37.3 (v) and local abstraction. The second conjunct is simply
Russell rephrased for level j. The third conjunct is a consequence of the
second one with (iii). The last statement follows by unboundedness of -~,
limit, persistence.
(vi). Assume k ~-i: then C L i C CL k by persistence and (v). As to
CLirlkCLk, apply local abstraction and 37.3 (v). 0
37.6. LEMMA.
9.5 (i)). Then:
Let A(u,x) be an L-formula elementary in x (see definition
(i)
T L - F Clk(x ) ---, (A(u, x) ~ Ak(U , x) ~ TkA(U , x)).
(it)
Closure of CL k under elementary comprehension:
T L - F C l k ( x ) ~ ( C l k ( { u : A(u,x)}) A Vv(vrl{u: A(u,x)} ~ A(v,x))).
(iii)
Closure of CL k under join:
T L - F (Clk(X) A f : x ---. C L k ) ~ Clk(~E(x , f ) ) A
A
3v3 (
=
A v,7 A
PROOF. (i) Assume that x is a k-class. The second equivalence holds by
37.2.1. We check the first equivalence by induction on A. If A is an atom
different from u~x, we are done by 36.4.2.1. If A ( u , x ) = urlx , urlkx implies
urlx by limit axiom. In the opposite direction, we easily reach a
contradiction from u~x and ~UrlkX (apply Clk(X), persistence, limit,
unboundedness and local consistency). If A(u, x ) i s a negation, a conjunction
or a universal quantification, we simply apply IH.
(it) If x is a k-class, so is {u: A(u,x)} by 37.2.1 (iii). The second equivalence
is an immediate consequence of (i) and 37.5 (iii).
(iii): by lemma 37.2 applied to the join principle 9.9 and (i) above. O
Up to this point, we have seen that the local approximations of T
satisfy the basic axioms for self-referential truth, but it is not clear whether
this happens for T itself, and hence whether there is a harmony between
global and local structure. The time is ripe to introduce a simple reflection
axiom which grants such correspondence.
Levels of Truth
224
[Ch.8
37.7. DEFINITION
(i)
The Reflection principle REF"
ViVyVz(Vx3j(x~liy ~ x ~ l j z ) ~ 3 k V x 3 j ( j ~ k A (x~liy--,x~ljz))).
(ii) T L R - - TL + REF and T L R - " - TLR minus class N-induction.
The reflection principle implies that there are enough levels for T, in order
to internalize universal statements about objects, and it is essential for
showing that MF c is a subtheory of TLR.
37.8. LEMMA
(i)
Positive reflection:
T L R - F V x 3 i T i ( f x ) ~ 3 i V x T i ( f x);
(ii) T L R - F V x T ( f x ) - ~ T(Vf);
(iii)
if A is acceptable and A is T-positive,
T L R - F A ~ 3i.T~A ~ T A ~ 3iA~ ;
(iv) /f
T-po it,v
acc ptabl ,
T L R - F Vu(uTl{x:A} ~-~ A[x "- u]);
(v) T L R - F C l ( a ) ~ 3iCli(a);
(vi)
if a C_ b "- Vx(x~Ta--, xob),
T L R - F a C_ b ~ Vi3kVx(x~lia ~ X~kb);
(vii)
a class of classes is always an i-class, for some level i:
T L R - F Cl(a) A a C C L ~ 3i. a C C L i.
PROOF. (i): apply reflection with y = { u : u - u) and persistence.
(ii): apply limit and positive reflection (i).
(iii) Let us consider the first equivalence. From right to left, it follows from
A-~-soundness (lemma 37.3). As to the reverse direction, we argue by
induction on A.
If A - - . T i t , choose k ~-j by unboundedness of -~" then T k - . T j t by
potential completeness, positive and negative soundness. If A := VxB, we
use IH, positive reflection and the local truth axiom for V. The other cases
are easy and left as exercises. The second equivalence is just a restatement
of the limit axiom. As to the third equivalence, T A ~ 3 i T i A ~ 3 i A
i (use
lemma 37.3 (iii)). Ai--, T A is inductively checked ((ii) above being used in
the case A := VxB).
Universes and Weyl's Principle
VIII.38]
225
(iv): by (iii) above.
(v): apply the limit axiom from right to left. The reverse direction is a
consequence of (iii), as the formula defining Cl is acceptable and T-positive.
(vi):=~ by limit and reflection; the converse is trivial.
(vii): assume that a is a class of classes. Then by (v), a is an/-class and
a C_ CL :v VjSkVx(x~lja --->xTIkCL) by (vi);
:~ Vj3kVx(x~lja-->Clk(X)), by localization and local abstraction;
:v 3kVx(x~lia --->Clk(x)) by logic;
==~a C_ CL k for some k,
as x~la ~ xTlia, by assumption on a and global consistency (37.3, T.6). O
37.9. T H E O R E M . If MF c F A, then T L R F A (A E s
P R O O F . By l e m m a t a 37.3 and 37.8, it only remains to check:
T L R F Cl(x)---. (Clos(x)---,N C x).
Since x is an/-class for some i, C l o s ( x ) ~ Closi(x ) by 37.6 (i) and hence we
can apply local class N-induction. 0
Of course, one may wonder whether the set of provable level-free
statements is really affected by the level structure, that is: what more do we
know about T within TL and TLR? Precisely this question is faced in the
next section.
w38. Universes and the Weyl extended iteration principle
We present two significant level-free statements, whose verification
essentially relies upon the level structure. The first principle roughly says
that, as soon as we deal with logical constructions depending on classes as
initial data, then we can always work within a nicely closed universe, which
is itself a class of classes and to which the initial data belong. To make this
idea precise, a few definitions are in order.
38.1. D E F I N I T I O N
(i) y I=J "- V/Vc((c
v ^ I"
f) v ^
A Vu(url2E(c , f)~-+ 3vqw(u -- (v, w) A vrlc A w~7(fv)))));
yl= J states that the join principle of 9.9 holds relativized to y.
tevels of Truth
226
[Ch.8
(ii) We say that y is elementarily closed iff y contains the singleton {x}
for arbitrary x, the classes ~ e e := { ( . . y . z ) : . y = z). u~):= { ( . . y ) : 9 = y}.
N := {x: N x } and is closed under intersection, complement and domain,
plus the combinatorial operations of expansion, converse, cycle, transpose
(see 9.11 and 9.13 in Ch.II).
In symbols, let Elemclos(y) be the universal closure of the conjunction of
the following two conditions:
EL.1
/~PP~y A Nrly A 0~)rly A Vz({z}rly);
EL.2
V u W ( u , y A v , y ~ ((u n v),y A ( - ~ ) , y A
dom(u),y
A
A Exp(u)rly A Cyc(u)rly A Trans(u)rly A Conv(u)rly ).
(iii) y is a universe of classes iff y is an elementarily closed class of
classes, which is also closed under join; in symbols:
Univ(y) := Cl(y) A y C_CL A Elemclos(y) A y I - J .
38.2. THEOREM
(i) TL-~-Vk. Univ(CLk).
(ii) T L R - ~- V y ( U n i v ( y ) ~ 3k(y C_e l k ) .
PROOF. (i) That CL k is closed under join and elementary operations
already follows from 37.6. CL k C CL and CLkrlCL are consequences of
persistence and lemma 37.5 (vi). (ii): immediate by lemma 37.8 (vii). El
38.3. COROLLARY. Let LIM := Vx(Cl(x)---~ 3y(Univ(y)A xrly)); then
T L R - F LIM.
PROOF. If x is a class, x is already a k-class (lemma 37.8), for some k and
CL k is a universe by the theorem. F1
38.3.1. REMARK. (i) LIM is stated in J~iger(1984a) (but see also Feferman
1982) in the context of a theory of total classes; LIM is false in O ( ~ ) .
(ii) Each CL k is closed under the basic type constructors of Martin-LSf's
type theory, and Martin-LSf's intuitionistic type theory with arbitrarily
finite universes without W-types can be interpreted in the theory TLR.
The second principle we deal with, extends to the present non-settheoretic framework the familiar transfinite recursion schema over
wellorderings. However, it is a priori unclear how to render the notion of
wellordering in the present context: shall we quantify over classes or
arbitrary partial predicates? We anticipate that the two alternatives yield
Universes and Weyl's Principle
VIII.38]
227
radically different notions and that the sharpest notion is obtained by
quantifying over classes. This point will be clarified in the next chapter and
with the help of proof-theoretic analysis.
38.4. DEFINITION
(i) Let w encode a binary relation, i.e. let w be a property of ordered
pairs. We keep using the infix notation x -< w Y in place of (x, y)~lW.
Field( ~ w ) i s the term {x" 3 z ( x - < w Z V z-<wX)} representing the field of
w, while the x-segment of "~w determined by x is defined by the term
"< ~F~ " - { u . u -< ~ ) .
(ii) LO( ~ w )
"-
:= wvyvz(-~(~ -< ~ ~) ^ (~ -< ~ y ^ y -< ~ z ~ ~ -< ~ z) ^ C o n ~ ( -< ~ )),
where Conn( -< w) is the formula
VxVy(x~?Field( -~ w) A y~?Field( ~ w ) ~ (x -~ w Y V x - y V y ~ w x)).
Clearly LO( ~ w) states that -~ w is a linear ordering.
(iii) Progr( -< w, b) "- (Vx~lField( -~ w))(Vy -< w x. y~lb ~ x~lb).
Progr(-< w , b ) i s to be read "b is progressive (relative to -< w)"" We also
define"
T I ( -< ~, b ) . -
P~og~ ( -< ~, b)--, F i e l d ( -< ~) c b.
(iv) A linear ordering < w is called a pseudo-well-ordering-in symbols
P W O ( ~ w), and, in short, -<: w is a p w o - iff Vb(Cl(b)---, T I ( -~ w, b)).
(v) Let A(u, x, y, z) be a formula with the free variables shown:
T R ( y , A, -< w, Z) "- VxVu(xqField( -< w) ---, (uTly(x) ~-~A(u, x, y[x, z))),
where y ( x ) " - {v" (x, v)qy} and y[x "- {(u, v)" u -< w x A v~ly(u)}.
38.4.1. REMARK. (i) The notion of pwo is deserved in the literature to
ttYP-wellfounded linear orderings of natural numbers (see Harrison 1968,
Friedman 1976). We maintain this terminology for a more general
situation, because of the analogy betwen hyperarithmetical sets and CL.
(ii) T R ( y , A , - ~ w,Z) roughly says that y encodes a sequence of predicates
{Yx} indexed by elements in -< w-order; each Yx is recursively computed by
application of the functional a H {u: A ( u , x , a , z ) } to the collection (encoded
by) y[x of previously defined predicates.
Now the main question is under which conditions there exists a class y
Leve]s of Truth
228
[Ch.8
satisfying T R ( - , A, -~ 9, z). If the given z is a class, "49 is a pwo and A is
elementary extensional in the relevant parameters, the answer is positive
and essentially requires the level structure of TL. For the reader's sake, we
recall from 20.9 that a formula A of s is elementary extensional in the list
Xl, . . . , x n iff A belongs to the least class of formulas inductively generated
by means of A, --, Vy (y distinct from Xl, . . . , x n ) , from atoms of the form
t - s, Nt, trlxi, provided Xl, . . . , x n do not occur in t, s. Then we know:
38.5. F A C T . If A ( u , x , y )
is elementary extensional in x, y and - e is
extensional equality with respect to rI (see 9.11), we can prove in pure logic:
A(u, x, y) A x - e x ' A y - -
e y'---, A ( u , x ' , y ' ) 9
38.6. T H E O R E M . Let A ( u , x , y , z ) be an s
elementary extensional
in y, z with the free variables shown. Then we have, provably in T L R - :
(i)
Cl( -4 w) A P W O ( -~ 9) A C l ( z ) ~ 3y(Cl(y) A TR(y, A, -4 w, z)).
(ii) Uniqueness: if y and y' are two classes satisfying T R ( - , A , -4 9, z),
then y and y' are pointwise extensionally equivalent, i.e. T L R - proves
[CI( -~ w) A P W O ( -4 9) A Cl(z) A T R ( y , A, -~ 9, z) A TR(y', A, -4 w, z) A
Cl(y) A Cl(y')] ~ Vx(xrlField( -~ 9) --* (y(x) = ~ y'(x))).
P R O O F . (i) Existence. Put gxzy " - { u " A ( u , x , y [ x , z ) } . Then by the fixed
point for operations we can find a term RC[g, -4 w] such that
RC[g, -4 w]zx - gxzE( -~ 9Ix, Au.RC[g, -~ 9]zu).
(1)
Also, if z and -4 w are classes, then z, -~ 9 and Field( -~ 9) are k-classes for
k large enough (by l e m m a 37.8 (v), 36.4.3.2, 37.5 (vi)).
Let us consider:
d "- {x" X~kField ( -~ w)A Clk(RC[g, -~ w]zx)}.
If we choose j ~- k, d is a j-class (its defining condition being a k-formula,
see 37.3(v)). Hence d is a class and we can apply induction on -~ w"
Assume x~Field( ~ w) and Vy ~ w x. yrld : then by 37.5 (iii), RC[g, -~ w]zy
is a k-class, for each y -~ w x. Hence by closure of CL k under join (37.6), the
term t : = E(-4 w[x, Au.RC[g,-~w]ZU) is a k-class and so is t[x. Since
A ( u , x , y , z ) is elementary in y and z and CL k is closed under elementary
comprehension, g x z t - R C [ g , - ~ w ] Z X is a k-class, which implies Xrljd ,
whence xrld. Therefore the class d is -~ w-progressive and we can conclude
that RC[g, -4 w]ZX is a k-class for every x in the field of -~ w, whence, again
by join, RC[A,-4 w,Z] := E(b, Au. RC[g,-4 w]ZU)is a k-class (where b is
Field(-~ w))" If x is in the field of ~ w, we have, with 37.6 and the
extensionality property of A:
Universes and Weyl's Principle
VIII.38]
229
u~?RC[A, -.< w, z](x) ~ u~RC[g, -.<w]ZX
,,,lg~z(~( ~ wF~, ~u.RC[g, ~ jzu))
A(u, x, ~( -.<w[X, )~u.RC[g, -.<wlZU)[X, z)
A(u, x, RC[A, -.<w, z]Fx, z).
{In the last step, we use the fact that if x is in b := Field( -< w),
(~, u),7(~( ~ wF~, ~u.nC[g, -~ ~]z~))r~ ~ (v, u>~(r~(b, ~u.RC[g, -~ ~]zu))[~).
In conclusion, we proved that the term RC[A,-< w,z] represents a class
satisfying T R ( - , A, -< w, z).
(ii) The uniqueness (modulo extensional equivalence) follows by applying
transfinite induction to S(x):-x~?Field(-<w)--,Vu(u~?y(x)+-~u~?y'(x)).
(Note that {u: B(u)} is a class if y, y' are classes and x~Field( -.<w))" [3
38.6.1. REMARK. An axiom, which requires the existence of an w-sequence
of properties, obtained by iterating a given predicative operation (here the
map x~-~{u: A(u,x,y[x,z)}), is clearly stated in Weyl's Das Kontinuum
(Weyl 1918, "Iterationsprinzip", p.27). Since the schema embodied in 38.6
can be naturally regarded as an extension of the Iterationsprinzip to pwos,
we name it WP( = Weyl's principle).
38.6.2. APPLICATION (see 40.5). Interpret ( - , - ) as a number-theoretic
pairing operation (i.e. an injection of N x N into N) and assume that (1) the
parameters "<w and z in the statement of WP are subclasses of N; (2)
{u: A(u,x,b[x,z)} C_N, whenever x~TField(-< w), z, y[x C_N (A elementary
extensional in b,z). Then we can find a subclass y of N such that
TR(y, A, -.<w, z).
Verification. First observe that, under the closure assumption of N with
respect to ( - , - / , the following version of join for numbers holds:
Cl(c) ^ Vu(u,~c ~ c l ( f u ) ^ fu c N) --+cl(~(c, f)) ^ ~(c, f) c_ N.
(.)
Then for gxzb "- {u" A(u, x, bWx,z)}"
Vx(x~?Field( -.< w)--* nC[g, -.<w]zx C N).
(**)
(**) readily follows by -<w-transfinite induction with (,) (apply the
hypothesis on A, the fact that "<w is a pwo, and N, RC[g,-< w]ZX are
classes if x is in the field of "<w)" Again by (,) we conclude that
b ' - ~ ( F i e l d ( - ~ w),)~u.RC[g,-<w]zu) is a subclass of N, which satisfies
T R ( - , A, -< w, Z). D
230
Levels of Truth
[Ch.8
w39. A recursion-theoretic model
We do not know yet whether our theory of self-referential truth with levels
is consistent" we are going to describe a model of T L R within a suitable
fragment of (powerless) set theory.
First of.all, we fix a countable w-model of the theory OP of operations,
i.e. a model Ml~, where N MI~ "- {a" a E IMl~lA Ml,l = / a } is isomorphic with w.
It is essential-especially for the consistency theorem of the final s e c t i o n - t o
identify Ml~ with an arithmetically definable model of OP; to be definite, we
henceforth deal with CTM, the closed term model of Ch.I, 4.10. This is
because C T M can be identified with a simple arithmetical subset of w; in
addition, application, equality and the interpretation of the basic constants
and predicates of OP in C T M are arithmetical by Ch.I, 4.13 and the
appendix to Ch. I. Alternatively, we may choose the recursive graph model
R E of w Then we inductively expand C T M with a suitable family
~ c t ' - { V ~ ' ~ < L} of truth predicates, indexed by the first recursively
inaccessible ordinal L; the level ordering is simply identified with the
standard ordering relation on ordinals < t. The existence of t derives from
ordinal recursion theory and admissible set theory; the results we
presuppose are covered by Richter and Aczel(1974), Barwise(1975),
Hinman (1978). However, in order to make the present treatment reasonably
self-contained, we define the essential notions and state the required results.
Additional details are also available in the appendix.
The pure first-order set-theoretic language L s is obtained from the settheoretic language L s of w
by omitting all individual constants and
predicates, except membership E . If X is a predicate symbol :/: E , L s ( X )
is L s U { X } ; Z s ( X ) contains new atoms of the form Xt; the intended
meaning of X is that X is a class (in set-theoretic sense).
L a is the collection of constructible sets up to the ordinal a, where
L o - 0 , L)~- U {L a 9a < A} (A limit), and L~+~ is the family of subsets of
Lf~ first-order definable with parameters in the standard set-theoretic
language over the structure (L~, E [Lz). L ' - U{L a" a E ON} is the
constructible universe. We mention that a set theoretic formula A is Z 1 iff
A has the form 3zB, for some bounded formula B; B is bounded if it only
contains bounded set quantifiers (i.e. of the form Vy E z, 3y E z; see again
w
When we deal with semantical notions (e.g. definability over L), we
tacitly assume that the set theoretic language is expanded with (distinct)
constants for (distinct) parameters from a suitably large segment of L; but
we use the same symbol for the object a E L and its name. Lower case
Greek letters will range over the class ON of ordinal numbers.
39.1. D E F I N I T I O N
(i) An ordinal a > w is admissible iff a is a limit ordinal and La is
closed under pairing, union, bounded separation and bounded collection (in
VIII.39]
A Recursion- Theoretic Model
231
other words, L , is a model of Kripke-Platek set theory KP; cf. 22.4, for the
axioms).
(ii) An admissible ordinal c~ is recursively inaccessible iff it is the limit
of the admissible ordinals < c~.
(iii) t "- the smallest recursively inaccessible ordinal.
(iv) If C is a class of ordinals and P is a (set-theoretic) class, a n-ary
relation R is uniformly ~1(L.) in P for c~ E C iff there exists a El-formula
A ( x l , . . . , xn, X ) such that, if (~ E C,
R M L , - {(Cl, ... , Cn): Ca,... , cn E L , and (L,, P M L , ) ~ A ( C l , . . . , Cn, X)},
where X is interpreted by P M L,.
R is uniformly A I ( L , ) in P for ~ E C iff R is uniformly E I ( L , ) in P for
c~ E C, together with its complement. As a special case, a n-ary relation R
on L , is ~ I ( L , ) ( A I ( L , ) ) iff R is uniformly E I ( L , ) ( A I ( L , ) ) in P - @for
c~ E C, C being the singleton {c~}.
(v) A (possibly partial) function F : L , - - , L , is uniformly ~ I ( L , ) i n P
for c~ E C iff its graph is uniformly ~1(L.) in P for c~ E C.
The notions of ~ I ( L , ) - and Al(La)-function are obvious by (iv) and the
preceding stipulation.
(vi) LEVe is the structure (~ , - , _ < ) , where - , _ < are equality and
less-than-equal relation, restricted to.ordinals < t (respectively).
(vii)
c~ is projectible iff there exists a ]El(La)-injection from ~ into w.
39.1.1. REMARK. A total El(L,)-function F ' L , - - , L , is always A I ( L , ) .
In general, every E l(L.)-function F" C --, L,, whose domain C C_ L , is
A I ( L , ) , can be extended to a total El(L,)-function. As a consequence, the
relation R F ( a , x ) "-- a E F(x) is A I ( L , ) , provided F is E I ( L , ) a n d t o t a l , or
defined on a Al(L,)-subset.
Obviously, if we let level variables range over ordinals below t, while
level identity and ~ are realized on ordinal theoretic - and _< (in the
given order), we have:
39.2. FACT. LEV~ is a model of the level axioms of 36.4.3. Indeed, the
model satisfies linearity and wellfoundedness of -~ .
For simplicity, it is convenient to identify closed terms of Lop with
their respective number codes in the arithmetized version of CTM and hence
to regard CTM as a subset of w. Since every arithmetically definable subset
becomes definable by a bounded formula on L a, for c~ > w, it follows, by
inspection of the definition of the term models in 4.10:
232
[Ch.8
Levels o f Truth
39.3. LEMMA. The sets CTM, N "-- {t" t E CTM, t >-~, for some ~} (cf.
4.1 for > , ~ numeral), the application function , " CTM •
the interpretations of the basic constants O, S U C , P R E D , P A I R ,
R I G H T , D of OP are all elements of La, for every c~ > w.
LEFT,
In order to interpret the local truth predicates of TLR, a delicate step is the
choice of a denotation for the function symbol LT; we must find an
injection I N of t into CTM, which does not spoil the self-referential abilities
of the T i's and is "reasonably" definable. Once I N is available, the model
building only requires the well-known closure of admissible sets under
El-recursion and El-inductive definitions (see Barwise 1975, pp. 24, 124,
208), plus the fact that L is an admissible ordinal, which is limit of smaller
admissibles.
39.4. LEMMA
(i) The predicate A d ( ~ ) " -
"c~ is admissible" is uniformly AI(L/3 ) for [3
limit > a).
(ii)
The operation
fl ~ fl+ -
the least admissible > fl
is uniformly AI(La) , for c~ limit of admissibles.
(iii)
Let r o - ~ and rc~ - least admissible 7 > r~3, for every fl < c~,
whenever c~ > O. Then the ordinal sequence (ra : c~ < i~) is uniformly
/kl(Lrs).
(iv)
t is the least c~ such that v a - c~. In particular the restriction of v
to L is ~l(Le).
PROOF. (i): by standard techniques of formal set-theoretic semantics and
the well-known uniform Al-definability of the operation 5~--~L~ (see Barwise,
cit.; Devlin 1977).
(ii): apply (i).
(iii): by (i)-(ii) and closure of admissible sets under ~l-recursion.
(iv): easy consequence of (iii). 1"1
39.4.1. REMARK. If w < a < t
and a is admissible, a - f l
+, for some
We can now state the main fact needed for interpreting LT: each admissible
fl with w < fl _<L, is projectible into w, uniformly via a El-ma p.
39.5. THEOREM.
There exists a function I N , uniformly E l ( i ~ ) for fl
admissible > ~, such that IN[rc~" r a ~ w is total and injective, for every
0 < a <L (here I N [ r ~ is the restriction of I N to r~).
For a full proof, the reader is sent to Richter-Aczel (1974), while a sketch is
A Recursion- Theoretic Model
VIII.39]
233
given in the appendix.
39.5.1. REMARK. (i) In the statement 39.5 we can assume that the range
of the projection I N is CTM. Indeed, it is enough to consider any
~l-bijection a between w and the set of numerals of CTM. Henceforth, we
still maintain I N as a symbol for the projection of t into CTM.
(ii) The uniformity of the function I N is not shared by all countable
admissible projectible ordinals, since there exist non-projectible ordinals
below projectible ones (see Barwise 1975; Hinman 1978, p.424).
Of course, if we extend the interpretation of 39.3 by realizing the
function symbol L T on the map I N , we have:
39.6. COROLLARY. The structure CTM t - (CTM, LEVi, I N ) is a model
of OP plus the level axioms and the projection axiom PROJ of 36.4.1.
We now proceed to expand CTM, with a family ~ of truth predicates,
indexed by the ordinal t, such that (CTMt, ~c) is a model of TLR. To this
aim. we assume that the language s
of T L R is enlarged to a language
s 4- with constants for ordinals < t; lower case Greek letters represent
both ordinals < t and their names in s +. a, b, c are used as metavariables
for arbitrary elements of CTM, while we keep i, j, k ranging over level
variables.
If t is a closed term of s +, possibly containing LT and ordinal constants,
CTM,(t) is the value of t in CTM: CTM,(t) is the closed term of s which
results from t by replacing each subterm of the form LT(a) by I N ( a )
(which is identified with a closed term of CTM by 39.5.1; of course, the first
occurrence of a stands for the name of a in the language).
We lift to the present context the simplified notations and conventions of w
and 36.3. If a, b belong to CTM and a < t, then ab, Va, -,a, a A b, tr(a,a),
id(a,b), tr(a), nat(a) denote the following elements of CTM (in the given
order)" Ap(a,b), ALLa, NEGa, ANDab, CTM,([T~a]), [a- b], [Ta], [Na].
Clearly by pairing, 39.5.1 and the remark 39.1.1 we have:
39.7. LEMMA
(i) tr(a, a) - (7,(IN(a),a)), and the operation (a, a) H tr((~, a) is
injective (in each coordinate separately);
(ii)
if fl < r , , and c, a E CTM, the relation R(c, ~, a ) " - (c - tr(~, a))
is uniformly AI(L ~. ) for every ~ <_ t.
Of
Hence if fl < r of, the function a~-~tr(~, a) is uniformly AI(L r ), for every
c~<t.
a
39.8. DEFINITION. If S C_ t x CTM, let, for a < t"
234
Levels o f Truth
[Ch.8
S(c~) "- (a: a E CTM and (c~,a) E S).
The structure Ct - ( C T M t , S) canonically determines a realization of s +,
in which Ta is interpreted by S(c~) (for c~ < L) and T is assigned the set
u
<
39.9. DEFINITION. If 6 < t, S c ~ x CTM, X C CTM, let F(6, S , X ) be the
subset of CTM, such that a E F(6, S, X) iff for some b, c E CTM, one of the
following cases holds:
a -- (-~)tr(t3, b) and b E S(t3)(b ~ S(fl)), for some/3 < 6;
o
a - (~)id(b, c) and CTM~-(-~)b -- c;
0
a = (-~)nat(b) and G'~Ml=(-~)gb;
.
a = (-~)tr(b) and b E X ((-~b) E X);
0
a = (-~)tr(6, b) and b E X ((-~b)E X);
.
a -- ~ b and b E X;
0
a = (-~)(b A c) and b, c E X (respectively (-~b) E X or (-~c) E X);
e
a = (-~)Vb and for every d E CTM, ( b d ) E X (for some d E CTM,
o
(-~bd) E X ) .
39.10. LEMMA
(i) F is monotone in the third variable: if 6 < t, S C_ 6 x C T M and
X C_ Y C_ CTM, then
s, x) c
s, Y) C_ CTM.
(ii) F(6, S , X ) is uniformly AI(La) in X , S, for ~ admissible with
w < ~ < t and 6 < ~. Hence L a is closed under F in the following sense: if
6 < a, X , S are AI(Lc~), then F(6, S , X ) E ~ ( C T M ) n L a ( ~ - the power set
operation).
PROOF. (i)" its defining condition positively depends on X.
(ii): by inspection of definition 39.9, lemma 39.7 and remark 39.1.1, we see
that F(6, S , X ) is uniformly AI(La) in X, S; so we can apply Al-separation
for La, since F(~,S,X) C CTM E La. l'1
39.11. LEMMA (Inversion). A s s u m e 6 < t, S _C 6 x CTM, X _C CTM and a,
b E CTM. Then, i f A has the f o r m a - b, -~a - b, Na, - N a ,
9
.
0
[A] E F(5, S, X ) iff CTMI=A;
tr(t3, a) E F(6, S , X ) iff either 13 - 6 and a E X or j3 < 6 and a E 5'(/3);
(-~tr(t3, a)) E F(6, S , X ) iff either t3 < 6 and a ~ S(t3) or
t3 - 6 and (~a) E X ;
A Recursion- Theoretic Model
VIII.39]
4.
( a A b ) E F(6, S , X ) iff a E X and b E X;
5.
(--,(aA b)) E F(6, S,X) iff (-~a) E X or (--,b) E X;
6.
(Va) E F(~,S,X) iff (he)E X , for all c E CTM;
7.
(-~(Va)) E F(~,S,X)
8.
tr(a) E F(~, S,X) iff a E X;
9.
(-~tr(a)) E F(5, S,X) iff (-~a) E X;
10.
(--,--,a)E r(6, s, x ) iff a E X .
235
iff (-~(ac)) E X , for some c E CTM;
PROOF: similar to Ch. II, 7.4; in addition, we apply 39.7 above. [3
39.12. DEFINITION. We introduce the /3-th iteration I t ( F , 5 , S , t3) of F, for
given 5 < t and S C 6 • CTM, by recursion on/3:
zt(r,~,s,0)
- 0;
it(r,~,s,/~+l)
- r(~,s,/t(r,~,s,/~));
for ~ limit, It(F, 6, S, ~) - U {It(F, ~, S, fl)"/3 < )~).
Clearly ~ < ( implies It(F, 6, S, ~) C It(F, ~, S, () by monotonicity of F.
Informally speaking, It(F, ~, S, ~) exactly determines the ~-th stage of truth
of level 6, uniformly in a sequence {S(7)" 7 < 6} of possible candidates for
truth predicates of lower level.
39.13. LEMMA. Assume that S C 6 x CTM and 6 < 5.
(i) /f a E CTM, fl < a, 5 < a, then P(a, 6, fl, S) "- "a E It(F, 6, S, fl)"
and the function ~3~It(F, 6, S,~3)are uniformly AI(L,~ ) in S, for a
admissible with w < ~ < 5.
Hence, if S is AI(La) , I t ( r , 6 , S , - )
. a--+ L~M~(CTM).
(ii) If 7 - a+, 6 < a and a is admissible with L > a > w,
I "- I t ( r , ~, S, a) satisfies"
I-r(6,s,I);
I E L.yM ~(CTM).
the set
(,)
(**)
PROOF. (i) I t ( F , 6 , S , - ) is recursively and uniformly defined by means of
the operation F, which is uniformly AI(La) in S by 39.10 (ii).
(ii) If La is admissible, the least fixed point of any given positive operator
which is ~l(La), is ~l(La) (this is Gandy's theorem, Barwise 1975, pp. 208210). Hence (,) is immediate by 39.10. As to (**), (,) implies that I is a
Al(Lw)-subset of CTM E L.y. [3
39.14. DEFINITION. If 6 < ~ and I t is the functional of definition 39.12, let
V(6) - I t ( r , 6, Vl6 , rr
(+)
236
[Ch.8
Levels of Truth
where
= { ( Z , . ) : z < ~ and a E V(/~)} and r
= 5 if 5 is a limit; else
r
= 5+1. ~ is well-defined on ordinals < t, by Al-recursion , 39.13 and
39.4.
In the following, T" denotes the unique function satisfying (+) above.
We can now associate to the structure (2t = (CTMt, ~r) the realization of
s +, in which T a is interpreted by ~r(a) (for c~ < t) and T is assigned the
set U {V(c~): c~ < t).
39.15. LEMMA
(i)
The relation R(5, a ) " - a E V(5) is uniformly AI(Lrr
every 5 < t. Hence:
'1('(5) E Lrr
(ii)
(iii)
1
a n d q['" t ---+L t
)
) for
+1
M z)(CTM) is AI(Lt);
if 5 < t, V(5) = r(~, v]~, v(~));
if S < t, either a ~ V(5) or (~a) ~ V(5), for every a E CTM;
(iv) for every 7 < 5 < t, a E CTM, either tr(7, a) E ~r(6) or
(--(tr(7 , a)) E ~r(5);
(v)
if 31 < 5 < t, ~(7) is a proper subset of ~(5).
PROOF. (i) and (ii) follow from 39.4, 39.13 and closure of admissible sets
under Avrecursion.
(iii)" by main transfinite induction on 5 < t, and secondary induction on
Tr , using (i) and inversion lemma 39.11 at the successor step (see 7.4).
(iv) If 7 < 5 and a E ~r(7), then by definition of F, monotonicity and (ii):
tr(7, a) E
Vl , O) g r(5, Vl , v(5)) g
For a ~ ~'(7), the argument is similar.
(v). If 7 < 5 and ~ ( 7 , ~ ) : =
induction on ~ < 7-r
V(7, ~ ) C V(5).
it is enough to verify by
(+)
If ~ = 0 or ~ is a limit ordinal, the proof is trivial. Assume (+) by IH and
a E F(7,~r]7,~r(7,~)): we show a E ~r(5) as a consequence of the inversion
lemma and the property (ii) above. We distinguish several cases according
to the form of a. Let a = (-~(tr(u,b)) for some u: then by inversion either
u < 7 and b ~ ~r(u) or u = 7 and (-~b)E ~r(7,~). In the first case, since
u < 5, a E F(5, ~r]5,0)C_ ~(5) by definition of F and (ii). In the second case,
(-@) E~r(7) by definition and hence b ~ ~r(7 ) by consistency (see (iii)
above). Since 3' < 5, a E F(5,~r]5,0)C_ ~r(5). Let a = (b A c); by assumption
and inversion b E ~r(7, ~) and c E ~r(7, ~), whence b, c E ,1('(5) by In. By
definition of F, a E F(5, ~r]5, ~r(5)) and a E ~(5) by (ii). The extant cases are
A Recursion- Theoretic Model
VIII.39]
237
easily checked as exercise. As to proper inclusion, consider the terms
R(7) = {x: -~xrl~x } and c = n ( 7 ) n ( 7 ) - [-~R(7)T/.~R(7)].
Then observe that (ii) and (iv) imply c E V ( 7 + l ) - V ( 7 ) (see 37.5 (v)). D
39.16. THEOREM.
Ct - ( C T M t , T ) I = T L R .
PROOF. LIND holds in r since CTM is an w-model, while corollary 39.6
takes care of the level axioms and LT-injectivity. Local truth axioms and
connection axioms (cf. 36.4) are straightforward consequences of the
definition of F, inversion, definition 39.8 and the previous lemma. As to the
reflection principle, assume, for a, b E CTM and 7 < t:
r
~ x~ljb).
(,)
By Al(Lt)-definability of R ( a , a ) " - a E ~/'(a) (see 39.15 (i)), condition (,) is
equivalent, by well-known absoluteness of Al-conditions , to:
L,I= (Vx e
CTM)(3~)A(x, 7, ~, a, b),
(**)
for a suitable Al-formula A(x, y, z, u, v); hence by El-reflection , for some
< t, we have itl=(Vx E CTM)(3~ < ~)A(x, 7,~,a,b), which yields by (,)
and (**), the required conclusion
e~l=SkVxSj(j ~_ k A ( x ~ a - ~ xnjb)).
39.17. REMARK
(i) C~ satisfies EA, the full schemata of N-induction and -~-induction.
Note also that the proof step from (,) to (**) still works if we replace
Ti(ax ) by any formula B, which belongs to the least collection E A + of
formulas containing atoms of the form Nt, t - s, Tt, Tit , closed under
bounded level quantifiers, object quantifiers and logical connectives (i.e. if
B E EA +, also Vj -< k.B E EA + "-- Vj(j -< k---~B)). Hence E~ makes true the
schema REFL+:
Vx3iB(x,i)--, SkVx3i ~ k.B(x,i) (B E EA+).
(ii) We might pursue the recursion-theoretic analysis of the subsets of
CTM, which are extensions of classes in r It should not be difficult to
show that the sets C_ w, which are representable by classes, are exactly the
sets C w, which are recursive in the Tugu~ functional El(Or hyperjump),
where E I ( F ) - 0 , if F codes a well-founded tree of number sequences, and
EI(F ) - 1, else, for F: w ~ w (see Hinman 1978).
(iii) If we replace t by ~ in the model construction of the theorem, we
obtain a model C T M ~ of TL, the theory without reflection.
238
t eve]s of -Truth
[Ch.8
w40. Levels of truth and predicatively reducible subsystems of second-order
arithmetic
Is there any standard mathematical system naturally related to the theory
TLR? It turns out that, although TLR is based on the logical notions of
truth and iteration of the reflection process, TLR is strictly connected with
an important subsystem ATR 0 of second-order arithmetic Z2; it is wellknown that ATR 0 has a non trivial mathematical content and is actually a
strong version of predicative mathematics. In order to clarify this point, we
are forced to digress into the classical realm of Z2.
As Hilbert and Bernays already showed in Supplement IV of the
Grundlagen der Mathematik, the language of Z2 is remarkably expressive,
and fundamental theorems of ordinary mathematics (including non trivial
parts of the theory of countable sets and ordinals) can be already derived in
Z2. On the other hand, proof-theoretic investigations of the sixties produced
a systematic classification of "natural" subsystems of Z2, which were mainly
suggested by definability criteria and limitation of logical complexity (see
Kreisel 1968 for a vivid picture of the intertwined technical and conceptual
motivations). In the seventies, this research thread progressively shifted
toward investigations of formal theories related to mathematical practice
(Feferman 1977, Takeuti 1978).
In particular, a new twist in the investigation of Z2 and its foundational
significance was impressed by Friedman (1976), and subsequently by
Simpson. Friedman discovered that, in order to formalize ordinary non-settheoretic mathematics, only five set existence principles formalizable in Z2
are actually needed; he further noticed that in a number of relevant cases,
the set existence axiom, applied to prove a given paradigmatic theorem, is
actually implied by the theorem itself over a weak basis theory (say, a form
of recursive analysis), whence the name of reverse mathematics to the
systematic development of the program. Since then, this phenomenon has
been widely investigated and Friedman's subsystems-accompanied by
sophisticated model-theoretic and recursion-theoretic t e c h n i q u e s - have
become an interesting tool to calibrate chapters of ordinary mathematics
(from calculus to countable algebra, logic included; see the announced
monograph by Simpson and several research papers).
We compare TLR with exactly the fourth level of Friedman-Simpson
reverse mathematics, namely the above mentioned ATR 0. We also apply a
theorem of Ch.V about choice principles, to give inner models of weaker
subsystems of Z2, based on corresponding choice schemata. In the final
section, we shall interpret the fifth level II~-CA 0 of reverse mathematics in a
natural extension of TLR.
Levels of Truth and Second Order Arithmetic
VIII.40]
239
40.1. DEFINITION. The language 2,2 of second-order arithmetic contains:
(i) a denumerable list of number variables Xl, x2, X3,... ;
(ii) a denumerable list of set variables Xo, X1, X2, ... ;
(iii) the individual constant 0, the function symbols ' (successor, 1-ary),
+ (addition, 2-ary), 9 (product, 2-ary);
(iv) the binary predicates < (ordering on w ) a n d c (membership);
(v) classical logical operations (say -1, V, A ) and - .
Terms are inductively generated from number variables and the constant 0
by application of the function symbols ' , . , +; thus, if t, s are terms, so are
t', t+s, t . s . The atoms of 2,2 have the form t - s, tcX, t < s, where t, s are
terms and X is a set variable. Formulas are inductively generated by means
of the following clauses" atoms are formulas; if A, B are formulas, -~A,
A A B, VxA, V X A are formulas.
An 2,2-formula is arithmetical if no set variable occurs bound in A.
E 1 = H I = the collection of arithmetical formulas. If A is II 1, then 3 Y A
(VYA) is E] (II]). If A is El, then V Y A (3YA)is II~ (El). An arithmetical
formula A is bounded (Ao) if it contains only bounded number quantifiers,
i.e. quantifiers of the form Vx(x < t --+...), 3x(x < t A . . . ) (x not occurring
in t). A is a E ~ (or II ~ formula iff A has the form 3xB (VxB) with B
bounded.
40.2. Second-order arithmetic (in short Analysis) and its subsystems
40.2.1. Z 2 is the theory in the language s
which contains:
(i) classical predicate calculus with identity for 2.2;
(ii) Vx (-~x~ - 0) A VxVy(x ~ - y'-~ x - y);
(iii) Vx(-~x < -0) A VxVy(x < y ~-, 2z(z'+x - y));
(iv) Vx(x + 0 - x) A Vx(x. 0 - O)A
^ v vy(
+
+ y)'A
+
(v) Induction axiom Ax-IND: 0cX A Vx(xcX--+ x'cX)--+ Vx(xcX).
(vi) Full comprehension schema CA:
where A(u, Y) is an arbitrary s
3XVu(ucX +-+A(u,Y)),
X does not occur in A.
40.2.2. If 05 is a collection of s
~5-CA is the schema CA restricted
to formulas A of 05; ~-IND is the schema of induction for every A E ~ (that
is, we replace ucX in Ax-IND with any A(u) E ~).
40.2.3. If ~ is a collection of s
A(~)-CA is the schema:
Vx(A(x) ~ - ~ B ( x ) ) - ~ 3YVx(xcY ~ A(x)),
Levels of Truth
240
[Ch.8
where A, B E ~ and Y does not occur in A, B.
40.2.4. Let ~(w) be the power set of w, the set of natural numbers. The
standard set theoretic model of Z2, also named ~(w), is given by letting the
set variables range over arbitrary subsets of w, while the individual
variables range over w and 0, ' , + , . ,
< , c are assigned the intended
meaning of zero, successor, addition and product, natural ordering and
membership E.
If P 1 , . . . , P k are sets C w, m l , . . . , m n C N, and B ( Z l , . . . , z n , Y 1 , . . . , Y k ) is a
formula of s
(with free variables occurring in the given list),
~)(w)l=A[ml,...,mn, P1,...,Pk] stands for "A is satisfied in the standard
model of Z2, whenever m l , . . . , m n , P 1 , . . . , P k ,
are assigned to x a , . . . , x n ,
Y I " " , Yk" (respectively).
40.2.5. If P C_ w, we say that R C_ w is arithmetical (~], II] respectively) in
P iff there exists an arithmetical ( ~ , I I ~ ) formula A ( u , X ) such that
R-
{n " ~(w)l=A[n,P]}.
40.3. For the sake of comparison with type-free systems, we explicitly
introduce the following subtheories of Z 2.
40.3.1 ACA 0 is the subsystem of Z 2 which only contains II1-CA; ACA o is
known in the literature as arithmetical analysis (with induction axiom).
40.3.2. Let A1-CA be the schema A(~I)-CA (see 40.2.3)" AI-CAo also
labels the subsystem of Z2, in which full CA is replaced by A ]comprehension;
this subsystem is known in the literature
as
hyperarithmetical analysis (with induction axiom).
40.3.3. ~ - D C is the schema of dependent choice:
V x V X 3 Y A ( x , X, Y) ~ V X B Z ( Z o - X A VxA(x, Zx,
Zx+ 1)).
Here A is any ~ - f o r m u l a , ueZ x stands for (u,x)cZ and (u,x) denotes any
fixed s
injective pairing function of w x w into w. ~ - D C o is the
theory ACA 0 plus the schema ~ - D C .
40.3.4 We now introduce the subsystem ATR o of arithmetical transfinite
recursion. Let WO( < x ) be the II~-formula, stating that X encodes a
linear ordering of w such that VY(Vx(Vy(y < x x ~ y c Y ) ~ xcY)---+ Vx(xcY))
{here y < x x " - ( y , x)cX}.
ATR is the schema:
V X V Z 3 Y ( W O ( < x ) ~ VyVu(ycYu ~ A(y, u, V[u, Z))),
where A is an arithmetical formula and Y[u is contextually defined by
(v, y)cY[u "- v < X u A (v, y)cY.
Levels of Truth and Second Order Arithmetic
VIII.40]
241
A T R o is the theory ACA o + ATR. ATR is a consequence of II~-CA, plus
the classical schema of bar induction, namely:
40.3.5.
BI := WO( < x)---* T I ( < x, B), where B is arbitrary.
40.3.6. We finally define the strongest subsystems we are going to consider.
A12-CA is the schema A(E~I)-CA; A1-CAo, II1-CAo are the subsystems of
Z2, where CA is replaced by A12-CA and II~-CA (respectively).
The weakest subsystem of reverse mathematics is recursive analysis
RCAo, i.e. the subsystem obtained from Z2, by replacing Ax-IND with the
induction schema for Y]~
and CA with A~
"-A(~~
the
schema of recursive comprehension.
Given two theories ~ 1 and ~2 in the language s
we set ~ 1 > ~ 2 iff
PA F-Cons(~
, where PA is Peano arithmetic (cf. appendix to
Ch.I) and Cons(OJ") is a standard formalization of the metamathematical
statement "~ is consistent" in the arithmetical language.
~ 1 and ~ 2 are proof-theoretically equivalent (in short, ~ 1 = ~2) iff zJ"1 > ~2
and ~ 2 > ~
~ 1 > r iff 03"1 > ~ 2 and not ~1 -- ~2" If ~ 1 > ~
we say
that ~1 is proof-theoretically stronger than ~ 2.
We now state without proof the following known results:
40.4. T H E O R E M
(i)
A1-CA 0 - II~-CA 0 and II~-CA 0 > ATR 0.
(ii) ATRo=_Predicative Analysis (in the sense
Feferman 1964, cf.Ch. XI). Moreover ATR o > ~E1-DCo.
of Sch~ttte
1977,
(iii) ~ - D C o > A~-CA o.
(iv) A1-CAo - ACA o - PA.
(v) RCA o - PRA.
(ii)-(iv) above can be obtained as a corollary of the proof-theoretical
analysis of Ch. XI (but see Feferman-Sieg 1981, for (i), (iii)-(iv), Friedman,
Simpson and Mc Aloon 1981, for (i) and Simpson(199?)for 40.4(v)). It is
to be mentioned that each instance of A]-CA is derivable in ATRo; however
E]~-DC is unprovable in ATR 0 (actually ATR 0 + ~ - D C is strictly stronger
than ATRo, by theorems of Friedman).
We are now ready to state the promised interpretation result. We recall
that PWp is obtained from the system PW c of 16.1 by replacing numbertheoretic induction for classes with number-theoretic induction for
properties.
Levels of Truth
242
[Ch.8
40.5. T H E O R E M
(i) A T R 0 is interpretable in TLR.
(ii)
(iii)
~ - D C 0 is interpretable in PWp + EA.
A]-CA 0 is interpretable in P W c.
PROOF. The proof is easy, as the essential work was already carried out in
earlier sections.
(i) We first define a translation * of Z2 into the level-free part of the
language of TLR. Informally speaking, we simply verify that N, plus the
subclasses of N, is a model of ATR in TLR. More formally, we choose
combinators 0, -, +, ~, in order to interpret the basic function symbols of
s (we adopt the same notation). Hence we can inductively assign to each
s
t a term t* in the language s
( - the operational fragment of
s
with the same free variables. Moreover, if t - s, tcX, t < s are atoms
of s
we put ( t - s ) * - ( t * - s * ) ;
(tcX)*-(t*rlx) (x fresh variable);
(t < s)* = (t* < s*) (the second occurrence of < being a canonically chosen
2.op-definition of < ; see 3.6). We then extend * to arbitrary formulas of Z2
by stipulating that * commutes with 9, A and
(VXA)*
-
Vx(CIN(X ) -+ A*), (VxA)* - Vx(Nx--. A * ) - VnA*,
where ClN(x ) := Cl(x)A Vu(u~lx--~ Nx). It is clear that * is a well-defined
translation of 2.2 into 2.. Let A be an s
with free variables in the
list X = X o , . . . , X n , y = YO,'",Yk: then we check by induction on the
definition of ATR0-provability:
if ATR 0 F A(y,X), then TLR F Ny A ClN(X ) -. A*(y,x).
(1)
It suffices to see that the *-translations of Ax-IND, HI-CA and A T R are
provable in TLR. Now (Ax-IND)*and (H1-CA) * become instances of class
N-induction and elementary comprehension and hence are provable in T L R
by theorem 37.9. Note also that, if (CIN(X) A WO( < X))* is assumed,
then < x encodes a subclass of N which is a pwo. Hence if z is any subclass
of N and A ( u , x , Y , Z ) is arithmetical, urlNAA*(u,x,y,z)is elementary
extensional in y, z (y, z fresh variables). Now the hypothesis of 38.6.2 are
trivially met, and there exists a subclass of N satisfying the *-translation of
the ATR-consequent.
(ii): apply the elementary dependent choice schema EDC of 20.10 (ii).
(iii): we simply apply A-comprehension 16.7 of Ch. IV to the translation of
hyperarithmetical comprehension. V1
To conclude, it is time to reconsider the opening problem of the section,
concerning the theoretical relevance of TLR and its ability to represent
significant parts of mathematical knowledge. The answer is implicit in the
VIII.40]
Levels of Truth and Second Order Arithmetic
243
interpretation result above. Here we freely rely on results of Feferman,
Takeuti, Friedman and Simpson (op. cit.).
First of all, significant parts of ordinary mathematics, like elementary
calculus and countable algebra, can be already developed in conservative
extensions of Peano Arithmetic (Takeuti, Feferman) and actually in
fragments of arithmetical analysis ACA0, which are not proof-theoretically
stronger than primitive recursive arithmetic (hence afortiori in fragments of
MFc).
A typical example thereof is the Cauchy-Peano theorem CP, asserting
the existence of solutions for ordinary differential equations; CP is indeed
equivalent to KSnig's lemma for binary trees WKL (modulo RCAo) by a
theorem of Simpson (1984). Furthermore, the very principle of arithmetical
comprehension is equivalent in RCA 0 to the statement that every Cauchy
sequence of reals converges to a limit in R and also to the existence of
maximal ideals for countable abelian rings or even to the KSnig lemma for
finitary trees.
On the other hand, ATR 0 has a good theory of countable ordinals and
it proves non-trivial classical results of descriptive set theory. In particular,
as Friedman and Simpson observed, ATR 0 is mathematically much more
effective then the subsystems of hyperarithmetical analysis and even
predicative analysis in the sense of Feferman-Schfitte: there are important
consequences of ATR0, which are false in the model of hyperarithmetical
sets and hence independent of Predicative Analysis. Here is a sample of
significant results. ATR is equivalent (modulo RCA0) to:
(i) comparability of well-orderings, i.e. countable well-orderings are
comparable;
(ii) the Lusin-Sierpinski theorem: every analytic set in the Baire space
ww of unary functions w---+w is either countable or has a perfect
subset);
(iii) the Gale-Stewart theorem: every open game C ww is determined;
(iv) the Ulm structure theorem for countable reduced abelian p-groups
(Friedman, Simpson and Smith 1984).
At the same time, the strength estimate of 40.4 (ii) assigns precise limits
to ATR 0 and hence, by the equivalence theorem of Ch. XI, to the theory
TLR of truth with levels. For instance, there is no way to prove in ATR o
that every arithmetical set of Dedekind reals has a least upper bound, nor
ATR 0 proves the classical Cantor-Bendixson theorem (every closed subset of
the Baire space is the union of a countable sets of reals plus a perfect set);
by contrast these two theorems are derivable in II~-CA 0 (and hence in the
extension of TLR of w
Levels of Truth
244
[Ch.8
~41. Consistency of a reducibility principle for classes
We wish to have a closer look to the recursion-theoretic model C~, in order
to investigate quantification on classes. In the usual inductive models,
like O(CTM) (see w
C L - { x " Cl(x)) is generally not closed under
quantification of classes: the best we can afford is a sort of A-comprehension
(see corollary 16.7). However, we shall verify that, if an elementary
predicate of classes, possibly depending on additional class parameters, is
non-empty, then the same predicate is already satisfied by some class of any
level ~- i (e.g. a solution is to be found in CLi+I, if i has a successor level
i + l ) , provided i is an upper bound on the level of the given class
parameters. Hence, at least for elementary predicates, quantification on
arbitrary classes is reducible to quantification on classes of a fixed level.
But we know that quantification on classes of fixed levels does not push
outside the realm of classes.
The result we hint at above, is in essence a consequence of the effective
nature of C T M and generalizes to the present framework the classical
Kleene basis theorem (Kleene 1959). Formally, we consider a reducibility
schema for classes RPC:
41.1.
i -~ k A Cli(x ) A 3y(Cl(y) A A(u, x, y))---, 3y(Clk(y ) A A(u, x, y)),
for every Z-formula A(u,x,y)
elementary extensional in x, y.
with the free variables shown, which is
The rest of the section is devoted to convince the reader that RPC holds in
Ct. The proof combines the classical analysis of 1-!l-sets by means of
recursive trees with a straightforward transfer argument from the standard
model of Z 2 to C~, coupled with representability of inductive predicates in
our language.
41.2. First, we recall the relevant recursion-theoretic notions and results. As
usual, n, m , k range over natural numbers; e, f, g denote indexes of partial
recursive functions. Par abus de langage, we keep using ( . . . ) for a fixed
number-theoretic primitive recursive function, which injectively maps finite
sequences of natural numbers into numbers; on the same par, (n)i will
denote the corresponding primitive recursive projection, such that, if n
encodes (n0,... , nk) and i < k, then ( n ) i - n i. lh(s) is the primitive recursive
function which computes the length of the sequence code s. Seq(x) is the
predicate "to be a (number which encodes a) finite number sequence", while
( ) " - 0 is the code of the empty sequence. If s, r, range over elements of
Seq, the concatenation of r with s is r , s "- (r0,... ,rn,s0, ...,sin) , where
r - (%, . . . , r n ) , s - (So,...,Sm). t is a subsequence of s (in symbols t C_ s)
iff s - t , r , for some r in Seq; if r # ( ) , we write t C s. If F ' ~ - - - , ~ , we
inductively define F ( 0 ) - ( )
and F ( n + l ) - F(n),(F(n)). The expression
A Reducibility Principle For Classes
VIII.41]
245
{e}P(n) " m means that the e-th partial function, which is recursive in the
set P C w, converges on n with value m; clearly the Kleene bracket relation
can be defined in the standard model by a E~
which is also
denoted by { e } X ( m ) ~ _ n and contains a second-order parameter X;
TotX(e) := Vn3m({e}X(n)~_ m) means that the function with index e is
total on natural numbers, grs is the index for the partial recursive function
defined by {g[s}X(s ') := {g}X (s.s') (s, s' in Seq). If s - (n), we simply
write gin instead of g[(n). In order to define recursive trees, we consider the
arithmetical formula
41.2.1
W ( e , X , Y ) :- TotX(e) A (Vs(-~{e}X(s) ~_ O) VVn((eVn)cY)) ).
Note that W is positive in Y, while X possibly occurs in negated atoms of
the form -~tcX. Then we can define:
coW(X) "- VY(Clos(W, X, Y) --+ ecY),
41.2.2.
where Clos(W, X, Y) "- Ve(W(e, X, Y) ~ ecY).
By standard arguments:
41.3. LEMMA
W.1
VXW(W( , X,
~(~)1= v x ( v e ( w ( e , x , B)---+ B(e))---+ Ve(ecW(X)---+ B(e))),
w.2
B(x) is an arbitrary s
and W ( e , X , B ) results from
W ( e , X , Y) by replacing each atom of the form toY by B[x := t].
where
If P is a fixed set C w, ~g(P) denotes the II]-set defined by the formula
W ( X ) in z)(w), when X is assigned P as value. In general, to any index e of
a total recursive function and any P C w, we can associate the tree
W'c
0},
which is closed under the subsequence relation C . ~ ( P )
well-founded trees which are recursive in P:
encodes the set of
41.4. LEMMA. If P C_ w, then e E ~ ( P )
to the converse of C,
iff T P is well-founded with respect
i.e. there is no function F : w---~w such that for
Let w w - {F" F is a unary function w---+w}; then II]-predicates enjoy a
simple, but essential property:
41.5. LEMMA (Normal form). If R C_ w is II] in a given P C_ w, then we
can find a primitive recursive predicate S R such that,
characteristic function of P, then"
if F p is the
ieveIs of Truth
246
[Ch.8
R - {n E w" (VG E ww)(3m E w)(<n, F p ( m ) , G ( m ) ) E SR) ).
II]-~t~). If R G ~ i~
can find a primitive recursive function T R E E R such that
41.6. T H E O R E M (Tree theorem for
rI]
in P c w, we
n E R iff T R E E R ( n ) E ~4r(p).
P R O O F . Let T R E E R ( n ) denote the primitive recursive index of the
characteristic function of the set
T R ( n ) -- {s" Seq(s) A Vs' C s.(n, Fp(lh(s')),s') ~ SR}
(we apply the normal form lemma and the related notations). Then T R ( n )
is a tree, which is well-founded with respect to the converse of C iff
T R E E R ( n ) E ~ r ( p ) iff n E R ( use 41.4 for the second equivalence), f'l
41.7. T H E O R E M (The Kleene basis theorem). Let F be the characteristic
function of P C_ w, fix n E w and let S be primitive recursive. Assume that
for some function G E w~o and for every m, (n, F p ( m ) , G ( m ) ) E S. Then
there exists H C ww which is recursive in ~7(P), such that, for every m,
(n, F p ( m ) , H(m)) E S.
P R O O F (after Shoenfield 1967). By the so-called leftmost branch selection,
one recursively defines"
H ( p ) - the least number k such that, for some G E ww and for every m,
(n, F p ( m ) , H ( p ) , k , G ( m ) ) E S. By inspection H is recursive in a set, which
is Y2~ in P; hence, by the tree theorem, it is recursive in ~ ( P ) . l-1
41.8. COROLLARY. Let A ( u , X , Y ) be an arithmetical formula with the
free variables shown and fix n E ~, P C_ w. If ~P(w)i=A[n, P, Q], for some Q,
then there exists R recursive in ~ ( P ) , such that ~(w)I=A[n,P,R ].
PROOF'by
41.5-41.7.[3
We finally apply the given machinery to r We adopt the notations of
w and assume that the language is expanded with names for ordinals < t
(for which we use lower case Greek letters). We fix a Ghdel numbering [-]
of CTM; by 10.11 of Ch.I, there is a term Val(x) such that if b E CTM,
Val([b]) - b. C T ( x ) is an abbreviation for "x - [b], for some b C CTM".
41.9. DEFINITION. If b C CTM, we set"
bct "- {x" Val(x)~b A CT(x)}; bct "- {[a]" E,]=Val(Va])rlb and a E CTM};
b(a) "- {x" xrlab};
b
.
-
^
CLt(a ) "- {b E CTM: r
.
-
If P C_ w, P* "- {~" n E P}; so P* C_ CTM.
VIII.41]
A Reducibility Principle for Classes
247
Clearly bct
by bct, b~ (respectively)
/~ ~ b~ are subsets of w, represented in r
9
41.10. LEMMA (Translation)
(i) Let A ( u , x , y ) be an L-formula with the free variables shown, which
is elementary extensional in x, y: then we can find an arithmetical formula
A2(u , X , Y ) of L2 such that, if a, b, c E CTM, then
C~l=A[c,a,b] iff ~(w)l=A2[[cl, a ctt , bct ].
(see w
(*)
for ~P(~)I=A).
(ii) Let A ( u , X , Y ) be an arithmetical L2-formula with the free variables
shown: then we can find an L-formula A*(u,x,y), which is elementary
extensional in the fresh variables x, y and satisfies, for a, b E CTM, n E w:
~(w)l=a[n, aT, bT] iff
C,l=a*[n, a w, bW].
(**)
(iii) Under the same assumption of (ii), if A * ( u , X , Y ) is the formula of
2. U { X , Y ) , which is obtained from A*(u,x,y) by replacing all atoms of the
form trlx, srly by t* E X, s* E Y (respectively), then
~P(w)I=A[n,P,Q] iff CTMI=A*(~,P*,Q* )
(***)
for n E w, P, Q c_ w. { The right member of (***) means that A * ( u , X , Y) is
satisfied in CTM by the assignement (~,P*, Q*)}.
PROOF. (i) Here is the recipe to obtain A2(u,X , Y ) f r o m a(u,x,y):
1) we replace each subformula of the form t~la, trlb in A respectively by
[tlca~t,[tlcb~t;
2) we replace the subformulas t = s, N t by the corresponding arithmetical
formulas of L2, which define - and N in the term model CTM;
3) we relativize the the universal quantifiers of A to CTM.
By assumption on A, we can inductively check that A 2 is arithmetical in
the sets a ct , b~ct and that (,) holds.
(ii)-(iii): apply the *-translation of 40.5 with a simple inductive argument
(using CTMI= t* = s*==r ~P(w)l= t = s and definition of P*). !-!
r
If a E CLt(a ) the set ~ ( a ~ ) of wellfounded trees, recursive in at,
definable in r by a term of Ly, which is a/?-class in r whenever fl > a.
41.11. LEMMA. We can find a term I ( W , x , i )
(where a E CTM, a < t ) , then
W(aT) - {n E w"
e,t=n%I(W,a~,~)};
Ctl=Vk(k )-- c~---,Clk({U: urlaI(W,a~,~)})).
is
such that if a E CLt(a)
(,)
(**)
PROOF. Consider the formula W*(e,x,y), obtained from W ( e , X , Y ) of
41.2.1 by means of 40.10(ii); then W* is elementary extensional in x, y,
and operative in y (see Ch. II, 9.5 and 10.3). If W i ( e , x , y ) is the/-transform
Levels of Truth
248
[Ch.8
of W*(e,x,y) (37.1), we have, for every e E w, a E CLt(a ), b E CTM:
~P(w)l=W[e,a~~
iff C,l=W*(-d,a~',b(a) ~) iff Ct~Wa(-d, aW, bW).
(1)
Choose FP(Ay.{e: Wi(e,x,y)} ) := I(W,x,i): since r
we apply 37.5,
37.2.1 under assumption that a E CLt(a), and we obtain, for every e E w:
Ct I=Er/aI(W, a w, a) +-+Wc~(E, a w, I(W, a w, a)).
(2)
Now set I : - (I(W, a w, a)(a)) 7 and F(Y) "- {e E w 9~P(w)I= W[e, aWt,Y]}"
Then I is F-closed by (2) and 40.10 (ii). On the other hand, I is the least
F-closed subset of w: apply the inductive generation of ~'a, (2), 40.10 (iii)
and adapt the proof of 13.4. As to (**), we use lemma 37.3 (v). [3
41.12. T H E O R E M . The schema RPC is true in Ct.
P R O O F . Let a E CL~(a), for some a < L, and ttl= 3y(Cl(y) A A(c, a, y)),
where c E CTM: then by lemma 41.10 (i) and 41.8, the set
{P" P C CTM and ZP(w)l-A2[[cl, a ct
t , P]} "- G(c, a)
contains an element P, which is recursive in the set ~,r(aCt)- ur((a~t)7 ). By
41.11, P is arithmetical in some d ctt, for d E CL,(fl), /3 > a, and by closure
of/3-classes under elementary comprehension, there is a fl-class e such that:
et
ct
--
P and ~(w)l=A2[[c],a ct
ect];
t, ,
again by lemma 41.10 (i), Ct~2y(Clf3(y ) A A(c,a,y)). [3
41.13. C O R O L L A R Y (Bar Induction)
If C~]- Cl(r) A PWO(r), then {(a, b): a, b E CTM and CtI- (a, b)•r} is a well-
ordering.
41.14. REMARK. If we replace t by w in the model construction of the
previous section, we obtain a model C T M ~ of TL, which makes RPC true,
once we replace Cl(x) by Cl~(x) := 3iCli(x ).
~42. Levels
arithmetic
of truth
and
impredicative
subsystems
of second-order
We are now ready to derive an analogue of IIl-comprehension for classes
from the reducibility principle for classes; in this section T L R - s t a n d s for
T L R without LIND, the local number theoretic induction axiom.
Let A(u,x,y)
extensional in x, y. Then:
42.1. T H E O R E M .
be an s
which is elementary
T L R - + RPC ~- Cl(x)---+ 3z(Cl(z) A Vu(urlz ~ V y ( C l ( y ) ~ A(u, x, y)))).
Levels of Truth and Impredicative Subsystems
VIII.42]
249
PROOF. Let i such that Cli(x ) and let i -~ k. Then by RPC, definition of ktransform and lemma 37.6 (i):
Vy(Clk(y)---, A k ( u , x , y ) ) ~ V y ( C l ( y ) ~ A ( u , x , y ) ) .
(+)
If we choose z "- {u" Vy(Clk(Y ) ---. A k ( u , x , y ) ) ) , then by lemma 37.3 (v), z is
a j-class for any j ~ k and hence a class such that, for every u,
u~z ~ UrljZ ~ Vy(Clk(y ) ---*Ak(u, x, y)) ~ Vy(Cl(y) ~ A(u, x, y));
(apply 37.5 (iii) for the second equivalence and (+) for the last one). V1
42.1.1. REMARK. (i) After remark 41.14, it is easy to see that the theorem
holds if we replace TLR + RPC by TL + RPC and we let C l ~ ( . . . ) occur in
place of Cl.
Numbers can be explained away in the classical style of Dedekind by
means of the above comprehension principle. Consider tile sublanguage Ly,
which omits the individual constants 0, SUC, P R E D , D, P A I R , L E F T ,
R I G H T and the predicate N. Let L T L - "Logicistic truth theory with
levels" be the restriction of TLR + RPC to s
thus LTL omits LIND and
the OP-axioms PAIR, NAT.I-NAT.2. Then we already know how to define
zero, successor and the other individual constants listed above in
combinatory logic (see Ch. I, 3.8). By 42.1 there esists a class c such that, if
Clos(y) stands for 0~y A Vu(urly ~ (u+l)rly), then:
Vu( c
vy(cl(y) ^ Clo
(y)
u y))
and we can interpret N x as XrlC, so that class N-induction for classes and
NAT.l-2 are derivable in LTL. Therefore we have:
42.2. THEOREM. TLR + RPC is interpretable in LTL.
We conclude by exploring two additional principles:
(i) we assume that levels are objects, and hence that the projection
function L T collapses to identity;
(ii) we apply reflective truth to expressions containing bounded level
quantification.
42.3. DEFINITION
(i) L~ is L y without LT. The definition of term is modified by
stipulating that level terms are terms tout court; all the rest is unchanged.
We introduce the abbreviations:
Vi -~ k.A "- Vi(i -~ k ~ A);
(ii) The
3j -~ k.A "- 3 j ( j -~ k A A).
class E A + o f extended acceptable formulas of L y
is the
Levels of Truth
250
[Ch.8
smallest class which contains all atoms of the form t - s, Nt, Tit and is
closed under bounded level quantifiers Vj-~ k, negation, conjunction and
universal object quantifiers.
(iii) We modify the map A H [A] for arbitrary A E EA t as follows.
First we let Vj-~ i . f " - ( l l , ( i , f ) ) (this makes sense now, since the level
variable i is a term). Then we inductively add the new clause
[Vj -~ i. A] "- (Vj -.~ i. (Ax. [A[j "- x ] ] ) ) ' - (]-1, (i, Ax[A])).
It is easy to check that [A] is a term of s
(iv) The bounded
formulas:
if A E EA+.
level quantifier axioms BLQ are given by the
Tk(V j -~ i. f ) ~ i -~ k A Vj -~ i. T k ( f j);
Tk-.(V j -~ i. f ) ~ i -~ k A 3j -~ i.Tk(-.(f j) ).
(v) The ontological axiom ONT is V i 3 x ( i - x).
(vi) TLR* - - TL + BLQ + REFL + + RPC + ONT;
(for REFL +, cf. 39.17), where the axioms containing L T are of course
omitted.
We leave to the reader the verification that the elementary facts of w still
hold for the extended notion of acceptable formula and with TLR* in place
of TLR. The definition of F(5, S , X ) in w
is easily adapted for
interpreting the new system: the idea is that level variables now range on
the set { I N ( a ) ' a < t}. The typical clauses which concern bounded level
quantification in the definitions of the modified operator (compare with
39.9) have the form:
1) if for some 7 < ~ with I N ( 7 ) -
i we have
VI3 < 7.Va E C T M . ( I N ( / ~ ) - a ~ f a E X),
then ( V j - ~ i . f ) EF(ti, S , X ) (where 6 < t ,
f E CTM);
S C _ ~ x C T M , XC_CTM, i, b,
2) if for some 7 < 5 with I N ( 7 ) - i ,
there exist / 3 < 7 , a E C T M with
I N ( ~ ) - a and ( - , f a ) E X, then (--Vj -~ i. f ) E F(~f,S,X) (where 5 < t,
S C_ 5 x CTM, X C_ CTM, f, i, b E CTM).
Since the relevant lemmas 39.10-15 still hold for the modified F, we
obtain:
42.4. THEOREM. There is a set ~* C CTM such that:
C~* . - ( C T M , ~ * > I - T L R * .
VIII.42]
Levels of Truth and Impredicative Subsystems
251
At this point one can proceed to strengthen the interpretation results.
AI(A,B) is the formula:
Vu(Vw(Cl(w) ---,3z(Cl(z) A B(w, z, u))) ,--, 3x(Cl(x) A Vy(Cl(y) ---, A(x, y, u))))
42.5. THEOREM. Let A(x,y,u) and B(w,z,u) be elementary extensional in
x,y and w,z respectively; then we can derive in TLR* without local
number- induction:
A~(A, B ) ~ 3v(Cl(v) A Vu(uTlv ,--, Vw(Cl(w)---, 3z(Cl(z) A B(w, z, u))))).
PROOF. Assume AI(A,B); then by 37.8 (v) we have, for arbitrary u"
VjVw(Clj(w)---, 3k3z(Clk(Y ) A B(w,z, u)))
3j3x(Clj(x) A VkVy(Clk(Y ) ~ A(x, y, u))).
(1)
By logic, we have, for every u,
3j3wVk(Clj(w) A Vz(Clk(Z)---,-,B(w,z, u))) V
(2)
V 3j3xVk(Clj(x)A Vy(Clk(y ) --, A(x,y, u))).
Hence:
3jVk3w(Clj(w) A Vz(Clk(z)---,-,B(w,z, u))) V
(3)
v 3jVk3x(Clj(x) A Vy(Clk(y)~ A(x,y,u))).
Again by logic:
Vu3jVk{3w(Clj(w) A Vz(Clk(Z ) ~-~B(w,z, u))) V
(4)
V 3x(Clj(x)A Vy(Clk(y ) ~ A(x,y, u)))}.
By (4), Vj3k(j ~ k) and persistence, we get:
Vu3k.3j ~ k.{3w(Clj(w) A Vz(Clk(Z)---,-,B(w,z, u))) V
(5)
V 3x(Clj(x)A Vy(Clk(y ) ~ A(x,y, u)))}.
But (5) has the form Vu3kC where C C EA+; hence by REFL + there is a
level p such that:
Vu.3k ~ p.3j ~ k.{3w(Clj(w) A Vz(Clk(Z ) ---,~B(w,z, u))) V
(6)
V 3x(Clj(x) A Vy(Clk(y ) ~ A(x, y, u)))}.
Choose c "- {u 9 3k -~ p.3j -~ k.3x(Clj(x) A Vy(Clk(y) ~ A(x, y, u)))}. Then
c is a q-class if p -~ q. Indeed, assume -,U~lqC, i.e.
-~Tq3k ~ p.3j -~ k. 3x(Clj(x) A Vy(Clk(y)---, A(x,y, u))).
(7)
Levels of Truth
252
[Ch.8
By the axiom BLQ of 42.3, we have:
Vk -4 p.Vj -4 k. ~Tq(3x(Clj(x)A Vy(Clk(Y)----, A(x,y,u)))).
(8)
Given k -4 p and j -4 k, we have j - 4 q and hence that CLj, CL k are qclasses (37.5 (vi)), whence we can erase Tq in (8):
Vk -4 p.Vj -4 k.Vx(Clj(x)---,3y(Clk(y ) A~A(x,y,u))).
(9)
In order to check that c is a q-class, we must prove:
Fq(3k -4 p.3j -4 k.3x.(Clj(x) AVy(Clk(y)~A(x,y,u)))) ,
(10)
i.e., by BLQ, Vk -4 pVj -4 k.Fq(3x(Clj(x) A Vy(Clk(y)~A(x,y,u)))), which
is easily seen to be equivalent to (9). We now claim:
Vu(uric ~ 3x(Cl(x) A Vy(Cl(y) ~ A(x, y, u)))).
(11)
=:~: by assumption, for some k -4 p, j -4 k and some j-class x, we get:
'r
) ~ A(x, y, u)).
(12)
Hence by RPC, we also have:
Vy(Cl(y) ---,A(x, y, u)).
r
(13)
were uric false, we should have:
Vk -4 p.Vj -4 k.Vx(Clj(x)--,3y(Clk(y ) A--,A(x,y,u))).
(14)
Hence by (6), we can find k -4 p, j -4 k and w in CLj such that
Vz(Clk(Z ) ~ B ( w , z ,
u)).
(15)
By RPC, we get
Vz(Cl(z)---,~B(w,z, u)),
(16)
3w(Cl(w) A Vz(Cl(z) ~ --,B(w, z, u))).
(17)
whence
But (17) implies, by assumption on A, B,
Vx(Cl(x) ---,3y(Cl(y) A -,a(x, y, u))). 0
By theorem 42.1 and 42.5, with application of the *-translation of 40.5, it is
straightforward to conclude:
42.6. COROLLARY
(i) II]-CA 0 is interpretable in T L R + RPC (actually LTL of theorem
42.2 is enough).
(ii) A~-CA o is interpretable in TLR*.
42.7. REMARK.
(i) The corollary holds true even if we add to the
Appendix
VIII.A]
253
subsystems based on 1-!~-CA and A1-CA the bar induction schema of 40.3.5,
and at the same time we strengthen TLR* with the schema
Cl(r) A P W O ( r ) - - ~ T I ( r , B ) ( B arbitrary). The new extension of TLR* is
obviously consistent by corollary 41.13.
(ii) Let Progr(r,c) be the formula Vx(Vy((y,x)~r~y~c)-~x~c). Then
T L R - + RPC proves:
Cl(r) A Cl(a)---+3c(Cl(c) A
A Progr(r, c) A Vd(Cl(d) A Progr(r, d) -~ c C_d)).
Since classes are closed under join and elementary comprehension in TLR-,
one can immediately verify that Feferman's system EM0[ + J + the nonuniform version of the so-called inductive generation axiom is interpretable
in T L R - + RPC (see Feferman 1979).
Appendix: on projectibility and stronger reflection
A (relatively) quick strategy for proving 39.5 (and actually far-reaching
generalizations of it) is to devise an arithmetical notation system N e for t;
this leads to reconsider work of the early seventies about non-monotone
inductive definitions.
The most comprehensive treatment we are aware of, is the long paper of
Richter-Aczel (1974), which is our reference text below for all details not
included here. There exists also a category-theoretic approach to generalized
recursion (e.g. Girard-Vauzeilles 1981, Ressayre 1982).
Coming to the point, we adopt the terminology and conventions of w
as far as the language of second-order arithmetic is concerned.
1. DEFINITION. If F: ~P(w)--, ~)(w), where F is possibly non-monotone, we
recursively define for a E ON:
(i) I(r, ~) = u {r(I(r,/~)):/3 < ~);
I(r) = u {I(r,~): ~ ~
ON);
if n E I(F), In] = least c~ such that n E I(F, e~+X).
(ii) Since the sequence (I(F,c~):(~ E ON)is non-decreasing with respect
to inclusion, it makes sense to define the closure ordinal IF] 6f r
IFI := least ~ such I(r,
~) =
I(r, ~ + 1).
(iii) Let A(u, X) be an arbitrary formula of second-order arithmetic with
Levels of Truth
254
[Ch.8
the only free variables shown; then A defines the operator:
r ( P ) - {m c ~ "~2(w)l=A[m,P]}.
If ~ is a class of formulas of Z 2 (e.g. zy is the class of arithmetical formulas),
an operator F is said to be 4, if F can be defined by a formula of zy.
2. LEMMA. / f F is arithmetical, the sequence (I(F,Z)./~ < ~) is uniformly
E I ( L a ) for ~ admissible > w. In particular, I(F, t3) E La, for each j3 < c~.
P R O O F : arithmetical formulas become
apply Al-recursion. El
A1
in La, if a > w; then we can
3. T H E O R E M
(i) There exists an arithmetical operator F whose closure ordinal is L,
i.~. Irl - ~.
(ii)
Indeed, F can be chosen so thai, for every n E w, P C_ w,
n e r(p)
where r 1 is a II~
iff n ~
r0(P ) or (r0(P)
g P and n E
rl(P)),
and r 0 is a bounded operator.
The proof can be found after lemmas 7.8 and 9.5 of Richter-Aczel (1974).
P R O O F of 39.5: if F is the operator given by the previous theorem, I(F, c~)
is a proper subset of I(F, c~+l) for every c~ < L. Hence the function:
I N ( a ) = least n E w with n E I(F, a + l ) and n ~ I(F, a)
is always defined on t and is trivially injective; lemma 2 grants that I N
satisfies the required definability conditions. El
This is not the end of the story, however; indeed theorem 3 is only the
starting point of a bold generalization, including ordinals, which immensely
overcome recursively inaccessibles. As a sample, we state a theorem of
Richter (1971).
4. D E F I N I T I O N
(i) A set-theoretic formula A is II 2 iff A is logically equivalent to a
formula of the form VXl... VXn3Yl... 3ykA , where A is bounded.
(ii)
If P is a class of ordinals, a reflects the sentence A on P iff
L aI=A implies L~I=A, for some/3 E P.
(iii)
a is II2-reflecting on P iff a reflects every II2-sentence on P.
(iv) a is recursively Mahlo iff a is II2-reflecting on Ad, the class of
admissible ordinals.
Appendix
VIII.A]
255
4.1. REMARK. a is H2-reflecting on ON, or H2-reflecting tout court iff c~ is
admissible > w. It is also easy to see that a is recursively Mahlo iff a is II 2reflecting on the class of recursively inaccessible ordinals.
5. T H E O R E M (Richter 1971)
(i) There exists an arithmetical operator F, whose closure ordinal is
the least recursively Mahlo ordinal #.
(ii) Moreover, there exist II~
n E w, P C w, then n E r(P) iff n E
6. COROLLARY.
r0(P)
or
I"l, 1-'o, such that for every
and n E F I ( P ) ) .
(r0(P) c_ P
The projectibility theorem of 39.5 holds if we replace
with #.
Below, we stick to the notations of 42.3 for bounded level quantifiers and
extended acceptable formulas of Zy" We now formalize the notions of
admissibility and inaccessibility in the language of truth with levels.
7. D E F I N I T I O N
(i)
Adm(i) is the conjunction of the following two formulas of s
Lim(i) := Vj ~ i.Vk ~ i.Hp -~ i.(j -< p A k -~ p);
II2-refl(i ) "- V fVg[Vj -< i . V x ( T j ( f x)---, 3p -~ i . T p ( g X ) ) ~
Vq -< i.3k(q -~ k -< i A Vj -~ i . V x ( T j ( f x) ~ Tk(gx)))].
(ii)
Inac(i) is the conjunction of Lira(i) with
T-Lira(i) := Vx(Tix--+ 3k -~ i.Tkx ) and
V I V g [ V x ( T ~ ( I x ) ~ Ti(gx))
---, Vj -~ i.3k(Adm(k) A j ~ k -~ i A V x ( T k ( f x ) ---, Tk(gx)))].
(iii) TMA, the Mahlo principle for truth, is the schema
V f V g [ V x ( T ( f x)--+ T(gx))-+ V j 3 k ( j -~ k A Inae(k) A V x ( T k ( f x ) -+ Tk(gx)))].
7.1. REMARK. The definitions are equivalent to the usual ones in the
intended model of w
(see the remark 4.1 above; if one chooses
f x = Ix = x] in the second condition of Adm(i), one readily has that, if
Adm(i) holds, then Vx3p -~ i.Tp(gX)-+ 3k -< i.VxTk(gx), i.e. positive
reflection relativized to i; cf. 37.8).
8. T H E O R E M . The theory TL + RPC + TMA is consistent.
P R O O F . We modify the construction of theorem 39.16 as follows; if a < / t ,
the least recursively Mahlo, we recursively define a new operation ~1 such
Levels of Truth
256
[Ch.8
that, for 6 < #:
Va(6 ) - U {~'l(fl)" fl < 6} if 6 is recursively inaccessible;
else, ~/'1(6)- I t ( r , ~ , ~f l[6, rr
).
Of course F, It, r are the operations of w
Observe that the notion of
recursively inaccessible ordinal is uniformly AI(La) for c~ admissible > w ;
thus the definition by cases used in ~1 does not lead out of the class of the
Al-operations. It is straightforward to check that, mutatis mutandis, all
conditions of 39.15 hold for ~1; hence we can define Cu - ( C T M , ~ I ) and
extend the interpretation of 39.16. Then TMA holds as immediate
application of II2-reflection of p over the class of recursively inaccessible
ordinals. D
On the axiomatic side, we propose a few easy consequences of TMA:
9. PROPOSITION. T L - + TMA proves:
(i)
Vi3k(i -< k A Inac(k)) (choose f - g - )~x.x in TMA);
(ii)
V x 3 i T i ( f x ) ~ V j 3 k ( j -< k A Inac(k) A V x T k ( f x));
(iii)
f " C L ~ CL. ~ 3 k ( A d m ( k ) A f " CL k ~ c n k ) .
10. PROBLEMS
1) Study the relation with recursion in the superjump functional sJ (see
Hinman 1978 for the relevant definition); for instance, is every set X C w
recursive in sJ, definable by a class in Ct,? Are the sets C w recursively
enumerable in sJ (which are known to coincide with IIl-sets), definable by
closed terms of Cu?
2) Charachterize the least ordinal for which the ~r-construction is no more
possible. There exist similar ordinals already below the first stable ordinal
cr0 (see Barwise 1975, Hinman 1978). For (r0 the lemma 39.5 fails badly: a0
is projectible into w, but there is an c~ < a0 which is not projectible into w.
3) W h a t is the proof theoretic strength of T L - + T M A ?
Rathjen (1991) is probably relevant here.
The work of
CHAPTER 9
LEVELS OF TRUTH
AND PREDICATIVE WELL-ORDERINGS
w
w
w
w
On well-orderings
Ramified hierarchies
Predicative well-orderings I
Predicative well-orderings II
In the previous chapter we announced that the system TLR of reflective
truth with levels is equivalent, as to its arithmetical content, to FefermanSchiitte predicative analysis (henceforth FS). In order to prove such a claim,
we are going to develop a proof-theoretic analysis of TLR. The first step is
to describe the standard primitive recursive well-ordering of type F0, the
Feferman-Schfitte ordinal, within the context of TLR. Indeed, we shall work
in a fragment MFR(p) of TLR, which includes: 1) the ground system MF c
of Ch. II with number-theoretic induction for classes; 2) axioms stating the
existence of the ramified hierarchy, generated by conditions of a fixed logical
complexity p, along suitable explicitly presented pseudo-well-orderings (in
the sense of 38.4).
In w we discuss two different notions in MF: pseudo-well-orderings
(orderings, which are well-founded with respect to classes, in short pwos)
and quasi-well-orderings (orderings which are well-founded with respect to
properties, in short qwos). We shall prove that MFp derives the analogue of
the Weyl principle for qwos and related transfinite recursion schemata. In
w44 we construct a formalized version of the second-order ramified hierarchy
%, the classical model-theoretic counterpart of predicativity. w167
contain an elementary presentation of the so-called predicative standard
well-ordering of type F 0 and a well-ordering proof within the fragment
MFR(p) of TLR. More precisely, we verify that for each c~ < Fo, the
segment of type c~ of the standard well-ordering is a pwo, provably in
MFR(p); in the special case where a = e0 (respectively Cw0, the first ~critical ordinal) MFR(p) can be replaced by MF c (MFp). The results are
optimal by the upper bound theorems of Ch. XI; there, we will establish a
constructive consistency proof of the theory TLR (MFc, MFp) within Peano
arithmetic extended by transfinite induction up to F 0 (Co, Cw0).
For the applicative-minded reader, we mention that, since a few years,
258
Levels of Truth and Predicative Well-Orderings
[Ch. 9
F0, as well as more powerful proof-theoretic ordinals, have found non-trivial
applications in the study of term rewrite systems.
w 43. On well-orderings
It is well-known that the notion of well-ordering (and more generally of
well-foundedness) is essentially second-order and it depends on the extension
of the universe 91 of second-order objects, be they sets, predicates or
functions over the ground level. In the familiar arithmetical case, this
dependence shows up in the non-absoluteness of the well-ordering notion,
with respect to the standard predicative interpretation (that is, suitable
segments of the hyperarithmetical hierarchy); there are straightforward
examples of primitive recursive linear orderings on w, which are HYP-wellfounded, i.e. well-founded with respect to hyperarithmetical sets of
numbers, and yet not truly well-founded (see Rogers 1967, Harrison 1968).
On the other hand, the whole power set of w is not necessary to test wellfoundedness. By the Kleene Basis Theorem of 41.7, if a linear ordering -< w
of w is well-founded within any collection 91 of sets C_ w, containing -<w
and closed under the hyperjump operation X ~ ' W ' ( X ) (for ~dY(X), see
41.2.2), then "<w is a well-ordering "in the real world" (compare with the
bar induction corollary of 41.13). {Such q.l.'s are properly included by
Friedman 1969 in the well-known collection of Mostowski's/3-models, the wmodels of second-order arithmetic Z2, which are absolute with respect to
II~-conditions}.
It is then natural to see how far we can p r o c e e d - w i t h i n the theories of
reflective t r u t h - in dealing with countable well-orderings. As we mentioned
in w38, there are at least two possible versions of the well-ordering notion;
and we already know that pseudo-well-orderings are pleasantly closed under
forms of transfinite recursion, provably in TLR.
Below we argue informally; nevertheless, it should be always clear how
to work out the results in the indicated axiomatic systems.
43.1. D E F I N I T I O N (we repeat 38.4).
(i) Let us first remind that if w defines a binary relation, i.e. w is a
property of ordered pairs, we keep using the infix notation x "~w Y in place
of (x,y)~lw.
Also, Field( -~w) is the term { x : 3 z ( X - ~ w Z V Z - ~ w x ) }
representing the field of "~w and LO(-~ w) means that "~w is a linear
ordering. If B(x) is a formula with the free variable shown,
Progr( -~ w,B) "- (VxrlField( ~ w))(Vy ~ w x.B(y) ~ B(x)),
On Well-Orderings
IX.43]
259
where Progr( -.4 w,B) is to be read "B is progressive" (relative to -4 w)" If
B ( x ) - xrlb, we write Progr( -4 w,b).
As usual, T I ( ~ w,B) "- Progr( -4 w , B ) ~ Vx(xrlField( -4 w ) ~ B(x)), while
T I ( -< w,b) stands for Progr( -4 w, b ) ~ Field( -4 w)C_ b.
(ii) We recall that -<w is called a pseudo-well-ordering-in symbols
P W O ( -~ w), and, in short, "<w is a p w o - iff Vb(Cl(b)~ T I ( ~ w, b)).
(iii) A linear ordering "<w is called a quasi-well-ordering-in symbols
QWO( -4 w), and, in short, ~ w is a q w o - i f f VbTI( -4 w,b)).
(iv) A qwo (pwo) -4 w is acceptable iff "~w is a class; -4 w is unbounded
(on its field) iff Vx(x~IField( -4 w ) ~ 3y(yTIField( -4 ~) A x -4 w Y))"
Given an acceptable unbounded pwo or qwo, we introduce the standard
notions of "zero" ( -4 w-least element), "successor" and "limit""
0.4 w " - the -4 w-least element of Field ( -4 w);
S .4 w(X)"- the -4 w-least element of {y" yT1Field ( -4 w) A x -4 w Y};
Lim(x) iff "x is nor 0 -~w neither a successor".
(v)
A q w o (pwo) -4wiS locally decidable via f and h, if h" a ~ a
and
f ' a ~ {0, 1, 2}, (where a - Field ( -4 w)), and for every x in a, we have
u
fx -
0 ifx - 0.~w
1 if x is a -4 w-SCCessor;
2 ifxisalimit;
(,)
hx - the predecessor of x, whenever f x -
1.
(**)
A pwo (qwo) is locally decidable iff it is locally decidable via some f and h.
If a pwo (qwo) -4 w is acceptable, its field and every initial segment of -4 w
(of the form {x: x -4 w Y}) are classes.
43.2.REMARK. The expression "quasi-well-ordering" is used by Crossley
(1969) for linear well-orderings of w, which are well-founded with respect to
recursive sets.
We now show that a strong transfinite
acceptable locally decidable qwos.
43.3. D E F I N I T I O N (cf. 9.14, for |
recursion
principle
below).
f " CL ~ CL "- Vx(Cl(x) ---+C l ( f x));
g. Field( -4 w) | CL | CL--, CL ":-- VxVuVv(x~lField( -.4 w) A Cl(u) A Cl(v)---+ Cl(gxuv));
holds for
260
Levels of Truth and Predicative Well-Orderings
.-
[Ch. 9
{u.
43.4. THEOREM (Special transfinite recursion along qwos). We can f i n d provably in M F - - a closed term ~h)~fi~w)~z)~x.Re[h,f, -4 w]ZX such that, if
h" CL---~CL, f " C L ~ C L
and -4w is an acceptable qwo, which is locally
decidable via gl, g2, then for every x in Field( -4 w),
m
gl x - 0 ---+ Rc[h, f, -4 w]ZX - hz;
(1)
D
gl x -- 1 ~ Re[h, f, -4 w]ZX - f(Rc[h, f, -4 w]z(g2x));
gl x - 2 ---, Re[h, f, -4 w]ZX - {u" 3v(v -4 w x A u~(Re[h, f , -4 w]ZV));
E(Field( -4 w),)~xRc[h,f, -4 w]ZX) is a class, whenever z is a class.
(2)
If hz C C L and f x C_ CL, whenever z, x are classes, then
Rc[h, f , -4 w]ZX C_ CL.
(3)
{NB: for simplicity, we omit the uniform dependence of Rc from gl, g2}"
PROOF. (1). Observe that the operation gl makes possible to define by
cases over the field of -4 w" Then the fixed point for operations implies the
existence of Rc[h, f -4 w] satisfying the given equations.
(2) It is enough to check that, if z is a class, then
Vx(xrlField( -4 w) ---*Cl(H(x, z))),
(,)
where H(x, z) - Rc[h, f , -4 w]ZX; then we can apply the join principle of 9.9.
But (.) is straightforward by -4w-induction applied to {x" C l ( H ( x , z ) ) }
with the assumption on h, f and -4 w"
(3) If G(z) - {x" Rc[h, f , -4 w]ZX C CL}, (.) implies
xrlG(z) ~ Rc[h, f , -4 w]zx C_ CL.
(**)
Hence we can proceed by -4 w-induction to show
Vx(xrlField( -4 w ) ~ xrlG(z)), under the additional assumption of (3). V1
We also mention that a version of the Weyl principle w holds for
qwos. With the notations of 38.4, T R ( y , A , -4 w,z) is a shortening for the
formula VuVxVu(xrlField( -4 w) ---*(urly(x) ~ A(u, x, yrx, z))).
Then we have, by an easy adaptation of 38.6:
43.5. THEOREM (WP for qwos). Let A ( u , x , y , z ) be a formula, which is
elementary extensional in y, z with the free variables shown. Then we prove
in MF-:
(i) if -4w is an acceptable qwo, z is a class, then there exists a class y
such that T R ( y , A, -4 w, z) holds.
Ramified Hierarchies
IX.44]
261
(ii) Under the same hypothesis, if y, y' are two classes satisfying
T R ( - , A, -.< w,Z), then for every x in Field( -< w), we have
Vu(u,y(.)
In general,
the
u,y'(.)).
hypothesis of 43.5 (i) above cannot
be weakened to
PWO( ~ w), unless we substantially enrich MF-, e.g. to TL-. Indeed, by a
theorem of Spector and Gandy, there is an elementary condition A such
that "<w is a well-founded recursive linear ordering of w iff a solution to
T R ( - , A , ~ w ) exists in HYP ( - t h e
collection of hyperarithmetical
sets C_ w). Thus no HYP-solution (in our case, no solution in CL) to
T R ( - , A, -< w) exists in general, whenever "<w is only HYP-well-founded;
furthermore, there are recursive pwos on w such that no solution at all
exists for T R ( - , A , - < w) (Friedman 1976). In positive form: postulating
closure of CL under elementary transfinite recursion along arbitrary pwos
is to require that CL is really much richer than in the simple inductive
models of MF-.
w44. IL~mified hierarchies
In this section we apply the special transfinite recursion theorem 43.4 to
prove the existence of the second-order ramified hierarchy % of classes of
natural numbers along any given acceptable unbounded locally decidable
qwo. The result can be extended to pwos in the formal setting of TL.
Let us first recall an informal definition of %. By L2 we understand the
language of second-order arithmetic, introduced in the previous chapter
(40.1). Thus L2-formulas are also closed under quantifications VX and 3Y
on sets of numbers, and atoms have the form t E X and t = s (t, s terms
built up from variables by means of + , 9 and successor function symbols).
44.1. DEFINITION. Let b~ be a family of subsets of w ( w - t h e
natural numbers):
(i) if
A
is
a
Xl,...,Xk, X1,...,Xn,
formula
of
s
with
free
variables
in
the
set of
lists
~f~A[nl,...,nk, P1,...,Pn] stands for the usual
satisfaction relation: it is understood that n l , . . . , n k E w are assigned in the
given order to X l , . . . , x k and number quantifiers range over w;
P1, " " , P n E ~f are assigned to X 1 , . . . , X k and set quantifiers range over the
family ~.
262
[Ch. 9
Levels of Truth and Predicative Well-Orderings
P C_ w is ~ - d e f i n a b l e (with set parameters) iff
P - {n E w" (w, ~f)I=A[n, Q I , ' " , Qn]},
for some s
A(x, X1,...,Xn)
Q1,'-', Q , E :f of set parameters.
(i) n e f ( ~ f ) -
and
a
(possibly
empty)
list
{ P C ~" P is ~f-definable}.
(ii) We define by recursion on countable ordinals an operation ~o such
that
% 0 - nef({w});
(where A is a limit),
zJ~a+ 1 --
Def
(%a) and
%~-
U %~3
f~ < ~
44.1.1. REMARK. (i) Def({w})extensionally coincides with the family of
the subsets C_ w, which are definable by s
with no set quantifier
and no set parameter. If P E %0, P is called a r i t h m e t i c a l .
(ii) Once one has accepted w and a segment A of ordinals, the definition
of % is predicative in the traditional sense: set quantifiers range over
countable collections, which are already built up.
(iii) We mention a few basic facts about % (for definitions and proofs,
see Apt-Marek 1974, Moschovakis 1974, Boyd-Hensel-Putnam 1969, Kleene
1959, Jockusch-Simpson 1975).
1.
% is a hierarchy, i.e. c~ < fl implies %a C_ %~.
2.
There exists a countable ordinal fl0 such that % ~ o -
%f~o+1 and %f~o
is the smallest fl-model of second-order arithmetic Z 2 (Gandy-Putnam).
3.
flo > w~k ( -
4.
Every set of %f~o is A~ and so is %~o"
5.
%
ck -- H Y P
the first non recursive ordinal) and flo is Al-definable;
- the collection of hyperarithmetical sets C_ w.
w1
44.2. DEFINITION.
Recall that a - e b " - Vx(xTla ~ x~lb); if we put
E m b e d ( g , ~1, ~2) "- Va(ar]~l --+ (ga~l~f2 A a - e ga));
then we define:
(i)
3'1 _C + if2 "- 3 g E m b e d ( g , ~fl, ~f2);
(ii)
~fl - + ~f2 : - 3'1 _C + ~f2 A ~2 C_ + :fl"
Any pair g, h such that E m b e d ( g , 3'1, 3'2) and E m b e d ( h , :f2, ~fl) is said to
w i t n e s s 5~
+:f2" Of course, C_ + is reflexive and transitive and, by
definition of E m b e d with - ~ , it follows that, if g and h witness ~ f l - + :f2,
Ramified Hierarchies
IX.44]
263
g is 1-1 onto with respect to - e" Indeed, we have VaVb(ga - e gb---, a - e b);
furthermore
Embed(g,~~
~~ ]k Embed(h, ~f2, ~fl )
---. Va(a~f 1---, h(ga) - e a) A Vb(br/~f2 ---. g(hb) - e b).
44.2.1. FACT. M F - proves:
C l ( l l ) A C/(~f2) A ~1 C C L A ~2 C_ C L . ~ T(~f2 C_ + 5'1) V F(~ 1 C e-t- ~2)"
( The same holds with
- + in place of
C_ +).
Of course, the relations - +, C_ + are natural strengthenings of extensional
equality and inclusion; we need them to capture the extensional features of
the ramified hierarchy in our non-extensional framework. We also remind
that MFp " - M F - + P-IND, and that property induction P-IND has the
form:
Clos(a) ---. N C a;
here C l o s ( a ) ' - Orla A Vx(x~?a---, (x + 1)r/a); N stands for the class {x" g x }
of MF-.
44.3. THEOREM (MFp)
We can f i n d - u n i f o r m l y
in any given acceptable unbounded qwo "~ w,
locally decidable via gl, g 2 - a n operalion Ax.~-POx such that, if x is in the
field of -~ w, then
(i)
g~-
0 ~ ~ -
D~f({N)) (i.~. iS 9 i~ th~ -~ w-Sight ~ l ~ , ~ t ) ;
gl x -- 1 ~ %x -- D e f ( % g 2 x ) U %g2x (if x is the ~ w-SUCCessor
of g2~);
g~
-
2~ ~
- (~- 3v(~ -~ ~ ~ ^ ~ ~ ) } ,
(if ~ i~ ~ -~ ,~-limit);
(ii) %= is a class of classes and %x C_ + %u' whenever x ~ w Y"
PROOF. The argument requires a number of separate steps and definitions;
the essential point is to introduce the operation D e f in our language.
First of all, we associate to each primitive function symbol f of s a closed
)~-term f*, which formally represents it (see Ch. I, 3.6).
We then define a satisfaction predicate S A T [ ~ , f l , f2 ] for s
uniformly in any fixed class ~f of classes C N, and in any pair of operations
/1 " N ~ N , f2" N--~ ~'.
To this aim, we fix a G6del numbering G D of s in such a way that all
the syntactical notions (term, formula, occurrence of a free variable, etc.)
define classes (provably in MFc); G O ( E ) ambiguously denotes the G6del
number of the expression E and the term, formally representing it. For2(x )
Levels of Truth and Predicative Wel/-Orderings
264
is the predicate "x is the Ghdel number of an s
predicate "x is the G6del number of an s
Put
var2(i ) "-- GD(Xi);
ins(GD(t);
[Ch. 9
Ter(x) is the
var2(i)) "- GD(t E Xi);
Iden(GD(t), GD(s)) "- GD(t - s).
If f" N---+~ ( ~ . - N or ~f) and b is in it;, then f(~)" N ~ !/; is the operation
defined by: f(~b)n- fn, i f i ~ - n , f ( ~ ) i - b ( here f ( ~ ) i s well-defined with
definition by cases on N). Moreover val(u, f l ) i s (the formal presentation of)
the operation, which associates to the term encoded by u its numerical
value under the assignment fl"
44.3.1. Let W ( ~ , f l , f2, b,k) stand for (the formalization of) the following
condition (as to logical complexity, cf. Introduction, 5.5)"
(i) every element of b has the form (GD(A),fl, f2) , for some
L2-formula A of logical complexity < k (k being a natural number), where
f~" N ~ N , f 2 " N ~ ;
(ii)
(GO(A), f l, f 2)~lb iff either:
1.1. A - (t - s) and val(GD(t),fl ) - val(GD(s),fl); or
1.2. A -
t E X i and val(GD(t),fl)~I(f2i); or
2.
A - -,B and not (GO(B), f l , f2)rl b; or
3.
A - B 1 A B 2 and (GD(Bi) , fi, f2)~ b, (GD(B2), f l , f2)rl b; or
4.
A - VxiB and VkrlN. (GD(B), fi(~), f 2)rlb; or
5.
A - 'r
and VP~I:f. (GD(B), f l,
i b.
f2(p))
Then we prove:
44.3.2. There is a term F -
F(~f, f l , f2) such that
(i) MFp F (Cl(~) A ~ C CL A f l " N ---. N A f2" N ~ ~)
--+Vn~IN. (Cl(Fn) A W(~f , f l, f 2, Fn, n));
(ii) for each n E w,
^
c CL ^ f l "
N ^ f 2"
-+ (Cl(F~) A W(~f, f l , f2, F~, ~)).
Verification of 44.3.2 (i). Let Co(fl, f2) , C l ( f l , f2) be the terms (in the
given order)"
{x" 3u3v(x -- (iden(u, v), f l, f 2) A Ter(u) A Ter(v) A val(u, f l) -- val(v, f l)));
IX.44]
265
Ramified Hierarchies
{z " 3u3i(N(i) A Ter(u) A x -- <ins(u, var2(i)) , f l,
f2) A val(u, fl)rl(f2i))).
By recursion on N, we define, uniformly in Y, fl, f2:
FO - Co(f x, f 2) u C1(/~,
f2);
F(n + 1) - Fn tJ {(GD(A), fl, f2)" A has logical complexity exactly n + 1
and
1)
2)
A - ~B and --(GD(B),
fl, f2)r/(Fn);
or
A - B 1 A B 2 and (GD(B1), fl, f2)o(Fn) and (GD(B2),
11, f2)~(Fn);
or
f l(~), f z)o(Fn); or
VP~Y. (GD(B), f l, f 2(~p))~(Fn).
3)
A - VxiU and Vk~?N. (GD(B),
4)
A - VXiB
and
Then we easily have, by hypothesis on ~f, fl, f2:
CI(FO) A W(~, fl, f2, F0, 0).
On the other hand, since the formula which defines F(n + 1) is elementary
extensional in ~f and Fn, then F ( n + l )
is a class and satisfies
W(Y, fl, f2, - , n + 1) (argue by N-induction on properties and elementary
comprehension).
As to the second part of 44.3.2, one applies metamathematical induction on
natural numbers instead of property N-induction. I-!
Let K o m " N--,N be the (primitive recursive) operation, which assigns
to each element of For 2 its logical complexity. Then we define"
44.3.3
(i) S A T n [~f,f l , f2] -- {x"
For2(x ) A (x, fl,
f2)~Fn};
S A T I n , f l, f2] - {x " For2(x ) A (x, f l, f 2)rlF(g~
(ii) By induction on A (A arbitrary L2-formula):
(t E Xi)[~, fl, f2] - val(t, fl)r/(f2i);
(t -
s)[~,f l, f 2] - (val(t, f l) - val(s, f a));
(-~B)[Y, fl, f2] - --((B)[Y, fl, f2]);
(B A C)[~f, fl, f2] - (B[Y, fl, f2] A C[~f, fl, f2]);
(VxiB)[Y, fl, f2] - Vkr/N. (B[Y, fl(B), f2]);
(VXiB)[Y, fl, f2] - VP(Pr/Y ~ B[Y, fl, f2(P)])'i
Then we obtain:
9
Levels of Truth and Predicative Well-Orderings
266
[Ch. 9
44.3.4
(i) MFp ~ (Cl(~) A ~ C CL A f l " N--~NA f2" N ~ )
---+Vn~IN. CI(SATn[y, f l , f2]) A CI(SAT[Y, f l ' f2])"
Moreover MF c F (el(Y) A ~ C CL A f l " N ~ N A f2: N ~ Y)
--* CI(SATn[ y, fl, f2]), for each n E w.
(ii)
For every a C L 2 of complexity n (n fixed),
^
c_ CL ^ fl: N
N^
(GD(A)~SATn[ y, fl, f2] ~ a[~f, fa, f2]);
(iii)
MFp ~ [CI(Y) A Y C_ CL A fl" N ---, N A f2" N---. Y A f3" N---, Y A
A VkrlN. (f2 k - ef3k)] ~ Vx(zrlSAT[Y, fa, f2] ~ xrlSAT[~f, fl, f3])"
The previous statement 44.3.4 (iii) can be proved in MF c if we replace S A T
by S A T n for fixed n G w.
Verification of 44.3.4. (i) holds by 44.3.2 (i), while (ii)follows by outer
induction on the complexity of A. As to (iii), proceed by property induction
on Kom(x), while the second part applies the second part of 44.3.4 (i). 0
Of course, we can define the operations:
extYn[x, i, f l , f2] -- {k" kr/N A xrlSATn[y, fl(/k), f2]}"
Defn(y) = {b'(3x)(3fl" N --+N)(=lf2 9N ---+Y)(3i)(For2(x ) A b - extOrt[x,i, fl,
f2])}.
Def(~f) -- {b: 3n(brlDefn(~))}.
44.3.5. Then we have in MFp:
(i) if ~f C_ CL and el(y), then Cl(Def(~)) and Def(~) C_ CL;
(ii) if ~f - + ~ and ~f, %t C_ CL are classes, then D e f ( ~ ) - +De f(%);
(iii) If D e f n, for n fixed, replaces Def in (i)-(ii) above, then the
corresponding statements are provable in MF c.
Verification of 44.3.5.
(i) follows by 44.3.4 (i). As to the subsequent point (ii), let g" :f---.cK,
h" ~---+ ~f be a pair of maps witnessing ~f - e+cU.. Then we can introduce two
operations g^" Def(~)--. Def(CK) and h ^" D e f ( ~ ) ~ Def(~), such that
g^b - ext~n [u,i, f l , g f 2] and h^c - extYn[v,i, f l, hf 3].
(*)
In (.), b - extYn[u,i, fl, f2]~lDef(~), c~lDef(U), u, v are in For2, i is the
IX.44]
Ramified Hierarchies
267
index of the individual variable free in u, v, f l " N--~N, f2" N--+Y,
f3" N ~ ~ and (gf2)J - g(f2J), ( h f 3 ) J - h(f3J) (where j is in N). That g^,
h ^ witness Def(~)--+e D e f ( ~ ) i s
checked by class N-induction on the
complexity of the formula (encoded by) u (or v), using 44.3.4(iii). As to
(iii), the proof is similar. 0
By 44.3.5 and the assumption on -< w, we are under the hypothesis of the
special transfinite recursion theorem of 43.4: hence there exists an operation
Ax.% z satisfying the equations of 44.3 (i). It only remains to check:
%x C_ e+ %y, for x -< wY"
Let
I x u b - {k" k~N A GD(v o e X o ) ~ S A T [ % x, f l ( Ok), f2( ~ )]}"
Clearly I z y embeds %x in %y, provided y is the
xqField( -< w)" Then we recursively define:
if gy - 0 or gy - 2, Cxub - b; else, if gy - 1, Cxyb
-
<w-SUCCessor of
I(hy)y(r
).
By -< w-induction for classes, we obtain
(Vy~lField( -< w))(Vx -< w y)Embed(r
%x, %y)" n
We next consider a refinement of 44.3, where we deal with a hierarchy
whose defining conditions have an a priori bounded logical complexity.
44.4. COROLLARY (Existence of the bounded ramified hierarchy, provably
in MFc). Let k be a natural number. Then:
(i) for every acceptable unbounded locally decidable qwo -~ w, there is
an operation Ax.~
such that:
1.
if x is in the field of ~ w, ff~kx is a class of classes C N;
1.1
~
1.2
%k _ D e f k ( %
1.3
(v.
2.
(ii)
- D~fk({N}), w h ~ ~
~ i~ Lh~ f/r~ ~ l ~ , ~
of -< ~;
U %y, if x is the -< w-successor of y;
y
c + % ,k whenever x-<
9
a
mi ;
y.
The same holds, provably in TLR, /f we replace qwo with pwo in (i)
above,
PROOF. (i) Simply replace D e f by D e f k in the definition of % and
observe that by 44.3.4-44.3.5(iii) we can proceed by N-induction on
classes, once we have a fixed bounded logical complexity.
(ii): similar to the proof of WP in w38. Fix a natural number k and choose
268
Levels of Truth and Predicative Well-Orderings
[Ch. 9
a level j such that Field( -~ w) is a class of level j. By 37.8 (vi), there exists
a level i, such that CLjrICLi, i ~- j ( ~- is the level ordering !) and hence
d - {x" xrljField ( -,1 w)A Clj(~k)}
is an /-class. By 44.3.5 and l e m m a 37.2, CLj is provably closed under
Def k, i.e. T L R proves
~f C_CLj A Clj(~)---, nefk(~) C_CLj A Clj(nefk(~)).
Hence d is -~ w-progressive and, being an /-class, is also a class. So the
conclusion follows by class induction on -~ w" n
44.5. Concluding Remarks
(i) Relativization. The construction of a)o is easily generalized to classes
other than N. Given any fixed subclass P of N, we can introduce the
hierarchy )~x.fJt,x(P ). The definition is analogous, except that s
is
expanded with a new predicate, whose interpretation is P itself, and we put
% x ( P ) - Def({P}), if x is the first element of the given qwo. It is also true
that if P - eQ, then ~t,x(P ) - + ~x(Q)"
(ii) Closure of limit segments under comprehension.
Let a be in a)ou for y -~ w x, where x is a limit of Field( -~ w)" Assume that
A is a s
with the free variables Xo, v0 and let
A(k, a) - A[~lt,y, fl(~), f2(~
(see 44.3.3),
where f l " N---,N, f2" N---,~,,y. Then we can find a class b - e{k'A(k,a)}
such that br]~t,x (apply 44.3.5(ii)); in such a case we simply say that
{k: A(k,a)} is in ~lt,x (this is not misleading, if we are working in contexts,
which extensionally depend on {k: A(k,a)}). Of course, the closure property
above also holds for limit segments of the bounded ramified hierarchy, as
soon as we deal with A ' s of the appropriate logical complexity.
(iii) ~,, only depends on the isomorphism type of the given qwo.
Indeed, let-~ 1, "~ 2 be qwos, which are acceptable and locally decidable.
"~ 1 is isomorphic to "~ 2 (in short "~ 1 -~ "~ 2) iff there are operations
f l : Field( -~ 1)---* Field( -~ 2), f 2 : Field( ~ 2 ) ~ Field( -~ 1), such that:
1. f 2(f l x) = x and f l ( f 2Y) = Y, whenever x belongs to Field( -~ 2) and
y belongs to Field( -~ 2);
2. u -~ iv implies f l u -~ 2 f l v and x -~ 2 Y implies f2 x -~ 1 f2Y, for every
u, v in Field( ~ 1) and x, y in Field( -~ 2)"
Then we can easily check, by induction and using 44.3.5 above:
if -~ 1 -~ "~ 2 via g, h, 9 t , ( 1 ) x - + O,(2)g x and "Jt,(2)y- + %(1)hv, where
%(i) is the ramified hierarchy along - ~ i ( i = 1, 2 ) a n d xyField(-~ 1),
y~Field( -~ 2)"
Predicative Well-Orderings I
IX.45]
269
w45. Predicative well-orderings I
The aim of w167
is to introduce a specific primitive recursive ordering
"~ w of N and to show that -~ w is isomorphic to a certain ordinal, which is
known in the literature as F o. F o is intimately connected with the theory
T L R of w36, and it is essential for the proof-theoretic investigation of the
formal systems, outlined in this book.
In the foundational literature there are classical and exhaustive
expositions of this matter (in particular Schiitte 1977); so we shall only
state the basic facts, needed for describing a notation system for F 0. Proofs
are mostly omitted or simply outlined. However, w will contain a detailed
well-ordering proof for F o in a fragment of TLR.
45 A. Preliminaries: Veblen hierarchies ( Veblen 1908)
In this subsection c~, fl, 7, 6, ~ range over elements of Q, the set of
countable ordinals; < is the natural ordering of Q.
45.1. D E F I N I T I O N
(i) f" f~--. f~ is increasing iff c~ < fl implies f(c~) < f(fl), for every c~, ft.
In particular, if f is increasing, f(c~) > c~ holds for arbitrary c~.
(ii) f 9f ~ f ~
is
continuous iff for every limit ordinal
f(A) - sup f(~). If f is increasing and continuous, f is called normal.
13<A
(iii) If Y C_ X C_ g2, Y is bounded in X iff there is a 7 E X such that, for
all~EY, fl<3'.
(iv)
in X.
X C fl is closed iff (supY) E X, for every Y C_ X, which is bounded
(v) XC_fl is unbounded iff for every a there is a f l E X such that
a < ft. If X C fl is closed and unbounded, we say for short that X is a club.
(vi)
Let X C fl: by transfinite recursion we may define
E x ( a ) - the least ~ E X such that for every fl < c~, Ex(fl) < ~.
E X is defined on a segment of fl and is called the canonical enumeration of
X. The following properties hold:
P.1. X C_ f~ is a club iff E x is normal. Conversely, if f is increasing, f is
normal iff f[fl] is a club.
P.2. Let f "
f~ ~ f~ be normal and consider f i x ( f ) - {a " f ( a ) - a}. Then
As usual, we write f ' for E f i z ( f ) and f ' is called the
f i x ( f ) is a club.
derivative of f. ft can be given a recursive definition, which displays the
Levels of Truth and Predicative Well-Orderings
270
[Ch. 9
iterative nature of ft.
Set f ~
a; f k + l ( c ~ ) - f(fk(o~)). By transfinite recursion, we define:
if(O) = sup
fn(f(O) + 1);
new
ft(a + 1) = sup fn(ff (a) + 1);
new
ft(A) = sup ft(fl).
~<~
P.3. Let { X a } a < 6 be a family of clubs such that, if/3 < 7 < 5, X.y C_ X~.
Then f3 X c~ is a club.
c~<6
P . 1 - P . 3 justify the introduction, again by transfinite recursion
hierarchy of normal functions, uniformly in any given normal F.
of a
45.2. The Veblen hierarchy starting with F.
V~ -- F;
V~ +1) --(V~F)t;
if A is a limit, V ~ -
E x, where X -
M (V~F)[~2].
~<
Here V~F(a) stands for the value of V~F applied to a. By construction, if
3' < 5, we have
VTF(V6F(a))- V6F(a).
45.2.1.
We write g(Z,~) for V~(~), whenever V is Clear from the Context; in the
speCial r
where r = ~ . ~ o ~, we use, following SCh~itte, CZ~ for g(Z,~).
r
is sometimes replaced by the more familiar notation ca; so, as a special
case, we have co = r
= the least solution of wa = a.
P.4. For all C~l, ill' C~2' f12 and R := < , = , we have
g(al, ~1) R g(a2,/32) iff
a I < a 2 a n d / 3 1 R g(c~2, f12), or
a 1 = a 2 and fll R/32, or
a 2 < a 1 and g(al, ill) R 132.
N . B . P . 4 does not depend on the initial normal function F.
Besides that, we have:
P.5. The function g(c~)= g(a, 0 ) i s normal.
45.2.2. D E F I N I T I O N : Fa is the a - t h solution of r
---/3.
Predicative Well-Orderings I
IX.45]
F r o m P.4, P.5, using the definition of f~(0) for f = Afl.r
F 0 is the least ordinal > 0 , which is closed under AaA~. Ca/?.
271
it follows that
45 B. Normal forms. Notations
The problem of constructing a notation system for a given countable ordinal
a is essentially algebraic: one has to find out a system of finitary operations,
which unambiguously represent the ordinals < a, starting with an effective
set of generators. In this section, we face the problem for the predicative
ordinals from an elementary point of view (for the category-theoretic
methods of IIl-logic, we send the reader to Girard 1982).
We first state the Cantor normal form theorem (for a proof, Schfitte 1977):
45.3. T H E O R E M . For every ordinal 7 Ys O, there are unique 71 >- ... >-7n,
n >_ 1, such that 7 - w'rl + . . . + w'rn.
Then we apply 45.A and extend theorem 45.3.
45.4. LEMMA. For every a E (0, Fo) of the form w ~, there are unique al,
a 2 such thai a = r
2 and a > a l , a 2.
P R O O F . The set X -- {~f: " / < c~ A c~ < r
is non-empty (c~ < r
by P.4
and P.5). Hence we can pick out its m i n i m u m c~1. If c~1 - 0 , choose c~2 - ~i;
since a < r
a2= ~< r
If c~1 r 0, we get: (V7 < c~1)(r
= a), i.e.
a - - C a l ~ , for some ~. By choice of al, Cal~ < C a l ( r
i.e. ~ < Cal~
(P.4). On the other hand, since a 1 _< a < F0, r
_~ r
> C~1.
As to the uniqueness, assume a = r
2 =r
and C a l a 2 > a2, al,
r
> ill' f12" By P.4, we must have a 1 = fll and hence f12 = a2" [:!
45.5. C O R O L L A R Y (Extended normal form for ordinals < Fo). For every
ordinal c~ E (0, Fo), there exist unique Oil, i l l , ' " , O~n, fin, n >_ 1, such that
(i)
r
i > ~i, c~i (i E [1, n]);
(ii)
r
1 ~ . . . ~ c/)O~nfln;
(iii)
c~ = r
fll -F... + r
P R O O F : apply 45.4 to the Cantor normal form. E!
Below, we temporarily adopt the symbol =
means that t and s coincide).
also for literal identity (t = s
Levels of Truth and Predicative Well-Orderings
272
[Ch. 9
45.6. A system of ordinal notations. We consider a term-language which
contains:
(i) a new constant e.
(ii) a concatenation operation 9 and a binary function symbol f .
The set T E R of terms is inductively generated from c by closing off under
the clause:
if tl, S l , . . . , tn, s~ (n >_ 1) are terms, so is f ( t l , sl) 9 . . . , f ( t ~ , s ~ ) .
The degree of t E T E R
gr(c)
-
is recursively assigned:
0;
gr(:f(tl, S 1 )) - - gr(t 1) + gr(s 1) + 1 and
gr(t 1 , . . . , tn) - gr(t 1 ) + ... A- gr(tn) (n > 1).
We recursively define the character cr(t) E T E R , for each t E T E R :
t
C if n > 1;
cr(c) - c; if t - f ( t l , s l ) , . . . , f(t,~,sn), let cr(t) (
t I i f n - - 1.
We
now
introduce
:< C _ O N x O N .
a
structure C o V - ( C N , :<,c), where C N C_ T E R ,
"_<sfor t ~ z s o r t - s , a n d t > _ s f o r s
"_<t.
Wewritet
45.6.1. Simultaneous recursive definition of C N ,
(i)
~
9
t E C N iff either t - c, or
t - f ( t l , s l ) , . . . , f ( t n , Sn), for some ti, s i , . . . , tn, s n E C N ,
such t h a t f ( t i , si) >_ ... >_ f ( t n , Sn) and cr(si) :(. ti, for i -
(ii)
1,...,n.
t ~: s holds iff s ~ c and one of the following cases holds:
1.
t - c , or
2.
2.1.
2.2.
3.
3.1.
3.2.
3.3.
t - t 1 , ... , tn, s - s 1 , . . . , s i n , t, s E C N , n - F r o > 2 and
n<mandt
i-si,foralliE[1,n],or
there is a j _< n, m, such that tj :(. sj and t i - si, for all
l _ < i < j , or
t - f ( t l , S 1), S - f(t2, 82) , t, s E C N and
t I - t 2 and s 1 ~: s2, or
t 2 ~: t 1 and t ~: s2, or
t l :< t 2 a n d s l ~ : s .
Clearly 45.6.1 can be reduced to a s t a n d a r d course-of-value recursion, and it
is s t r a i g h t f o r w a r d to check"
45.7. L E M M A . C N , ~. are primitive recursive.
T h e n we have:
IX.45]
Predicative Well-Orderings I
273
45.8. LEMMA. (i) C N is linearly ordered by :<.
(ii) /f t, s E C N , cr(s) < t iff s g: f ( t , s ) ;
(iii) /f f ( t , s ) E C N , then s ~ f ( t , s ) and t g: f ( t , s ) ;
(iv) f f t , s E C N a n d t : < c r ( s ) , t h e n t : < s .
P R O O F (Hint). The proof of (i) entirely relies on the recursive definitions of
45.6.1 and is carried out by appropriate induction on the (sum of the)
degrees of the terms involved (details can easily be supplied by the reader or
adapted from Schiitte 1977, Feferman 1968). (iii)easily follows from (i)-(ii).
Verification of (ii). ::~: by induction on gr(s). If s = c, the claim is trivial.
Let s - f ( t l , Sl) c: C N and c r ( s ) - t 1 <_ t. As s E C N , we have tl, s 1 ~_ C N ,
cr(sl) <_ t I and hence also cr(si) ~ t. So, by IH, since gr(sl) < gr(s),
s 1 < f(tl, Sl);
(la)
s I < f(t, sl).
(lb)
If t = tl, (la) suffices for s k: l ( t , s ) . Let t 1 k: t: then ( l a ) - ( l b ) and kZtransitivity yield 81 < f(t,s), whence s :< f ( t , s ) .
If s - f ( t l , S l ) , r e C N , t h e n / ( t l , Sl) E CN; so tl, s I E C N and cr(sl) ~ t 1.
As in the previous case, (la) holds by IH. We then apply ~:-linearity. If
t I = t, we have s 1 ~ f ( t l , Sl) 4 : / ( t l , s l ) , r , whence s :< f ( t , s ) . If t I ~: t, also
(lb) above holds and we can repeat the same argument. If t ~ tl, we have
by definition of k:, f ( t l , S l ) < f ( t l , S l ) . r , and hence s k: f ( t , s ) .
r
if s - c
or s - t 1 , . . . , t n with n > 1, the claim is trivial. Assume by
contradiction s - f ( t l , Sl) , s k: f ( t , s ) and t k: c r ( s ) = t 1. Then by 45.6.1 (ii)
s k: s: against (i).
Verification of (iv). t k: c r ( s ) i m p l i e s s = f ( t i , si); so, by definition of cr(s)
and (iii), t k: t I k: f ( t l , s i ) - s. [-1
The proof of 45.8 is formalizable in a a weak fragment of OP.
45.9. DEFINITION. By recursion on gr(t) we define a m a p F : C N - - . F o ,
such that:
F(c) = 0 ;
F ( l ( t l , S l ) ) . . . . . l ( t n , Sn) ) = t F ( t l ) F ( s n ) + ... + t F ( t n ) F ( s n ) .
45.10. LEMMA. If t, s e C N , then t g: s implies F(t) < F(s).
P R O O F : by induction on the definition of ~ with P.4. E!
45.11. C O R O L L A R Y .
~: is well-founded on C N .
P R O O F : any strictly descending sequence in C N
(via F ) a strictly descending sequence in F 0. [-1
t o :~ t I ~ t2... becomes
Levels of Truth and Predicative Well-Orderings
274
45.12. T H E O R E M . F is an isomorphism between C ~ and F -
[Ch. 9
(F0, < , 0).
PROOF" F is already .~-preserving by 45.10. If t, s C C N and t r s, we
have t~: s or s ~ t
(45.8); hence, by 45.10, either F ( t ) < F ( s ) o r
F(s) < F ( t ) , i.e. F is injective. But F is surjective, i.e., for every c~ < F 0
there is a term t C C N such that F(t) - ~. We argue by induction on c~. If
c ~ - 0, put t - c. If c~ :/: 0, let by lemma 45.4, c~1, i l l , - ' - , C~n, fin such that
r
> fli (1 < i _ n), a > r
1 >__ ... ___ r
n and
a - - Caafl 1 - ' ] - . . . +
Canfl n.
By IH, there are tl, S l , . . . , tn, s n in C N with F ( t i ) - c~i, F ( s i ) - fli for
every 1 _< i _< n. Since F is <-preserving and <: is linear, f ( t i , s i ) ~ s i for
every 1 _< i _< n and f ( t i , si) >_ ... ~ f ( t n , Sn). Hence by 45.8 (ii) we have
cr(si) <_ t i for i - 1,..., n and we can choose
t -- f ( t i , S 1) * . . . * f ( t n , Sn).
!-1
We do not develop ordinal arithmetic in CN. It is possible to represent
the standard operations of ordinal sum, product, exponentiation and the &
function, as primitive recursive operations on the terms of C N . As a byproduct, the elementary properties, which are usually proven by transfinite
induction, become provable by N-induction. As samples, we present the
definitions of ordinal sum, successor, &function, w-exponential.
45.13. D E F I N I T I O N . Let t,s C C N .
(i)
t+c-t-c+t.
Let t - t l , . . . , t m ,
S-Sl,...,s
n (re+n>2),
are principal terms, i.e. of the form f(a,b); then
t+s-
~ s
i f t 1 ~: Sl;
tI ,...,
(ii)
(iii)
](t,
(iv)
(v)
where t l , . . . , t m , s l , . . . , s ~
tj 9 s I , . . . ,
sn, for j - the largest j such that tj ~ s 1.
The successor of t" t + 1 - t + f(c, c).
The internal &function j~ 9C N • C N ~ C N :
8)
f
l ( t , s),
if cr(s) ~ t;
[
s
otherwise;
w t - f(c, t);
wt.0_c;
wt(n+l)-wt.n+wt.
Predicative Well-Orderings I
IX.45]
45.14. LEMMA. I f t, s E C N ,
275
then t + s, t + l, f ( t , s ) ,
w t, 03 t . n are in
CN.
If we apply 45.8 (iii)-(iv) and the basic definitions, we readily see that je
behaves like r and has a useful syntactical property:
45.15. LEMMA. Let t, s, ti, s i E C N ( i -
1,2).
(i) s <_ f ( t , s ) and t < f ( t , s ) ;
(ii)
s < r =:Vf(t, s) < j~(t, r);
(iii)
if R is -
or ~ , f and R satisfy the analogue of P.4:
f ( t l , S l ) R f ( t 2 , s2)
iff t 1 - t 2 and s i R s 2 ,
or t 2 :< t 1 and tRs2, or
t 1 ~. t 2 and s i r s .
(iv)
j ' ( t , r ) + l ~: s ~ j~(t,r+l) ::r g r ( f ( t , r ) + l ) : <
gr(s).
45.15 implies that the Cantor normal form theorem and its extended
version
(cf. 45.3-45.5) have natural counterparts for notations. So, in
particular, by definition of C N and w-exponential, we get:
45.16. LEMMA. For every t C C N ,
if t ~ c, there are unique canonical
terms t 1 > . . . > tn, n >_ 1 , such that t -
t
w tl + . . . + w n.
45.17. DEFINITION. If t , s E C N ,
c, if s w t. S
--
WtTsl
and
c;
- t - . . . Jr- o ) t T S n ,
provided
S - - Ws l + . . .
-[- r
,
8 1 ~__... ~__ 8 n.
Of course we have:
45.17.1. I f t, s E C N ,
then w t. s E C N .
45 C. Enriching the ordinal structure of C N : f u n d a m e n t a l sequences
From a constructive point of view, a (countable) limit ordinal )~ is given
only if we have an effective sequence ()~[n] : n E w) such that
(i) ~[n] < ~ [ n + 1] < ~, for each n E w;
(ii) if fl < A, we can find-effectively in fl and ~ that fl < A [k].
a number k such
We now assign to each limit term a primitive recursive sequence
(tin]: n E w) of CN-terms, which converges to t. First, we need a few
Levels of Truth and Predicative Well-Orderings
276
[Ch. 9
preliminary notions.
45.18. D E F I N I T I O N
(i) If t E C N , t is a successor iff t has the form s + 1, for s E C N .
(ii) I f t E C N ,
tELim ifftr
(iii)
We "also define, by induction on n E w:
f(O)(s,r) - r;
f ( i + 1)(s,r) - f ( s , f ( i ) ( s , r ) ) .
45.19. D E F I N I T I O N OF (t[n]: n E w) (t E Lim M C N ) . Such definition is
carried out by main induction on gr(t) and secondary induction on n E w:
(i)
t = f ( c , s + 1): then
(ii)
t-
t[O] = w s + 1 and t[n + 1] = t[n] + ws;
f ( s + 1, r + 1): t i n ] - f ( n + 1)(s, f ( s + 1, r ) + 1);
(iii)
t - f ( s , r + 1) and s E Lim: then t i n ] - f ( s [ n ] , f ( s , r ) + 1);
(iv)
t - f ( s + 1, c)" t[n] - f ( n + 1)(s, c);
(v)
t = f(s, c), s E Lira: t[n] = f(s[n], c);
(vi)
(vii)
t-
f ( s , r) and r E Lim: t[n] - f(s, r[n]);
t = s 9 r and r = f ( r l , r2): tin] = s 9 r[n].
For technical reasons (cf. the proof of 45.20 (ii), 3), we set (t + 1)[n] = t.
45.20. T H E O R E M
(i) If t E C N M Lim, then for every n E w, t[n] ~ t[n + 1] ~ t and
t[n] E C N .
(ii)
If
1.
2.
3.
4.
t E C N M Lira, then
gr(t) <_ gr(t[O]) + 1;
s E C N , n E w, t[n] ~ s ~ t ==Vgr(t[n]) < gr(s);
s E C N , n e w , t[n] ~ s ~ t ~ t[n] <_ s[O];
s E C N , s :< t ~ s <_ t[gr(s)].
P R O O F (Hint). (i): by main induction on g r ( t ) a n d secondary induction on
new.
(ii): 1 is easily checked by induction on gr(t).
The verification of 2 runs by induction on g r ( s ) + gr(t). The essential step
is to check the claim whenever t = f ( t l , s l ) and s = f(t2, s2). There are
three main cases to distinguish, according to t 1 - t 2 , t 1 <: t 2 or t 2 < tl,
together with several subcases, depending on the form of t and s; one
repeatedly applies l e m m a 45.15. 1-2 imply 3-4, as noticed by BuchholzCichon-Weiermann (1993). Indeed, let s E C N , n E ~, t[n] ~ s ~ t and
assume s[0] ~: t[n]. Then, by 1-2, gr(t[n]) < gr(s) _< g r ( s [ 0 ] ) + 1 _< gr(t[n]):
IX.46]
Predicative Well-Orderings II
277
contradiction. So we have 3. As to 4, 2 implies
gr(t[O]) < gr(t[1]) < gr(t[2])...,
whence n <_gr(t[n])for all n E w. In particular, gr(s)~_ gr(t[gr(s)]), which
together with 2 yields (s :(. t=V s ~: t[gr(s)]). [1
45.20.1. REMARK. For a uniform approach to fundamental sequences and
their relations with hierarchies of number-theoretic functions, the reader
should consult the cited paper of Buchholz-Cichon-Weiermann. According
to their terminology, 45.20 (ii) 1-2 states that the fundamental sequences for
F o form a normed Bachmann system; by 3-4 the system is regular and has
the so-called nesting (or Bachmann) property.
45.21. ENCODING. Since we aim at a constructive well-ordering proof of
C ~ in the next section, it is convenient to fix an arithmetical copy of C ~
by standard Ghdel numbering. So there a r e - within a fragment of O P closed A-terms, f l , f2 representing the number-theoretic characteristic
functions of ~: and of CN*, the set of Ghdel numbers of CN-terms. To
simplify the matter, we also assume that CN* coincides with N (fix a
primitive recursive bijection from N onto CN*). It is immediate to prove:
45.22. LEMMA (MFc). ~. is acceptable (i.e. {(n,m)'n ~. m} is a class) and
locally decidable.
We can find an operation distinguishing zero, limits and successors,
because these notions correspond to syntactic properties of CN-terms.
Similarly a predecessor operation on CN-terms can be produced.
45.23. CONVENTION. Henceforth we systematically identify F o with its
arithmetical copy CN*. In particular we use lower case Greek letters,
ordinal predicates and operations within the language of MFc, instead of the
corresponding codified notions and notations. So, in particular, r
will
henceforth stand for f(t,s) where t - c ~ and s - fl; A E L I M will formalize
the property of being a limit term, etc.
w46. Predicative well-orderings H
While the elementary properties of the natural well-ordering of type F 0 can
be dealt with in MFc, the verification of well-foundedness for arbitrary
segments of F 0 is non-elementary and essentially requires a transfinite
iteration of elementary abstraction. To this aim, we single out a subsystem
of TL, which is tailored for carrying out the well-ordering proof.
278
Levels of Truth and Predicative Well-Orderings
[Ch. 9
46.1. DEFINITION
(i) Fix a natural number p; we introduce the following abbreviations:
CL N "- {x" x C NA Cl(x)};
Good(~oP(z)) "- Vfl(fl < c~---+Cl(~o~(z)) A ~o~ (z) C CLN);
TI(fl, B) "- Progr( < , B) ~ Vc~ < ft. B(c~);
IU(fl) "- Vb(b~U ---+TI(fl, b)).
Ce0 - - W;
O~n+ 1 - - r
%P(z) is the term constructed by corollary 44.4 and 44.5 (i); if U write PWO(o~) instead of ICL(o~).
(ii)
CL, we
R A M ( p . . ) ' - P W O ( . ) A Cl(z) A z C_N--. eood(%~ (z));
mFa(p..)
is the theory MF c + RAM(p..);
MFR(p) "- W { M F R ( p . . ) ' .
< ro}
We stick to convention 45.23; thus a < fl is a shortening for the formula
that says: a, fl are codes for CN-terms ordered by ~:, the term ordering of
type F 0.
46.1.1. FACT. For each p, M F R ( p ) C TLR.
PROOF: immediate by 45.22 and the corollary 44.4 (ii). E!
Let I~1 denote co (r
Fo), if ff is MF c (MFp, MFR(p) with p_> 10,
respectively); we are going to prove"
46.2. THEOREM.
For each c~ < I~rl, ~r ~ PWO(a).
The core of the proof for ~ = MFR(p) hinges upon an idea of Feferman
(1982) and is based on the fact that, if the ramified hierarchy is well-defined
up to wa + 1, then IV(r
holds for any class U of classes C N, extending
% a + l ( X ) , where X is an arbitrary class CN. Actually, we obtain a
r
sharper result: we show that only the bounded ramified hierarchy is
required. We begin with two simple observations:
46.2.1
MF c F Iu(O)A ( I U ( a ) ~ (Vfl < a)IU(fl));
MF c F A E L I M ~ (VnIU(A[n])---+ Iu(A)) (apply 45.20 (ii), item 4).
A basic step in the proof of the main theorem is the introduction, due to
Schfitte, of the *-operation, where
A * ( a ) - Vfl(V6 < f l . A ( 5 ) ~ V~ < fl + wa.A(5)).
Predicative Well-Orderings II
IX.46]
If A ( a ) -
279
aTla, a* - {x" A*(x)}. Then we observe:
46.2.2. LEMMA
(i) MF c F Cl(a)--, Cl(a*);
(it) Progr( < , A ) ~ Progr( < ,A*) is provable in any subtheory of MF,
which derives N-induction on formulas positive in A (hence MF c proves
46.2.2 if {x" A(x)} is a class).
(i) follows by elementary comprehension. The verification of (it) is
immediate if a - 0 or a is a limit. If a - 7 + 1, one applies the hypothesis
and N-induction to (V~ </3 + ~'Y-k)A(~) (k in N).
If we define Clos(U;,)"-Vx(xrlU-~3y(yrlU A y - e x*) ( - - V is ,-closed),
we have:
46.2.3. LEMMA. The statements (i)-(iii) below are provable in MFc:
(i) Cl(a) ---, (TI(a, a*) --+ TI(w ~, a));
(it) P W O ( a ) ~ PWO(w~);
(iii)
(Cl(U) A U C_CL A Clos(U; ,)) ~ ((IU(~)---, IU(w~)) A IU(%));
(iv) PWO(wk) , where wo - 1 and Writ 1 -- wWn;
(v) MFp ~ QWO(wk), for each k > O.
PROOF. (i)Assume Progr( < ,a), TI(a, a*) and aTICL. By 46.2.2, a* is a
progressive class, whence Y5 < a.5~la* (with TI(a,a*)) and c~/a* (again by
Progr( < ,a*). If we c h o o s e / 3 - 0, we obtain V5 < wa.5~la.
(it): immediate by ( i ) a n d 46.2.2.
(iii) The first part is immediate by (i) and assumptions on V. As to IU(%),
apply class N-induction to VXTIU. TI(wn, X ) - B ( n ) ; this is possible since
U is a class of classes.
(iv)" metamathematical induction on k with (it) at the successor step and
class-N-induction for k - 0.
(v): metamathematical induction on k. We apply QWO(w) and the fact
that if wkm < u < wk(m + 1), u = wkm + y, for some y < wk. E]
It is convenient to adopt a stronger notion of progressiveness:
Pr+(A) "- A(0) A V~(A(13)~ A(/3 + 1)) A VA(Lim(A) A YnA(A[n]). ~ A(A)).
Pr+(A) : - " A is strongly progressive". Clearly Pr+(A)implies Pr(A); the
reverse implication holds whenever Vc~(A(a)--,(V~ < c~)A(fl)) holds (e.g. if
A(a)- TI(a,B)).
46.2.4. LEMMA. Let U be a class of classes C N closed under *. Then:
(1) Au((~) "- Vfl(IU(fl)~ IU(r
implies that
280
Levels of Truth and Predicative Well-Orderings
s u ( a ) - - {/3" IU(r
[Ch. 9
+ 1)fl)} is strongly progressive, provably in MF c.
{Formally: ME c F el(U) A U c_ CL A Clos(V; , ) ~ (AU(~)--~ Pr+(Su(~)))}.
(2) If Vn.AU(A[n]), where A is a limit, then Lu(A)"-{/3" IU(r
strongly progressive, provably in MF
is
C ~
P ROO F. Ad (1).
1.1: by hypothesis on U, we can apply class-N-induction to check
VnIU(h[n]) (this suffices by 46.2.1), where h [ 0 ] - r
h[n + 1 ] - r
and r
+ 1)0 - lim h[n]. The case n - 0 is trivial. If IU(h[n]) holds by IH
and we c h o o s e / 3 - h[n] in AU(a), we obtain IU(h[n + 1]).
1.2 Assume
IU(r
+ 1)fl).
(,)
We check (again by N-induction for classes): IU(f[n]), for every n in N,
where f[0] = r162
+ 1)fl + 1), f[n + 1] = r
If n = 0, we are done by
(,), 46.2.3 (iii) and 46.2.1; in the successor case, apply AU(a) and IH.
1.3 If A is a limit and we assume Vn. IU((r
we get
VnIU(r + 1)A[n]) by definition of the fundamental sequences and hence
IU(r
+ 1)A) by 46.2.1.
Ad(2): as in (1), we distinguish three subcases and we make use of the
fundamental sequences for r
I-I
If p C w and p _ 10, we define- uniformly in X, 7 - the formula
AX(o~) "-- (Vt~ > 0) (wc~+ 1. ~ < r
where U - U(a, ~, X) -- r
__., Vfl(iu(/3) ~ iU(r
'
5(X).
46.2.5. LEMMA (provably in MFc). Let p >_10. If X C_N is a class and
~
) is good (see
46.1), then {•" A ~ ( , ) )
is a strongly progressive
class.
PROOF. By hypothesis {c~" AX(c~)} is a class. We need a simple fact of
ordinal arithmetic (provable in OP; see Schiitte 1977, p.93), which explains
the choice of the ordinal terms:
ift,<w a.5,~<c~,thenwecanfindah'witht,<w
~.5'<w a.5.
(1)
Let c ~ - 0: once we note that %P~_~(X)is closed under * (by remark 44.5(ii)
and since p >_ 10), we get Ax(0) as a consequence of 46.2.3 (iii) and the fact
that ~ Pj y + l ( X ) is good. Next, we show:
X(c~ + 1) (X C hi class).
Fix 5 > 0 and assume
(2)
Predicative Well-Orderings II
IX.46]
k-
281
(3)
(4)
(5)
6o c~+2. ~ < w-,/+l.
zX(a);
IW(fl),
If c is in ~
where W -
%~(X).
and is progressive, we want
(,)
1)fl).urlc.
crI%P(X),
(Vu < r
+
By (1), choose 6 ' > 0 such that
where V - - w a + l ' ~ ' < k.
Obviously %P(X)C_ CL N is a class, being a segment of %~(X), and it is
closed under the map * (u being a limit). If U - % P ( X ) i n 46.2.4, we have
with (4):
sU(a) is strongly progressive and hence progressive.
But sU(a) is in ~ ( X ) because u < k, v is a limit and
logical complexity < 10 < p; hence by (5),
IU(r
(6)
+ 1)fl)
has
(7)
w, < ft.
By progressiveness, flrlSU(a), i.e. IU(r
+ 1)fl), which implies (,), as c is
progressive and in U. As to the limit case, we may assume, for X C N, X
class:
Vn. AX(A[n])(A limit);
W-
aJ~oPA+l. 6(X) and t r
(8)
r "k+l" ~ < r "y+I (where 5 > 0);
(9)
IW(fl).
(10)
As in the successor case, let c be a progressive class of W, i.e. c is in
U - %P(X), where u - ,))'.6' < n, for some 6 ' > 0. In order to apply the
second part of 46.2.4, we show:
VnVfl(#(fl)
Indeed, fix n arbitrary, assume
Then
a~l%~(X), for
IU(fl)
(11)
and pick a progressive class a of U.
some ~ < v and by (1) we can find a 6n such that
~n < b', i.e. ar]CJ}oPA[n]4.1.r
< c~
- Un
Hence, since U - %~P(X)_D Un, we have, by downward persistence,
But (8) and (9) imply"
IUn(fl)---+ IUn(r
IUn(r
Vu < CA[n]fl. urla; this
umon
over the Un's.
"
Therefore
holds and
completes
But (11)
progressive. Since u < n, Lu(A)is
for every ft.
IUn(fl).
(12)
by progressiveness of a, we conclude that
the verification of (11), because V is the
and 46.2.4 yield that L v (A) is strongly
an element of W (see (9)), whence by (10)
Levels of Truth and Predicative Well-Orderings
282
[Ch. 9
Vu < fl. urlLU(A).
(13)
But Lu(A)is progressive, hence flrlLU(A), i.e. Iu(r
since c is progressive. Vi
which yields (.)
46.2.6. First part of theorem 46.2:
MFp ~ PWO(a), for each a < tw0.
MFp F- Cl(X) A X C_N ~ Good(% m(X)), for each m C a~.
P R O O F . Let a < tw0, and choose m such that a < t m 0 . Since wm+2 is a
qwo (46.2.3 (v)), %Pm+2(X)is good by 44.3 (X is assumed to be in CLN).
x 1(m); for 6 - 1 , we get Vfl(IU(/3) ---. IU(r
Then 46.2.5 implies Am+
,
where U - %Pm+I(X).
If we choose f l - 0 and we remark that X belongs
0~
to U, we get TI(r
X); but X is an arbitrary class, whence the
conclusion follows by 46.2.1. V1
Proof of the theorem 46.2 (conclusion). By 46.2.3 and 46.2.6, it remains to
check
the
case of i f - M F R ( p )
with
p > 10. By the axiom
schema
M F R ( p ) F (PWO(a) A X~?CL A X C_N)--,Good(%P(X)).
(1)
RAM(p), we have for arbitrary a < F 0"
We prove (i)-(ii) below by metamathematical induction on n, where a o - w
O~n+ 1 - - r
(i)
M F R ( p ) F VX(X~?CL N --, Good(%Pn(X)));
(ii)
M F R ( p ) F PWO(an).
If n - 0, (ii) simply reduces to class N-induction, while (i) follows from (1).
Let n - m + 1. By IH we may assume PWO(am) , whence PWO(a m + 2),
which implies by 4 6 . 2 . 3 - p r o v a b l y in MF c - PWO(wam+2). Hence, if X is
any subclass of N and ~ -
wC~m+ 2, we have by (1) that ~o~(X)is good. But
we can apply lemma 46.2.5 and we get that A X + l ( a ) is progressive, and
ITS
hence with PWO(wam+2), AaXm+l(am) , i.e. PWO(r
A final application of (1) yields (i) for n - m
) -PWO(am+X).
+ 1.0
46.3. REMARK. F 0 and the applications. H.Friedman showed that the wellfoundedness of the standard well-ordering ~: for F 0 follows from a theorem
of Kruskal about well-quasi-orderings (see Gallier 1991); hence Kruskal's
theorem is unprovable in Predicative Analysis. On the other hand,
IX.46]
Predicative Well-Orderings II
283
Kruskal's theorem is a powerful tool for investigating term rewrite systems
used in computer science. Thus Friedman's result suggests that there may
be connections between (segments of) the standard well-ordering ~: of type
F 0 and the term orderings involved in termination proofs of term rewrite
systems. Indeed, Dershowitz and Okada established interesting relations
with proof-theoretic ordinals; for instance, it can be shown that the ordertype of the so-called multiset path ordering on the terms of an alphabet
whose precedence ordering is w, is exactly Cw0. For a survey on term
rewrite systems, the reader can consult Dershowitz and Jouannaud (1990).
The relevance of F 0 for combinatorics and computer science is discussed by
Gallier (1991), where the results of Dershowitz and Okada are also
reviewed.
This Page Intentionally Left Blank
C H A P T E R 10
REDUCING REFLECTIVE TRUTH WITH LEVELS
TO FINITELY ITERATED REFLECTIVE TRUTH
w
w
w
w
w
w
A sequent calculus STLR for a theory of reflective truth with levels
Basic properties of STLR
Elimination of the full level induction schema
Elimination of unbounded level quantifiers
The infinitary sequent calculus I T ~ of n-iterated reflective truth
Embedding STLR n into I T ~
In semantic form, the main theorem we are going to establish sounds as
follows: the first recursively inaccessible ordinal can be replaced by w in the
construction of the recursion-theoretic model of w39, insofar as we deal with
TLR-consequences of the form
Vi3jVz(Cli(z)---, 3y(Clj(y)A A(z,y))) (A elementary extensional in x, y).
Indeed, something stronger will be true: as a consequence of proof-theoretic
analysis, we shall prove that the theory T L R of reflective truth, with
variable levels and full transfinite induction schema on level ordering, can
be constructively reduced to a family {ITS" n E w} of theories of arbitrary
finitely iterated truth predicates. In each system I T S , level variables and
quantifiers are explained away in favour of a sequence {Tk: k <_n} of selfreferential truth predicates of increasing logical complexity. The reduction is
carried out in four steps that we summarize below.
1) T L R - t - T I ( l e v ) ~ STLR: we give a sequent style presentation STLR
(= sequent calculus of reflective truth with levels and reflection)of a system,
which contains T L R and the full transfinite induction schema on levels (see
w167
2) S T L R H STLR~176STLR is embedded into an infinitary system STLR ~176
where TI(lev) is eliminated in favour of an w-rule, which forces the level
variables to range over finite standard ordinals. Since STLR ~176
contains a
reflection principle for levels, STLR ~176
cannot have w-standard models; yet,
due to the weak number-theoretic induction, STLR ~176
is consistent. STLR ~176
enjoys a crucial quasi-normalization property (w
cut-rule can be
286
Reduction to Finitely Iterated RetTective Truth
restricted to formulas, which
existential level quantifiers.
contain
only
unbounded
[Ch.10
universal
or
3) S T L R ~ 1 7 6
E w}: this is the central step of the constructive
interpretation. First, we define a sequence of finitary approximations
STLR n to STLR ~176in which only bounded level quantifiers are allowed and
where we can explicitly refer only to the first n levels. The main theorem
49.18 ensures that STLRC%theorems can be suitably interpreted in the
STLRn's. The result is based on an asymmetric treatment of unbounded
universal and existential level quantifiers; the informal idea is to reinterpret
unbounded quantifiers on levels according to a "potentialistic" point of
view, so that Vj only refers to arbitrary finite segments of the level
ordering. As a consequence, the meaning of 3j depends on the given initial
segments, and this dependence is expressed by majorizing functions, whose
complexity depends upon the transfinite ordinal height of the given quasinormalized STLR~
4) STLRn~-~IT ~. In w167
we carry out a complete elimination of level
quantification and level structure: each STLRn-system is embedded into a
level-free infinitary system ITS, where the number-theoretic induction
schema is replaced by an infinitary rule for N.
w 47. A sequent calculus STLR for a theory of reflective truth with levels
We describe a sequent calculus STLR in the style of Tait (1968), which
strengthens TLR.
47.1. The syntax of STLR. The language L+y of STLR is, in essence, L V.
For technical reasons, it is convenient to adopt a different set of logical
constants and new predicate symbols for falsehood. Here is the list of
primitive symbols:
(i)
(it)
(iii)
(iv)
(v)
(vi)
(vii)
individual variables x_0, Zl, x 2 . . . (x, y, Z are metavariables);
individual constants 0, SUC, P R E D , P A I R , L E F T , R I G H T , D;
function symbols Ap (binary)and L T (unary);
predicate symbols = , Yr, Fl, -~, = i (binary); T, F, g (unary);
level variables i0, i l , . . . ;
countably many individual constants {m: m C w} for levels;
the logical constants V, A, -1, V, 3.
The L-terms are exactly the L(evel)-variables and the L-constants; i, j, k
will ambiguously range over L-terms. The terms of 2,+ form the least
collection which is closed under the following clauses: individual variables
X.47]
287
A Sequent Calculus for Truth with Levels
and constants are terms; if j is an L-term, LT(j) is a term;
terms, Ap(t,s) is a term.
if t, s are
Atoms and formulas of L+v. If t, s are terms, t - s, Nt and their negations
are e-atoms ( e - elementary); if i, j are L-terms, i ~_ j, i - l J, and their
negations are L-atoms. If t is a term and j is an L-term, then Tt, Ft,
Ur(i, t), Fl(i, t) and their negations are T-atoms. A is an atom iff either A
is an e-atom or A is an L-atom, or else A is a T-atom. An atom A is
positive if -. does not occur in A; an atom A is negative iff A has the form
-.B, where B is a positive atom.
The collection of L+-formulas is inductively generated from atoms by
closing off under A, V and quantification over either variable sort.
NB" in the previous chapters "atom" was used for what is here called
"positive atom"; this change of terminology should not cause any trouble,
since it only concerns the languages for sequent calculi.
47.2. Preliminary definitions and conventions
47.2.1 We assume the conventions and notations of Chapters I, w1 and VIII,
36.2; T i t , F i t , trlis, t-~is, trls , t-~s and Cli(t ) abbreviate Yr(i,t), Fl(i,t),
Vr(i, ap(s,t)), Fl(i, Ap(s,t)), T(st), F(st) and Vx(xrlitVx-~it ) (in the
given order). If i, j are L-terms, i - j stands for i - l J; we also write i -~ j
for (-"i - j) A (i 5 J). We set:
Vi ~ j . A :- Vi(i ~_ j ~ A) and 3j ~_ i.A "- 3j(j _ i A A); Vi -~ j and 3 j _ i
are called bounded level quantifiers.
47.2.2. Negation is inductively
according to the following clauses:
extended
to
arbitrary
-"-"Z - A, if A is a positive atom; -"(A X B) -
-"( ~ v . A ) -
L v+-formulas,
(-"A)vA (-~B);
(~ v.-"A)(provided v is an individual or level variable).
The other connectives --,, ~-~ are introduced according to their classical
definitions. Observe that, if A is arbitrary, -"-"A and A coincide.
47.2.3. E is the smallest collection of Lv+-formulas, which includes e-atoms,
L-atoms, T-atoms of the form Ft, Tt, -.Tit , Tit and is closed under
conjunction, disjunction, existential quantification on level variables,
bounded
universal
level quantification,
universal
and
existential
quantification on individual variables.
We also define: II "- {-"A" A E E} and A 0 "-- II M E.
Following 36.3, the collection A+ of acceptable formulas of L + is the
smallest collection which includes e-atoms, T-atoms and is closed under
conjunction, disjunction and universal and existential quantification on
288
Reduction to Finitely Iterated Reflective Truth
[Ch.10
individual variables.
47.2.4. Once the combinators ID, TR, NAT, NEG, AND, OR, ALL,
E X I S T of Ch. II, w are given, we inductively extend the map A~--~[A] to
arbitrary formulas of A + (see w
we recall that [Tit ] "-17,1LT(i),tll)
and in addition we stipulate: IF i r := [Ti-,t ] and [rt] := [T-,t].
If A E A +, TiA "- Ti[A] and TA "- T[A]. )~-abstraction is extended to 2.+terms by adding ~x.LT(i):= K(LT(i)) (where / i s an arbitrary L-term).
Hence, if A E A +, it makes sense to set {x: A) := )~x. [A].
47.2.5. As usual in proof-theoretic investigations, we partition variables of
either sort into bound and free variables: for instance, we may use variables
of even index as bound variables, while variables of odd index are only used
as free variables, or, in short, parameters.
We say that a formula of s + has the variable separation property ( - VSP)
iff every free variable of A has odd index, while every bound variable of A
has even index. Henceforth, we stick to the following convention VSP
(which is not restrictive, by trivial logical considerations):
(i) by s
we always mean a formula which satisfies the
variable separation property;
(ii) by s
we always understand a term whose free variables are
actually parameters, i.e. have odd index.
If F is a set of s
F V ( F ) is the set of parameters occurring
in the formulas of F; of course, FV(A):= FV({A}).
In the special case of individual variables, x, B, z range over bound
variables, while a, b, c stand for parameters.
47.2.6. Inductive definition of Lc(A) (A arbitrary Z+-formula):
Lc(A) = 0, for A atom;
if o = A, V;
Lc(A o B) = max(Lc(A), Lc(B))+I,
Lc(QxA) = Lc(A)+I, if Q = V,3.
Lc(A) is the logical complexity of A. Clearly Lc(A)= Lc(~A).
47.2.7. Inductive definition of rk(A) (A arbitrary L + -formula).
rk(A) = 0 if A E E U II; else
rk(A o B) = max(rk(A), r k ( B ) ) + l , if o = A, V;
rk(QxA) = r k ( A ) + l , if Q = V, 3.
rk(A) is called the rank of A.
47.2.8. For a compact presentation of STLR-axioms, it is convenient to
introduce the following abbreviations:
A Sequent Calculus for Truth with Levels
X.47]
289
T i - C l a u s e ( t ) "- 3x3y((t - [x - y] A x -- y) V (t -- [Nx] A N x ) V
(i)
V ((t -- [Tx] V t - [Tix]) A T i x ) V 3 j ( j -~ i A t -- [Tjx] A T j x ) V
V (t - (-~x) A F i x ) V (t - (x A y) A T i x A TiY ) V (t - (Vx) A Vv. T i ( x v ) ) ).
(ii)
F i - C l a u s e ( t ) "- 3x3y((t - Ix - y] A-~x -- y) V
V (t -- [Nx] A -~Nx) V ((t - [Tx] V t - [Tix]) A F i x ) V
V 3 j ( j -~ i A t -- [Tjx] A -~Tj x) V (t - (-~x) A T i x ) V
V (t - (x A y) A (Fix V FiY)) V ( t - (Vx) A 3v. Fi(xv)) ).
47.2.9. R E M A R K . T i - C l a u s e ( t ) and F i - C l a u s e ( t ) formalize the conditions,
which are necessary and sufficient for t to fall under T i and F i respectively.
The choice of T i - C l a u s e and F i - C l a u s e is motivated by the definition of
the operator which generates the recursion-theoretic model for T L R (39.9).
47.3. The system S T L R
The language of S T L R is L+y" We present S T L R as a Tait-style sequent
calculus, where sequents are derived instead of formulas. Sequents are finite
sets of s
denoted by capital Greek letters F, A, . . . . The
intended meaning of a sequent F - {A 1 , . . . , A n } is the finite disjunction
A 1 V . . . V An. The expression "F, A" stands for the set-theoretic union of F
with A. If p is a parameter of either sort and t is a term of the same sort of
p, F[p := t] denotes the sequent obtained by substituting each occurrence of
the parameter p by t in F; F[p := t] is called a substitution instance of F.
A set ~f of sequents is closed under substitution whenever
F E 30 implies r [ p - - tl e ~, for arbitrary p(arameter), t(erm).
The substitution closure of tf is the smallest set of sequents, which contains
:f and is closed under substitution.
47.3.1. A x i o m s of STLR: they form the substitution closure of the following
sets of sequents.
A.1. Logical axioms. Let p, q be parameters of the same sort and let =
denote the corresponding equality predicate. Then we postulate"
(i)
(ii)
(iii)
p - p;
- ~ p - q,-~A(p), A(q), where r k ( A ) --,A, A , where r k ( A ) - O.
A.2. Operational axioms
(i)
Kab-
a and S a b c -
ac(bc);
0;
290
Reduction to Finite@ Iterated Reflective -I-ruth
(ii)
(iii)
(iv)
(v)
--C(( al,
-~Na,
-~Na,
[Ch.10
C' (C, C' distinct individual constants);
a 2 ) ) i - a i ( i - 1, 2);
-~Nb, -~a - b, D a b c d - c;
-,Nb, a - b, Dabcd - d.
A.3. Peano axioms
(i) NO and -~Na, N(a+l);
(ii) -~Na, -~a+l - 0;
(iii) -~Na, P R E D ( a + I ) - a.
A.4. Level axioms
(i)
(ii)
(iii)
(iv)
(v)
i _ i;
~i~j,~j_k,i-4k;
3k(i -4 k A j -4 k);
--i_j,-~j~i,i-j;
- ~ L T ( i ) - LT(j), i - j.
A.5. Persistence" -~i-4 j, -~Tia , Tja;
A.6. Consistency: ~Tia , -~Fia;
A.7. Limit axioms:-~Ta, 3i. Tia; -~Tia, Ta; -,Fa, 3i. Fia; -~Fia, Fa.
A.8. Fixed point axioms
(i) -~Ti-Clause(t), Tit and-~Tit , Ti-Clause(t);
(ii) -~Fi-Clause(t), Fit and -~Fit , Fi-Clause(t ).
A.9. Ao-N-induction: -~A(O), - ~ V x ( A ( z ) ~ A ( z + I ) ) , -~Nb, A(b) (A arbitrary
Ao-formula ).
A.10. Ao-Reflection: -~Vx3i. A(x, i), 3j. Vz. 3i -K j. A(x, i) (A arbitrary A oformula).
A.11. Level Induction: -,Progr(-4 ,A), ViA(i), where A is arbitrary and
Progr( -4 ,A) abbreviates Vi(Vj(j -4 i ~ A(j))---,A(i)).
47.3.1.1. REMARK. All formulas occurring in the axioms, except for A.11,
are at most ~ or II (see A.4 (iii), A.7, A.10).
47.3.2. Rules of STLR. STLR contains the following finitary rules:
(A) F,A
r,
F B
;
(v)
F,A
F, A V B
F, VxA
A(a) with a ~ FV(F,A); (3x) F,
(w) F,
F, A(t)
3xA
F,B
and F, AVB;
( t individual term);
Basic Properties of 5TLR
X.48]
(V j)
r, A(k)
F, V j A
291
F, A(i)
F-, 3 j A (i L-term);
with k ~ F V ( r , A ) ; (3j)
(Cut) F, A F, -~A
F
Terminology. ( A ) , ( V ) , (Vx), (Vj), (3x), (3j) are called logical rules. In a
given rule %, the elements of F are called side formulas. A formula which
occurs in the premises (in the conclusion) of %, but is not a side formula, is
called minor (active) formula of the inference. The minor formulas of (Cut)
are called cut formulas; observe that they have the same rank. A sequent F
is said to be initial iff F _3 A, for some axiom A.
w48. Basic properties of S T L R
We introduce a notion of STLR-derivability.
48 1 Inductive definition of the derivability relation S T L R F-m F (for m,
"
"
n
new).
48 1 1. If F is an initial sequent, S T L R F- m F, for every m, n C w.
"
"
n
48.1.2. Assume that (i) F is the conclusion from the premises F i of a logical
rule, or of a cut rule of rank < n; (ii) S T L R b mi Fi (i _< 2) and m i < m.
Then S T L R b
1~
n
m
12
S T L R b m F is read as "F is STLR-derivable with length < m and cut
rank < n". We immediately have from the definition:
48.1.3. F A C T . If S T L R F- m F and m < k, n < p, then S T L R F k F.
n
m
m
p
By 48.1.3, it is not restrictive to assume that the premises of a rule are
derivable with the same length.
48.1.4. R E M A R K . S T L R ? m F iff there exists a finitary tree ~ of sequents
with root F, such that (i) every top sequent of ~ is initial; (ii) every other
sequent S occurring in ~I" is obtained by means of a rule of S T L R from
sequents standing immediately above S; (iii) the height of ~" is < m and
every cut formula occurring in the tree has rank < n. ~1" is usually called
derivation of the given sequent.
n
We now state a few elementary properties of the derivability relation.
Reduction to Finitely Iterated Reflective Truth
292
[Ch.10
48.2. LEMMA (Substitution)
(i) If STLR F- nm F(a), then STLR f- nm F[a "- t] (t individual term);
(ii)
/f STLR F- nm r(i) (i L-parameter), then STLR F- m
n r[i "- j]
(j L-term).
48.3 ~ LEMMA (Weakening) If STLR F- m
F, then STLR F- m
F, A.
n
n
48.4. LEMMA
(i) Tautology: STLR ~ ~ r, A,--A (A arbitrary), for every
m > 2. rk(A).
(ii)
Substitutivity: STLR F- ~ F , - , t a bit
y),
s,-,A[x "- t], A[x "- s] (A
m > 2.
We show that STLR is not weaker than the axiomatic system T L R of
Ch. VIII.
48.5. N O T A T I O N . STLR F- A stand for STLR ~- {A}; we also adopt the
abbreviations: S T L R ~ - n [ " - S T L R ~
nmF, for some m, and we write
STLR ~ F "- STLR F- n F, for some n.
48.6. LEMMA (Independence)
(i)
Let C, D be distinct elements of the set
{ [ a - b], [Na], [Ta], [Tja], a A b,-.a, Va}.
Then STLR f- - - ( C - D).
(ii)
STLR ~ [Tjt] - [Tis ] ---, i - j A t -- s.
P R O O F . Apply A.2(iii), A.4(v) and S T L R b ~ ( ~ - ~ ) ,
from A.3, if ~ - ~
are distinct numerals). [:]
48.7. LEMMA. Each substitution
derivable in STLR:
instance
(which follows
of the following sequents
(i) -,T i-Clause([A]), A and -~F i-clause([A]), -,A, if A is a - b, We;
(ii)
-.Ti-Clause([Ta]) , Tie;
(iii)
-~Ti-Clause([Tja]) , j ~_ i A Tja;
(iv)
-~Fi-Clause([Tja]) , i - j A Fja, j -~ i A-.Tja;
(v)
~Ti-Clause(~a), r i d and -~ri-Clause(-~a), Tia;
(vi)
-~Ti-Clause(a /~ b), T i a/~ T i b; -~Fi-Clause(a A b), F i a, Fib;
is
Elimination of Level Induction
X.49A]
(vii)
-,Ti-Clause(Va), VxTi(ax);
293
-~Fi-Clause(Va), 3u. Fi(au ).
The straightforward verification makes use of the previous independence
lemma and is left to the reader.
Now observe that every formula A of the language of TLR has a
canonical equivalent NF(A), written in the language of STLR; NF(A)is
the so-called negation normal form of A, i.e. the formula obtained from A
by pushing -~ in front of atoms with the help of De Morgan's laws and
standard equivalences between -~V and 3-~, -,3 and V-~, and by deleting
double negations. Consider the schema:
TI(lev) := Vi(Vj(j ~ i ~ A(j))---, A(i))---, ViA(i) (A arbitrary).
Then we can state:
48.8. THEOREM. /f TLR +
TI(lev)F A, then STLR F NF(A) (A arbitrary
formula of s
PROOF: it suffices to prove that the non-logical axioms of TLR are
derivable in STLR. First of all, observe that the local N-induction axiom
LIND of 36.4.1 and the reflection axiom of 37.7 are special cases of A.9 and
A.10 (respectively). The operational and number-theoretic axioms (36.4.1),
the level axioms 36.4.3, local consistency, limit and persistence (i.e. 36.4.2.6,
36.4.4.1, 36.4.4.2) are disposed of by means of A.2, A.3, A.4, A.6, A.7, A.5
(respectively). As to the other axioms of TLR, which concern T and T-~,
they follow from lemma 48.7 and the fixed point axioms A.8. To this aim,
observe that, by use of logic only, we can "invert" the sequents of 48.7, e.g.
STLRF-,Tia, Ti-Clause([Ta]) , etc. The schema of level induction is
derivable with A.11. D
Henceforth, we generally disregard the fact that TLR and STLR have
different logical primitives, and we simply identify A and NF(A).
w49 A. Elimination of the full level induction schema
The idea is to let the level variable range over finite
leads to the following infinitary variant of STLR.
49.1.
(i) The language s
of STLRr
ordinals. Formally, this
it is obtained from
L+v by omitting
free level variables.
(ii) L-terms are exactly the level constants m, for each m E w.
(iii) The notion of formula is inductively defined in the standard way,
as well the classes of E-, II-, A0-formulas; the notions of logical
Reduction to Finitely Iterated Reflective Truth
294
[Ch.10
complexity and rank are lifted without change to s
49.2. Axioms of S T L R ~ : they are obtained from the axioms of S T L R by
means of the following changes:
(i) all level parameters occurring in the axioms of w are replaced by
level constants;
(ii) level induction A.11 is omitted; the logical axioms A.1 (ii)-(iii) are
restricted to atoms of s
(iii) the level axioms of A.4 are replaced by
A'.4.1"
--,LT(i) - LT(j) if i # j (i, j E w);
A'.4.2:
if A is a true L-atom of s
then A is an axiom;
("true" refers to the structure with support {k: k E w}, where level identity
and level ordering are interpreted by number-theoretic identity and natural
ordering respectively; of course, the axioms of A.4 become provable from
A'.4).
(iv)
A'.5.
The persistence axiom A.5 takes the form:
-~Ti t, Tk t, if i < k.
49.2.1. REMARK. The formulas occurring in the axioms of STLR ~176
have
rank 0.
49.3. Rules of STLR~176STLR ~176
has the same rules as STLR, except that
(Vi), (3i) are now replaced by
(Vw) "" .F, A(i)...F,vj.f~
Below lower case Greek
< Eo - r 10 (cf. Ch. IX).
49.4.
i E w ; (3w) F,
letters
A(i),F,for3j.someAi
E w .
a,/3,.., range over arbitrary ordinals
Inductive definition of the derivability relation
S T L R ~ F ~ r (k E w, ~ < Co).
1.1. If F contains (as a set) an axiom of STLR ~, STLR c~ F ~ I' for every c~
and every k E w;
1.2. Assume that I' is the conclusion from the premises I' i of a finitary
logical rule, or of (Vw), (3co), or else of a cut rule applied to formulas of
c~i
rank<kEw;
assume also that STLRCCFk F i ( i _ w ) and a i < a . Then
S T L R ~ F ~ I'.
49.5. N O T A T I O N S
(i) S T L R ~ F ~ s "- I' is STLR~-derivable with length < a and cut
Elimination of Level Induction
X.49A]
295
r a n k < k.
(ii)
(iii)
S T L R cr ~- < ~ F " - S T L R ~176
F ~n F, for some finite m, k.
STLR ccFr'-
STLR ~176
There are two m a i n reasons for introducing S T L R ~ : the s y s t e m satisfies a
weak cut e l i m i n a t i o n property; S T L R is e m b e d d a b l e into S T L R ~ .
49.6. L E M M A
(i)
(ii)
If S T L R ~ 1 7F6- na r ( a )
~
then S T L R F- an F[a "- t] (t individual term);
S T L R ~176
~- na r , A 1 A A 2 implies S T L R ~ }- n F , A i ( i - 1 ,
provided rk(A 1 A A2) > 0;
2),
(iii)
S T L R ~ t- na F, A 1 V A 2 implies S T L R ~176
F- an F, A 1, A 2, provided
rk(A 1 V A2) > 0;
(iv)
S T L R ~ F ~n F, VxA implies S T L R ~ F- ~n F, A[x "- t] (t individual
term), provided rk(VxA) > 0;
(v)
(vi)
S T L R ~ 1 7F6 na F, V j A implies S T L R ~ 1 7F6- n F, A[j "- t~ (i E w),
provided rk(VjA) > O.
If STLRCr ~- ~n F, then S T L R ~ 1 7F6- na r , A.
49.7. L E M M A
(i) Tautology: S T L R ~ }- ~ F, A,-~A (A arbitrary), for every
a > 2. rk(A).
(ii)
Substitutivity: S T L R cr F 0 F, - , t - s , --,A[x "- t] , A[x "- s] (A
arbitrary), for every ~ > 2. rk(A).
Proofs of the l e m m a t a can be easily carried out by a p p r o p r i a t e inductive
a r g u m e n t s ; the inversion properties need the proviso, essentially because we
allow f o r m u l a s of logical c o m p l e x i t y > 0 in the a x i o m s (e.g. logical axioms,
A.9). If we a p p l y the usual cut e l i m i n a t i o n procedure to S T L R ~176we see
t h a t it is not restrictive to replace full cut rule by cut on E- or H-formulas.
49.8. T H E O R E M
( W e a k Cut Elimination)
If S T L R ~ 1 7}-6 F, then S T L R ~ 1 7t-6 al
F, for some a < %.
T h e t h e o r e m is a consequence of the so-called reduction a n d e l i m i n a t i o n
lemmata:
49.8.1. L E M M A (Reduction). If r k ( A ) -
n + l , STLRCr F an+l F, A and
S T L R ~176
F- ~n+l [',-,A, then S T L R cr F- a#f~
n+l r
Reduction to Finitely Iterated Reflective Truth
296
[Ch.10
( # is the natural ordinal sum; see below w53).
49.8.2. L E M M A (Elimination)
ot
If n > 0 and S T L R ~ F ~n + l
F, then S T L R ~176
F ~n
F"
To avoid repetitions, proofs will be given for the infinitary systems of the
next chapter.
49.9. L E M M A
(i) If A is an arbitrary s
STLR~176
b 0~o--,Progr(-.4 , A) , ViA(i);
{Progr( -~ , A) abbreviates Vi(Vj(j -.< i ~ A ( j ) ) ~ A(i))}.
(ii)
S T L R ~176
F- 1< ~' F, provided F is a sequent of the following form:
{-~i ~_k,-,T~t, Tkt};
{i5i};
{-~i~j,-~j_k,i_~k);
{--,i ~ j , - , j ~ i, i -
{3k(iqkAj-<
j ) ; {--,LT(i) - L T ( j ) , i -
k)};
j};
( i, j, k E w arbitrary).
P R O O F . (i)" let A induction on i C w:
{-~Progr(-~ ,A)}. Then it is enough to check by
S T L R ~ ~- < ~A, A(i).
(1)
If i - 0, (1) follows with the derivability of { - n -< 0}( - {-~n ~ 0 V n - 0)).
In the induction step, we get by IH and tautology (respectively):
S T L R ~ F- o< ~ A, Vj -~ i + l . A ( j )
and S T L R ~176~- 0< ~ A , - - A ( i + I ) , A ( i + I ) .
F r o m this we get S T L R ~176
F o< ~ A, -~A(i+I)A Vj -~ i+1. A(j), A ( i + I ) by
( ^ ). Now STLR ~176
e 0< ~ A, A ( i + I ) follows by (3~).
(ii)" by A'.4.1-4.2, A'.5, logical axioms and (3w). Vi
49.10. D E F I N I T I O N . Let r be a set of s
F' is a [0, m]-instance
of F, iff F' is obtained from F by replacing the free level variables occurring
in F with level constants of value _< m; clearly, once F' is a [0, m]-instance of
F, F' is a set of s
49.11. T H E O R E M (Embedding). If S T L R F- F, there exist a < w 2 and k < w,
such that, for each m E w and each [O,m]-inslance F' ofF, then
S T L R ~ F- ~ F'.
P R O O F : straightforward by application of 49.9. lq
Elimination
X.49B]
w
of Unbounded Level Quantif~ers
297
B. Elimination of unbounded level quantlfiers
We define a sequence { S T L R n ' n E w} of subsystems of S T L R c~, such that
STLR ~176
is locally embeddable into U {STLR n ' n E w}, in a sense to be made
precise below.
49.12.
Syntax of
S T L R n (n E w)
49.12.1. The language .5"
the unary predicates T and
{m'm < n}. Thus L-terms
each m _ n E w, and Tt, Ft
of STLR n is the fragment of s
which omits
F, and only contains the first n level constants
of s coincide with the level constants m, for
are no more atoms of s
We also define
:-
u
n e
49.13. The axioms of STLR n are obtained from the axioms of STLR c~ by
means of the following changes:
(i) all level constants occurring in the axioms of w
constants of value _< n E w;
must be level
(ii) the limit axiom A.7 and the A0-Reflection axioms A.10 are
omitted; the logical axioms A.1 (ii)-(iii) are restricted to atoms of s
STLR n has the same rules as STLR c~, except that
replaced by
(v)b ...F, A(i)...F,vjf~
A i < k _< n ;
(3)b F,
(Vw), ( 3 w ) a r e now
A(i), for some i _< k _< n
r, 3j~_k.A
We assume that the (finitary) notion of STLRk-derivability is made precise
by rephrasing it in the style of the definition 48.1. To save space, we also
assume that the obvious analogues of substitution, weakening and tautology
l e m m a t a 48.2, 48.3, 48.4 (i) have been stated and checked for STLR n.
49.14. S T L R k k mn F "- "F is STLRk-derivable with length _< m and cut
rank < n";
STRL~ k F "- " F is STLRk-derivable for some k E w".
We now proceed to a systematic translation of the language with
unbounded level quantifiers into the language of STLRn, which can only
deal with quantification on levels < n. The result is that we can associate to
each provable statement of STLR cr a family of "approximations", each
provable in some STLRk, for k big enough.
Reduction to Finitely Iterated Reflective Truth
298
[Ch.10
49.15
(i) Inductive definition of A[m, n], for each L~-formula A (m, n E w).
1. (Tt)[m, n ] - 3i ~ n. Tit and (-~Tt)[m, n ] - Vi ~ m. (~Tit);
(Ft)[m, n ] - 3i ~ n. Fit and (-~Ft)[m, n] - Vi ~_ m. (~Fit);
2. A[m, n ] - A, for every other atom of L~;
3. [m, n] commutes with A, V, Vx, 3x and bounded level quantifiers;
4. (ViA)[m, n ] - k/i ~_ m.(A[m, n]) and (3ia)[m, n ] - 3i ~_ n.(a[m, n]).
NB" an occurrence of Tt, Ft within a term of the form [A] is not affected
by the [m, n]-transform; for instance:
(FiTt)[m , n] - F i T t and (TTt)[m, n] - 3i ~_ n. T i T t .
(ii) If A is a L~-formula and k is a level constant, then A k is the
expression, which is obtained from A by replacing each unbounded
quantifier Qj of a by Qj ~ k (Q - V, 3).
49.15.1. FACT. (i) If A E L~r and all the L-constants of A have value < k,
then A[m, n] is a formula of L*, provided k, m _< n.
(ii) Let A e s
thenA[m,n]-A
A [ m , n ] - A, if A C A o.
If
F
--
m, i f A E I I ; A [ m , n ] - A n, i f A e ~ ;
{A1,... , Ak} is a set of L~-formulas,
F[m, n ] - {Al[m,n],...,Ak[m,n]}.
We state a simple property, which motivates the Ira, n]-transform.
49.16. LEMMA (Persistence)
(i)
Let A be an L~-formula with L-terms of values < k, and assume
m' <_ m <_ n <_ n' < k. Then STLR k F --,(A[m, n]), A[m', n'].
(ii)
Let F be a finite set L~-formulas with L-terms of value < k, and
assume m' < m < n < n' < k. Then
STLR k F Fire, n] implies STRL k F F[m', n'].
PROOF. (i) Induction on a . If A is an atom with A[m, n ] - A, we are done
by tautology lemma. If A - T t (or Ft), then we must check
STLR k F Vi -~ n.-,Tit , 3i -~ n'. Tit. But for each i < n, STLR k k - , T i t , T i t
(logical axiom), and the conclusion follows with (V) b and (3) b. If
A-BAC,
B V C , 3xB or VxB, we simply apply IH and the rules
corresponding to the principal connective of A; if A - V j B or 3jB, we
apply IH and the bounded level quantifier rules.
Elimination of Unbounded Level Quantifiers
X.49B]
299
(ii) follows by repeated application of (i) and cut rule. Vi
49.17. Preliminaries. We fix the initial segment -~ of order type co of the
primitive recursive well-ordering of type F0, which was defined in w
Lower case Greek letters range over elements of Field(-~ ); c~ < fl is a
shortening for "a, fl E Field( ~ ) and a -~ fl'. We also remind'that
w 0 - 1,
r
03rot 1 --
~
m.
To each derivability statement STLR~162
b ~ F, we can naturally associate a
construction tree ~, labelled by formulas and locally correct with respect to
the rules of STLR cr to be regarded as a derivation of F. By standard proof
theory, it is not restrictive to assume that infinitary derivations of STLR cr
are represented by primitive recursive trees and hence encoded by numbers.
The exact choice for encoding derivation trees is largely unessential; it
suffices that we can primitive recursively read off from ~ all relevant data
(final sequent, information on final inference and immediate subderivations
of ~, ordinal length and cut rank). Details are given in the appendix of the
next chapter for the ramified system RSn, and they can easily be adapted to
the simpler STLR cr As a temporary notation, ~ F- g F will stand for "~ is
a derivation of F in STLR 0r with cut rank k and length < a". For the sake
of simplicity, we do not distinguish the derivation ~ from its code. If F is a
sequent of STLR ~162
or STLR~, define:
49.17.1.
LevPar(F) "-- {k: k occurs as a level constant in r};
Irl :-
max
LevPar(F).
49.18. T H E O R E M (Asymmetric interpretation of STLR cr into STLR~).
We can find a partial recursive function F such thai if ~ [-1 F, then for
m,
m)
d4in d, m <
m) and
n >_
m),
I r[m,
]l <
STLR n F- F[m,n].
PROOF. We define F(~, m) by induction on the length c~ of ~.
Case 1: F is an axiom. We choose F(~, m) - max(m, I r I). If F ( ~ , m) < n,
then trivially n > I r[m, n]l - mac{m,n, I r I}. It suffices to check that the
[m, n]-transforms of STLR~176
are derivable in STLRn, for
n > F ( ~ , m) (we apply persistence and weakening, if necessary).
If F is an instance of A.1 (i), A.2, A.3, A'.4, A'.5, A.6, A.8, A.9, the
verification is trivial, because F[m, n ] - F is an axiom of STLR n. If F is an
instance of A.l(ii)-(iii), the conclusion follows b y t a u t o l o g y , substitution
and a simple persistence argument:
if rk(B) - 0 and B E E, (-~B)[m, n ] - --,Bm and B[m, n ] - Bn; but
Reduction to Finitely Iterated Reflective Truth
300
[Ch.10
STLR n f- ~B m, B n.
The [m, n]-transforms of instances of A.7 have the form -~T i t, 3i -< n. Tit
and Vi ~ m.-~Tit, 3i -< n.Tit , for m ~ n; they are easily derivable in STLR n
by use of logical axioms ~Tia, Tia and ~Fia, Fia with bounded level
quantifier rules. As to the first sequent, we need that the parameter i has
value <_ F ( ~ , m ) (this is ensured by the definition of F at the outset). If we
have an instance {-~Vx3iA( x, i), 3jVx3i ~_ j.A(x,i)} of A.10 (A is Ao), its
asymmetric transform becomes
{~Vx3i -~ m. A(x, i), 3j ~_ n. Vx3i -'< j. A(x, i)},
which is trivially STLRn-derivable as m _<n.
Case 2" (Cut). Then for some fl < a, we can assume (by the analogue of
48.1.3) that ~1 F 1~ F,B and ~2 F-1~ F,-~B, where ~1, ~2 are the immediate
subderivations of 2). Since r k ( B ) - 0, we may assume without restriction
that B E E and hence (-~B) E II.
Fix m, n > F ( ~ , m) - F(~)2, F(~)I, m)) and choose Pl - F(~)I, m). By IH
we obtain:
F ( ~ , m) _) F ( ~ I , m) ~ m;
n)Pl)__
IF[ m ,Pl], Bpll and n)_ I F[Pl,n],~B pll;
STLR n ~ F[pl, n], -~BPl;
STLR n ~ F[m, Pl], Bpl"
Since m < Pl-~ n, we conclude, by downward and upward persistence
(respectively),
STLR n ~- r[m, n], -~B pl and STLR n F- r[m, n],B pl,
whence STLR n ~- F[m, n] by (Cut).
Case 3. (3w)" then ~1 b ~1 F, A(i), for some fl < a, i e w, F - A, 3jA and ~1
immediate subderivation of ~). Fix m and n >_ F(~), m ) - F(~)I, m). In fact,
by IH,
F ( ~ , m ) >_m and n >_ I F[m,n],A[m,n](i) l >_i;
(1)
STLR n F- F[m, n], A[m, n](i).
(2)
By the second part of (1), we conclude with (3) b, applied to (2),
STLR n F- A[m, n], 3j ~ n. A[m, n](j).
Case 4. (Vw): we have, for each k E w, if F - A, ViA(i)
~k F- a1 k
r,A(k).
X.49B]
Elimination of Unbounded Level Quantifiers
301
Fix m and n > F ( ~ , m ) - max{F(~o,m),...,F(a~m,m)}; then we have by
IH F(a~, m) > m and for each k _<m, n > IF[m, n],A[m, n](k) I and
STLR n b F[m, n], A[m, n](k).
The conclusion follows by the rule of bounded universal level quantification.
The other cases are disposed of with similar arguments (persistence and the
appropriate logical inference). Finally, if we consider derivations as
primitive recursively encoded objects, the defining conditions of F can be
regarded as effective equations, having thereby a partial recursive solution
by the second recursion theorem. D
49.19. COROLLARY
(i) /f STLR F F and F' is the [O,O]-instance of F, we can effectively
find a natural number k such that STLR n F F'[0, n], for all n > k.
(ii) If TLR + TI(lev) F A and A E Lop, then STLR~ F A.
PROOF. (i) Apply 49.11, 49.8 and 49.18.
(ii) Apply 48.8 and (i). D
49.20. FINAL REMARKS
(i) Let STLR(-) be STLR without transfinite induction on levels A.11.
Then STLR(-) still contains TLR and enjoys weak cut elimination (i.e. if
STLR(-) F qk F, then also S T L R ( - ) F ~ F for some p and every F). Thus we
can avoid the detour through STLR c~ and we obtain a simpler version of
49.18: if S T L R ( - ) b k F, then STLR n F F'[m,n], for all m, n >_ m+2 k, and
each [0, m]-instance F' of F.
(ii) So as it stands, the index for F in the proof of 49.18 is not
primitive recursive. Indeed, if ~ ends with (Yw), the definition of F involves
a non-primitive recursive enumeration function U 1 for primitive recursive 1ary functions: for F satisfies
F ( ~ , m ) = max{r(Ul(h('~),i),m):i <_m},
where h ( ~ ) = e is a primitive recursive index, primitive recursively read off
via h from ~, and Ul(e,n ) encodes ~n" Of course, this does not exclude the
existence of sharper upper bounds on the complexity of F; we leave this as a
problem: is there any primitive recursive function F satisfying 49.18 ?
(iii) As a refinement, observe that the partial recursive function F of
theorem 49.18 can be made total and indeed "wk-recursive" (for k E w
sufficiently big). In fact, if we adapt to STLR ~176
the notions of derivation
code and label from the appendix of Ch. XI , we can directly introduce F by
combining definition by cases on primitive recursive clauses, course-of-value
302
Reduction to Finitely Iterated Reflective Truth
[Ch.10
recursion and recursion on the (possibly) infinitary tree ~ of height a such
that ~F-~F'I For the reader's sake, we define wk-recursion (after
Schwichtenberg 1977)"
49.20.1. % ( w k ) - t h e class of wk-recursive number-theoretic f u n c t i o n s - i s the
least class which contains the primitive recursive functions and is closed
under composition and the schema of (nested) wk-recursion (for k > 0)"
e(0, if) - H(fi');
G(/3, i f ) - t(fl, ~, Gift), for 0 < fl < wk;
here 5" stands for a finite list of natural numbers, t(y,s ~) is a term built
up from the number variables y,s function symbols for elements of %(wk)
already introduced, and the function variables ~ (of suitable arity);
further (Gift)(5, ~) - ~ G(6, ~) if 6 </3
( 0 , otherwise.
By 46.2.3 we know:
49.20.2. For each n E w and A E Lop, OP proves
TI(wn, A)'-
Va(V/3(/3 < a ---+A(/3))---+ A ( a ) ) ~ Va(a<w,---, A(a)).
As a consequence of 49.20.2 and proof theory, we can also show:
49.20.3. A function F is wk-recursive iff F is provably recursive in OP.
{If F is a m-ary number-theoretic function (m > 0), F is provably recursive
in OF iff there is an B(+)-formula A ( X l , . . . , X m , Y ) of OF (see Ch. I, 4.13),
such that:
F ( n l , . . . ,nm) -- p implies OP t- A ( ~ I , . . . ,rim,p);
OP F- VXl ... V x m ( N x 1 A . . . A N x m --+ 3!y(Ny A A(Xl, . . . , Xm, y)))}.
Now 49.18 can be formalized in OP + T I ( w k ) , for each given k > 0 and
derivations of length <w k. Therefore the partial recursive function F of the
asymmetric interpretation theorem should be at most wk-recursive. These
remarks are useful for the conservation theorems of Ch. XI.
We conclude with a second problem, concerning a requirement of
greater uniformity on F: is it possible to define a majorizing function F,
which satisfies the conditions of the asymmetric interpretation theorem
49.18 and only depends on the height a of ~ and m ?
X.50]
A Calculus for n-lterated Reflective Truth
303
w50. The infinitary sequent calculus IT~~ of n-iterated reflective truth
We devise a new system I T S , in which level variables, bounded level
quantifiers and level atoms are omitted.
50.1. The syntax of IT~n
50.1.1. The language s of I T ~ is s without" ( i ) b o u n d level variables
and bounded level quantifiers; (ii) the function symbol i T ; (iii) the
predicate symbols - / , _ .
The L-terms of Ln are the level constants of value _ n; the individual terms
of s coincide with Z-terms (they are generated without the clause: if i is
an L-term, L T ( i ) i s a term).
The atoms of s have the following form" N t , t - s, -~Nt, . " t - s (e-atoms);
T i t , Fit , ."Tit , ."Fit (T-atoms; t, s are s
i is a level constant of
s
s -f~
are inductively generated from s
by means of A,
V, Vx, 3x.
50.1.2. Since L T is omitted, we must redefine the map A ~ [A], for
we stick to the previous definition (see 36.3), except that
[ T i t ] " - ( 7 , ( i , t ) ) , where the boldface occurrence of i stands for
constant of s
while the overlined occurrence denotes the
( - closed term of .5; see Ch. I), whose value is the value of i.
A Es
we let
a level
numeral
CONVENTION. Unless it is unclear from the context, we keep using the
same symbols i, k, j, n, m for level constants and their values.
50.1.3. Simultaneous inductive definition of P O S n and N E G n.
(i) P O S n is the least class of s
which contains every A of
the form t - s, . " t - s, N t , ."Nt, Fit , ."Fit , T i t , -.Tit , for i < n, T n t , F n t
and is closed under A, V, V, S; if A E P O S n, A is said n-positive.
(ii) N E G n is the least class of s
which contains every A of
the form t - s ,
-.t-s,
N t , -.Nt, Fit , -.Fit , T i t , -,Tit , for i < n , ."Tnt ,
-.Fnt and is closed under A, V, V, 3; if A E N E G n, A is said n-negative.
(iii)
50.1.4.
(i)
(ii)
(iii)
A is n-separated iff A E P O S n or A E N E G n.
DEFINITION of K n ( A ) ( = n-complexity of A E Ln)"
K n ( A ) = 0 if A is n-separated; else:
K n ( B o C) = m a x ( K n ( B ) , K ~ ( C ) ) + I ( o = A, Y );
K n ( Q x B ) = K n ( B ) + I (Q = V, 3).
For a proper statement of the axioms, it is convenient to recall the obvious
finitary generalizations of A and V"
50.1.5.
~ { A i" i <_ O} - A o -
U { A i" i <_ O};
304
[Ch.10
Reduction to Finitely iterated Reflective Truth
{ A i" i <_ n + l } - ( ~ { A i" i _< n}) A An+l;
U{A i" i _ < n + l } - ( U { A
i ' i _ < n } ) V A n + 1.
Then we can define the "finitary" versions of T i- and F i - C l a u s e ( t ) for the
language L n.
50.1.6.1
T n - C l a u s e * ( t ) "- 3 x 3 y ( ( T - Ix - y] A x -- y) V (t -- [ i x ] A N x ) V
V ((t -- [Tx] V t - [Tnx]) A T n x ) V ( U {t - [Tjx] A T j x "j < n}) V
V (t - (-,x) A F n x ) V (t - (x A y) A T n x A T n y ) V (t - (Vx) A Vv. Tn(xV)) ).
50..1.6.2
F n - C l a u s e * ( t ) "- 3x3y((t - [x - y] A ~(x -- y)) V (t - - [ g x ] A ~ g x
)V
V ((t - [Tx] V t - [Tnx]) A F n x ) V ( U {t - [Tjx] A - , T j x "j < n}) V
V (t - (-~x) A T n x ) V (t - (x A y) A ( F n x V FnY)) V (t - (Vx) A 3v. F n ( x v ) ) ).
50.2. The system I T ~ is a sequent calculus in the language L n. In order to
avoid repetitions, we shall maintain, as far as possible, the same
terminology of the axioms and rules for STLR and STLR n (w
w
50.2.1. A x i o m s of ITS. We assume the substitution closure of the following
sets of sequents.
Logical axioms:
(i) a - - a;
(ii) ~ a - - b,~A(a),A(b);
(iii) ~A,A.
Proviso: in (ii)-(iii) A is an arbitrary Ln-formula with K ~ ( A ) Operational axioms:
(i)
(ii)
(iii)
(iv)
(vi)
K a b - a and S a b c - ac(bc);
- . C - C' (C, C' distinct individual constants);
(( al, a 2 ) ) i - a i ( i - 1, 2);
D - ~ - ~ c d - c;
(v) D k ~ c d - d ;
-~(m+l) - 0 ;
(vii) P R E D ( m + I ) - ~ .
Proviso: in (iv)-(vii) k, ~ are arbitrary distinct numerals.
Persistence:
-.Tia , T j a (i < j).
Consistency:
-.T i a, -.F i a.
Fixed point axioms:
(i) - . T i - C l a u s e * ( t ) , T i t a n d - T i t , Ti-Clause*(t);
O.
Embedding STI_Rn into 17~
X.51]
(ii)
~Fi-Clause*(t ),Fit
and
305
~ F i t , F i - C l a u s e * ( t ).
NB" the formulas occurring in the ITn~-axioms are all n-separated.
50.2.2. Rules of I T s : I T ~ contains the logical inferences ( A ) , ( V ) , (Vx),
(3x), (Cut), and, in addition, the rules (~N), (N) below:
(-~N)
"'" r ,
(N)
F, t = ~ (for some m E w)
F, N t
-
... (for every m E w)
F,--,Nt
;
ITn~-rules and IT~-axioms determine, as usual, ITn~-derivability for finite
sequents F of Ln-formulas. Below we let lower case Greek letters a, fl,...
range over arbitrary ordinals < F 0 (cf. Ch. IX).
50 92.3. Inductive definition of the derivability relation I T ~ t- p r (n e
1.1. If F contains (as a set) an axiom of 50.2.1, I T ~ F- p F, for every c~ and
every p _< w;
1.2. Assume that:
(i) F is the conclusion from the premises F i of a logical rule, (N), (-~N)
or a cut rule applied to formulas of n-complexity < p < w ;
(ii) IT~~ t- pai F i (for i < w ) and a i < a.
Then I T s~176
t- C~l?"
p
" - F is IT~-derivable with length < c~
50.2.4. NOTATIONS . (i) I T ~ F-p~ F
and cut complexity < p.
(ii) I T ~ F- ~ ~ F "- I T ~ ~ ~ F, for some fl < a and some k < w.
{The choice of n-complexity will be clear from the partial cut elimination
theorem for ITs~176
of Ch. XI}.
w51. Embedding STLR n into I T ~
*
We define a translation e n of Ln into Ln, which transforms provable
sentences of STLR n into provable sentences of I T S . If n is fixed, we simply
write e instead of e n.
Reduction to Finitely Iterated Reflective Truth
306
51.1. Inductive definition of e(t), for each s
[Ch.10
t.
(i) t e - t, if t is an individual parameter or an individual constant;
(ii) ( L T ( j ) ) e - j {remind that the boldface j is an L-constant, while
the overlined one is the numeral having the value of the given constant};
(iii) (Ap(t, s) e - A p ( t e, se).
51.1.1. REMARK. By (ii) above and 50.1.2, if [Tit ] is the term which
encodes T i t in s
([Tit]) e - (7, <7,t~)) is the term which encodes Ti(U ) in
~n"
51.2. Inductive definition of ca(A), for each formula A of s
We write e(A) for en(A), while _1_ stands for the s
0 - 1.
(i)
(t = s) e = (t e = se); (Nt) e = N(te);
( T i t ) e = Ti(t e) and ( F i t ) e = Fi(te);
(A) e - - 1 _1_ if A is a positive L-atom which is true in the standard
number-theoretic interpretation; else, ( A ) e = _1_;
(--A)e =--,(A) e if A is a positive atom;
(ii)
(AoB) e-A
(iii)
coB e(o - V,A);
(QxA) e - Q x ( A e ) ( Q - v , 3 ) ;
(Vj -< k. B(k)) e - n {B(j)" j < k) and
(3j ~ k. B(j)) e - U {B(j)" j -z, k}.
It is immediate to check that:
51.3. LEMMA
(i) If t is a term of Z*, t e is a term of Z n ( - Z )
with the same
parameters of t; if A is a formula of s
then A e is a formula of s with
the same parameters of A.
(ii) If A is a formula of s
Ae - A.
51.4. FACT. I T ~ satisfies the analogues of substitution, inversion for ( A ),
( V ), (Vx), weakening and tautology lemmata of w
A proof of 51.4 will follow from w
chapter).
{and the appendix of the next
51.5. LEMMA. (i) If A is an arbitrary s
I T ~ F 0mA,_,A;
ITn~ F 0 m - , t -
(ii) If A(x) is an arbitrary s
and m >_ 2. K n ( A ) ,
s,~A[x "- t],A[x "- s]
and k -
gn(A)+l ,
m
IW~ F ~ - - A ( 0 ) , - , V x ( A ( x ) ~ A ( x + l ) ) , ~Nt, A(t).
(iii) I T ~ F < ~~ (~ _ ~ ) or I T ~ F < ~ - - ( ~ - ~ ) , for every p, m e w;
307
Embedding 5TLR n into 17~n
X.51]
(iv) I T ~ F- <<a,,oF, provided F is a sequent of the following form:
{-,Na, -,Nb, -,a - b, Dabcd - c};
{
Sa,
sb, a -- b, D a b c e - e}; { s o } ;
{~Na, N(a+l)}; {-,Na,--(a+l)-
0}; {~Na,
PRED(a+I)-
a}.
PROOF. (i): induction on n-complexity of A.
(ii): similar to 49.9 (apply (i) and the rule (--N) of 50.2.2).
(iii): apply the logical axiom (i) and the operational axioms (vi), (vii) of
50.2.1.
(iv): the proof is easy by use of ( g ) , (--N) and the operational axioms
(iv)-(vii). We check I T ~ F- < ~ ~ g a , F R E D ( a + I ) - a. First of all, for each
p E w, we have:
ITn~ ~ o P R E D ( p + I )
- -~;
I T ~ F- 0 _ , ~ _ a, - , P R E D ( p + I ) - -~, P R E D ( a + I ) - a.
An application of (cut) yields I T ~ F-~ -~-~- a, P R E D ( a + l ) p E w and (-~N) yields the conclusion. Vl
a, for every
51.6. THEOREM. If STLR n F- F, then I T ~ ~- ma Fe, for some finite m and
some a < w2 (of course F e - { A l e , . . . , A k e} for F -
{ A 1 , . . . , A k } ).
PROOF. By induction on k such that STLR n F- k F.
Case 1. Assume F is an instance of a STLRn-axiom.
1.1. F is an operational axiom (a Peano axiom, a fixed point axiom, a
consistency axiom, an instance of A0-N-induction , persistence, or a logical
axiom): then F e is IT~-derivable by 51.5 and the corresponding axioms of
ITn~.
1.2. F is a level axiom of STLR n. Then either F = A , - , L T ( i ) = L T ( j ) , for
i:/=j or F = A , A ,
where A is a true atom of the form ( i = j ) , - - ( i = j ) ,
(i ___j), --(i _ j) (i, j _<n). It follows that either F e = A e, --(i = j), or else
F e - A ~, --1 _I_; both sequents are derivable in IW~ by 51.5 (iii).
Case 2" F is not an axiom. By IH the e-translations of the premises of the
logical rules and cut are IT~-derivable" since the e-map commutes with the
corresponding logical operators, we conclude by use of ( A ) , ( V ) , (V), (3),
and (Cut). As to the bounds on the length and cut complexity, only the
interpretation of the A0-N-induction schema requires the transfinite ordinal
w; in the other steps, we only finitely increase length and cut complexity. [q
Reduction to Finitely Iterated Reflective Truth
308
[Ch.lO
51.7. COROLLARY
(i) If B(x) is an L-formula which is elementary extensional in x and
TLR + TI(lev) F V x ( C l ( x ) ~ B(x)),
then we can effectively find a natural number n such that, for some k < ~o,
c~ < w2:
ITn~ F ~ Vx(Clo(x)---,Bo(x));
(here Bo(x ) results from B(x) by replacing T with To).
(ii) In particular, if A is a formula in the operational fragment Lop
and T L R + TI(lev) F A, then ITn~ F ~ A, for some n, k < w, c~ < w2.
P R O O F . By 48.8 and 49.19, we can find n such that
STLR n F Vx(Clo(x)---,(B(x))[O,n]).
(1)
By induction on B, using consistency and persistence axioms, we have
STLR n F--,Clo(a),--,B(a)[O,n], Bo(a ).
(2)
By (1)-(2) with inversion, cut and (Vx), STLR n F Vx(Clo(x)---,Bo(x)); since
( V x ( C l o ( x ) ~ B o ( x ) ) ) e = ( V x ( C l o ( x ) ~ B o ( x ) ) , the conclusion follows with
51.6.
(ii): in addition to the previous argument, recall that the [m, n]-transform is
the identity map on the operational language. 0
We conclude the chapter with a model-theoretic application; we assume
that the reader keeps in mind the recursion-theoretic model Ct of Ch. VIII,
w39 and the related notions.
Let C~ be the first order structure (CTM, { ~ k : k C w}), where CTM
denotes the closed term model of OP and { ~ k : k < w} is the function ~ of
39.14, restricted to w. Now let ~i, T'~o:= U {~fk:k < ~o} be the intended
interpretations of Ti, T (respectively), while level variables range over w. It
is clear that an obvious adaptation of 39.16 yields:
51.8. PROPOSITION. C~ is a model of I T S , for each n E ~o (i.e. if A is a
sentence of s and I T ~ F A, then E~I=A ).
51.9. T H E O R E M
If T L R + TI(lev) F Vi3jVx(Cli(x ) ---+3y(Clj(y) A A(x, y))), where A(x, y) is
an L-formula, elementary extensional in x, y, then
e~,l=Vi3jVx(Cli(x ) ~ 3y(Clj(y) A A(x, y))).
X.51]
Embedding 5TLR n into IT~n
309
PROOF. Let B "- Vi3jVx(Cli(x )---+3y(Clj(y)A A(x,y))), where A(x,y) is a
formula of s elementary extensional in x, y and TLR + TI(lev)F B. By
use of 48.8, 49.8, 49.11 and 51.6, we can find, for each m, a number n > m
such that
I T ~ F Vx(Clm(x ) ---,3y(Cln(y ) A Amn(x,y))),
(1)
where Amn(Z,y) results from A(x,y) by replacing each occurrence of t~lz
(t~ly) with t~lmx (t~lnY). By 51.8 and (1)we have
C~l=Vi3jVz(Cl~(z) ~ 3y(Clj(y) A A~j(z, y))).
(2)
By assumption on A, we also have, for m, n E w"
^
cl.(y) ^ Amn(Z , y)---, A(z, y)).
(3)
(2)-(3) yield the required conclusion. 0
However, theorem 51.9 is still unsatisfactory, because it does not make
any deep use of the constructive information, associated to the proof tree of
the given theorem of T L R + TI(lev). In chapter XI, we shall give a finer
proof-theoretic interpretation.
This Page Intentionally Left Blank
CHAPTER 11
PROOF-THEORETIC INVESTIGATION
OF FINITELY ITERATED REFLECTIVE TRUTH
w
w
w
w
w
~57.
w
The ramified system RS n
Cut elimination
Some derivable sequents of RS n
Embedding I T ~ into RS n
The upper bound theorem for I T ~
Upper bound theorems for TLR and its subsystems
Conclusion: the conservation theorems
Appendix: primitive recursive cut elimination for RS n
In chapter X we embedded the theory of truth with levels into infinitary
systems I T ~ with iterated truth predicates, where level variables and level
quantifiers are explained away. We proceed further ahead in investigating
the arithmetical content of the systems I T ~ and TLR.
We first design an infinitary system RS n in which T n is split into a
family {Tna'a < F0} of approximations. The Tna's are linked together by
natural recursive conditions, which can be encoded by symmetric
introduction rules with the cut elimination property (w167
Then we embed I T ~ into RS n by a modified version of the asymmetric
interpretation technique of w (see w167
54-55). The analysis of cut-free RS nderivations readily implies that RSn-theorems of level< n are already
derivable without Tn-rules , of course at cost of greatly increasing the
derivation length. This increase is given an upper bound with the main
theorems of w167
56-57.
As a final step, if we formalize the whole reduction procedure, we realize
that the proof-theoretic analysis only involves OP-principles, except for the
schema T I ( < Fo) of transfinite induction for operational formulas along
each a < F 0. We can conclude-with the help of the main result of
Ch.VIII-that
the operational consequences of TLR are already
consequences of OP + T I ( < F0).
Similar results hold for the systems MF, MFp, MF c of Ch. II. In
particular, the operational consequences of MF (MFp, MFc) are axiomatized
by OF + T I ( < r
(OP + T I ( < Cw0), OF respectively).
Proof Theory of Finitely Iterated Reflective Truth
312
[Ch.ll
52. The ramified system llS=
To a certain extent, the system RS n is meant to be a constructive
simulation of the n-th stage in the recursion-theoretic model of Ch.VIII (see
w
The truth predicates of level < n are assumed as given and satisfy
closure conditions corresponding to the IT~-axioms for the predicates T i
and F i with i < n; on the contrary, T n and F n are built up in stages (here
ordinals < Fo) , and there are rules for passing from stage a to stage a + l
and for collecting information at limit stages )~.
The essential fact is that the corresponding rules can be symmetrically
arranged as introduction rules for T a+l and -~T a+l (similarly for F a + l and
-~Fa+l). As a consequence, the standard predicative cut elimination
procedure of Schfitte applies, and this property grants elimination of level n
statements from derivations, whose conclusion refers only to lower levels.
As the reader will see, each RS n is an infinitary calculus; however, it is
well-known how to represent infinite derivations by suitable finite data
structures (Mints 1975, Schwichtenberg 1977, Buchholz 1991). It follows
that the cut elimination procedure is indeed primitive recursive and
"tractable", within a fragment of OP, possibly expanded by transfinite
induction principles.
52.1. The syntax of RS n. The ramified language Ln, r of RS n contains the
following primitive symbols"
(i) individual variables x0, Zl, z2... ( Z, y, z are metavariables);
m
(ii) individual constants 0, SUC, P R E D , P A I R , L E F T , R I G H T , D;
the function symbol for application Ap (binary);
(iii) predicate symbols True n and Falsen, - ,
N(unary);
Vr, FI (binary) and
(iv) level constants for each level < n (i, j, k syntactical variables);
(v)
(vi)
ordinal constants ~, for each ordinal a < F 0 ;
the logical constants V, A, --, V, 3.
52.1.1. The L-terms are exactly the L-constants; the o-terms are exactly the
ordinal constants; i, j, k range over L-terms, while we ambiguously use
lower case Greek letters c~, /~, 7, e t c . . . , both for o-terms and the
corresponding ordinals. Ln, r -terms are inductively generated from
individual constants and variables by use of applications; thus they coincide
with the terms of the underlying combinatory logic, and we stick to the
previous conv~n~ion~ and definitions. Expressions of the form t - s, - ~ t - s,
Nt, -~Nt are called e-atoms, while Yr(i, t), -~Yr(i, t), Fl(i, t), -~Fl(i, t),
Truen(t, ~), -~Truen(t, ~), Falsen(t , ~), -~Falsen(t , ~) are called T-atoms
The Ramit~ed System R5 n
XI.52]
313
(t, s range over individual terms, i is an L-term and a is an o-term).
s
are inductively generated from s
by use of the
logical constants A, V, V, 3. It is understood that we adopt the variable
separation property VSP of w
We also adopt the more perspicuous
abbreviations Tit "-- Yr(i, t) and Fit "- Fl(i, t), while T~(t) "- Truen(t , ~)
and Fan(t)"- Falsen(t, ~). We generally omit the subscript n and we simply
write Ta(t) and Fa(t), whenever n is clear from the context. Similarly
t~l% := Ta(st) and t~as := Fa(st).
NB. The m a p A ~ [A] is left unchanged (see 50.1.2); so it is well defined for
Ln-formulas and formulas of Ln, r not containing the new predicates T ~ and
F~; the expression {x:xrl~y} does not make sense.
52.2. If A E Ln, r, L c ( A ) ( - the logical complexity of A) is the number of
logical symbols occurring in A (-~ excluded).
52.3. Level and n-stage o f A (A formula s
(i)
(ii)
Let o -
A, V, Q -
V, 3.
Lev(Tjt) = j and Lev(Tan) - n ; i f A is an e-atom, Lev(A) - 0;
Lev(B o C) - max(Lev(B),Lev(C)) and Lev(QxB) - Lev(B).
Stn(A ) = 0, if A is an e-atom or Lev(A) < n;
Stn(A ) = ~, if A is a T - a t o m of level n and ~ occurs in A;
Stn(B o C) = max{Stn(B),Stn(C)} and Stn(QxB ) = Stn(B ).
52.4. n-rank of A (A formula of Ln, r)"
Rn(A ) = 0, if n = 0 and A is an e-atom or Lev(A) < n; else:
n.(T
t) =
Rn(B o C ) =
=
max{Rn(B), R n ( C ) ) + I and Rn(QxB ) = Rn(B)+I.
52.5. The a-transform Aa of a formula A E s is the Ln, r-formula, which is
obtained from A by replacing each occurrence of the atoms (~)Tnt , (~)Fnt
respectively with (-~)Tat, (-~)Fat respectively. Similarly, the c~-transform of
A E L (L = language of the basic systems of Ch. II) is obtained from A by
replacing each occurrence of (-~)Tt in A with (-~)Tat.
52.6. L E M M A
(i)
If A is a formula of Ln, r, Rn(A) - w. S t n ( A ) + m , for some
mew;
(ii)
If A is a s.~-fo~mut~ ( o~ a formula of L), Rn(Aa) < w ( a + l ) .
Before specifying axioms and rules of the new system, we define, in analogy
Proof Theory of Finitely Iterated Reflective Truth
314
[Ch.ll
with 50.1.6"
52.7
(i) T%Clause(t) := 3x3y((t = [x = y] A x = y) V (t = [Nx] A g x ) V
V ((t - [Tx] V t - [Tnx]) A T a x ) V ( U {t - [Tjx] A T j x " j < n}) V
V (t = (~x) A Fax) V (t = (x A y) A T a x A Tay) V (t = (Vx) A Vv. Ta(xv))).
(ii) F%Clause(t) := 3x3y((t = Ix = y] A-~x = y ) V (t = [gx] A-~Nx) V
V ((t - [Tx] V (t - [Tnx]) A Fax) V ( U {t - [Tjx] A ~ T j x " j < n}) V
V (t = (~x) A T a x ) V (t = (x A y) A (Fax V Fay)) V (t = (Vx) A 3v. Fa(xv))).
RS n is a Tait-style sequent calculus, like the systems STLR and I T ~ of
w167
so, we have to specify axioms in sequent form and introduction
rules for the logical and mathematical primitives.
52.8. Axioms of RS n. We assume the substitution closure of the following
sets of sequents.
52.8.1. LOG ( - Logical axioms)"
(i) t - - t;
(ii) - ~ t - s, ~A(t), A(s) ( R n ( A ) - 0 ) ;
(iii) -~A, A ( R n ( A ) - 0 ) .
52.8.2. O P E R ( - Operational axioms):
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
K a b - a and S a b c - ac(bc);
- ~ C - C' (C, C' distinct individual constants);
((al, a 2 ) ) i - ai, where i - 1, 2;
D-~ ~ c d - c;
Dk -~ c d - d;
-~(~+1) - 0 ;
P R E D ( - ~ + I ) - -~.
Proviso: k, ~ stand for distinct numerals.
52.8.3. P E R S i j
52.8.4. C O N S i ( -
( - Persistence)" -~Tia , Tja, for i < j < n;
Consistency)" -,Tie , -~Fia (i < n);
52.8.5. F I X i ( - Fixed point axioms for level i < n):
(i) -~Ti-Clause*(a), Tia and-~Tia , Ti-Clause*(a);
(ii) -,Fi-Clause*(a), Fia and-~Fia , Fi-Clause*(a ).
The Ramified System R5 n
XI.52]
Ti-Clause*(t),
315
Fi-Clause*(t ) are defined in the previous chapter (see
w
52.8.6. I N I n ( - initial Tn-axioms): -~T~ and -~F~ if n - 0; else, if n > 0:
-~T~ ,Tn_ I t ; -~F~ , Fn_ it;
-~Tn_lt , TOt; -~Fn_lt , F ~
NB: the level terms occurring in the axioms are all < n; if n - O , C O N S i ,
F I X i , P E R S i j must be omitted, for i < n. Every formula occurring in the
axioms has n-rank 0.
52.9. Rules of RS n. They include:
(i) the standard logical inferences ( A ), ( V ) , (Vx), (3x), (Cut);
(ii) the N-rules:
(-~N) " " F ' - ' t - ~ " "
(for e a c h m E w )
F,-~Nt
F, t - m ( f o r s o m e m E w )
; (N)
F, N t
(iii) Successor rules for T n and F n (remind that
(Ta+l)
F, Ta-Clause(t)
F, Ta+lt
;
(Fa+l) F, Fa-Clause(t)
F, Fa+lt
T a, F ~ stand for Tn~,
(-~Ta+l)
F,-~Ta-Clause(t)
F, ~ T a + l t
(~Fa+l)
F,-~Fa-Clause(t)
;
;
F, -~F~+lt
(iv) Limit rules for T n. Let c~ < F 0 be a limit:
(T_LIMa)
F, Tf3s
F, Tas ' for some ~ < a;
(-~T-LIM a)
F, -~T ~ s . . .
F, ~Tas
for every fl < a.
(F_LIMa)
F, Ff3s
F, Fas ' for some fl < c~; ( ~ F - L I M a)
F, ~ F f~s...
F,-~Fas
for every fl < c~.
The rules and axioms of RS n induce a relation of RSn-derivability for
finite sequents F of s
Remind that low Greek letters a, fl,...
range over arbitrary ordinals < Fo, but also over ordinal constants of RS n.
Proof Theory of Finitely Iterated Reflective Truth
316
[Ch.ll
52.10. Inductive definition of the derivability relation RS n t- Olp F (n E w).
DER.1. If (the finite set) F _~ F' and F' is an axiom of RSn, RS n F- pOt F, for
every c~ and every p;
DER.2. Assume:
(i)
(ii)
RS n F- pZ F~3, for every t3 < 6;
F follows from {F~"/3 < 6} by means of the rule ~, where :1 is an
inference of RS n with 6 premises (0 < 6 < F0);
(iii)
%3 < c~ for every/3 < 6;
(iv)
sup{p~'/3 < 6} < p and p < p, where # "- R n ( A ) + I , if ~is a cut
with cut formula A; else p := 0.
Then RS n F- C~l-,"
p
The previous inductive definition immediately implies:
52.11. L E M M A (Monotonicity of ordinal assignments).
If RS n ~ ~ F and cr < fl p < 6, then RS n F - ~ F .
52.12. N O T A T I O N
(i)
(ii)
RS n F- ap F "- F is RSn-derivable with length _< c~ and cut rank < p.
RS nF- < po~ F . - R S n F - ~ r , f o r s o m e / 3 < a a n d s o m e 6 < p .
w53. Cut d i m i n a t i o n
Following the classical method of Schfitte, we show that every sequent
derivable in RS n is already RSn-derivable with cut rank at most 1. As a
preliminary step, we collect a few simple properties of the derivability
relation for RS n.
53.1. L E M M A
(i)
(ii)
Weakening: if RS n ~ ogp F, then RS n t- ~p F,A.
Substitution: if RS n F- p r(a), then R S . ~- p r [ a " - t]
P R O O F : by induction on c~. (i) is a consequence of clause DER.1 of the
definition of derivability for RSn; (ii)follows from the fact that RSn-axioms
are closed under substitution. !-1
Cut Elimination
XI.53]
317
53.2. D E F I N I T I O N
(i) A formula A is reducible to A iff one of the following conditions
holds:
1. A = B A G and A = {B} or A = {C};
2. A = B V C and A = {B,C};
3. A = VxB and A = {B[x := t]}, for some t free for x in B;
4. A -
(~) T~+l(t) and A -
{(-)T~-Clause(t)};
5. A -
(-~)F~+i(t) and A -
{(-,)F%Clause(t)};
6. A - - ~ T ~ t (-~F~t), a limit and A -
{-~T~t} ( A - {-~F~t}), for
some ~ < a.
(ii)
A is reducible iff A is reducible to some A.
Clearly, a formula of RS n is reducible iff it can occur as active formula
in one of the following inferences: ( A ) , ( V ) , (V), ((-~)T~+I), ((-~)F~+I),
(-~T-LIM~), (-~F-LIM~), (fl limit).
53.3. LEMMA.
If Rn(A ) > 0 ,
A is reducible to A
and B E A,
then
Rn(B) < Rn(A).
The verification is obvious by definition of n-rank (52.4).
53.4. LEMMA (Inversion). Let RS n Fpa F, A with Rn(A ) > 0 and let A be
reducible to A. Then
R S n F pa F , A.
P R O O F . Induction on c~.
Case 1: F,A is an axiom. Since no reducible formula with n-rank > 0 is
active in the axioms of RSn, F, A is still an axiom.
Case 2: A = YxB and A is active in the inference ~ = (V) which concludes to
F,A. Then we must have, possibly by use of 52.11 and weakening lemma,
RS n F p~ F, YxB, B(a) (where a is an eigenparameter not in F,A), for some
/3 < c~. Moreover A has the form B[x:= t], for some t. Then by IH,
RS n F p~F, B(t), B(a), whence RS n F ~ F, B[x "- t] by the substitution
lemma 53.1 (ii) and monotonicity.
Case 3: A is active in the inference 5 which concludes to F, A, but ~ ~ (V).
Then F, A follows by applying IH to the premise of F, A, which is obviously
determined by the given reduction A.
Case 4: A is not active in the inference :J which concludes to F, A. Then by
IH we can replace every occurrence of A in the premises of ~ by means of A
and finally conclude with ~. FI
Proof Theory of Finitely Iterated Reflective Truth
318
[Ch.ll
We proceed to the crucial step in the proof of cut elimination; but we
first need the notion of natural ordinal sum. We know that, by the Cantor
normal form theorem 45.3, every ordinal 7 is uniquely representable in the
form 7 1 + . . . +Tn, with 71 >--'" >--7n, where each 7i has the form J ( i ) f o r
some ~(i). If O~1 + . . . -t-O~k and c~k+1 + . . . +C~k+m are the normal forms of c~
and fl respectively, we define:
O~#fl "-- Cr
) -I-...-[-
O~Tr(kTm),
7r being the permutation of { 1 , . . . , k + m } such that a,rll ) >_ ... _> .a,r(k+m ).
Clearly # is commutative and strictly increasing in each variable: 1.e. o < 7
implies both a:]/:6 < a # 7 and 6 # a < ")'#a.
53.5. LEMMA (Reduction)
q rts
O~
r,a
RSn F A, A
>_ 1,
P R O O F . We argue by induction on c~#f~.
Case 1" I', A and A,-~A are axioms. Then F, A is already an axiom because
neither A nor ~A can be active formulas of an axiom, having n-rank > 0.
Case 2: We may assume that, say, I',A is not an axiom. (This is not
restrictive: since Rn(A ) -Rn(-~A), the whole argument is symmetric with
respect to F, A and A,-~A}.
2.1" A is not active in the application of ~, ~ being the rule applied to infer
F, A. Then ~ has the form:
Pk F', A, Bk,... ~ for each k C (0, 5),
infer RS n ~- pa F, A from RS n t- ak
where 5 is the number of the premises of ~. Since ~k#fl < ~=]/=fl, we get by
IH, for every k E (0, 6),
RS n [-- ; k~/3 r', Bk, A.
The conclusion follows by application of 1t with length a k # f l # l <_a#fl and
rank _ p.
2.2: A, -,A are active formulas of the inferences 110 and ~1, which conclude to
r,A and A,-,A, respectively. Again by symmetry, we may assume that A is
reducible and not disjunctive. Then ~1 can only have one premise and it has
the form:
from RS n ~- p A,--A,--C infer RS n b ~
o A,--A,
where -,C is the minor formula of ~1" Since a#fl I < a#fl, we get by IH 9
RS n ~ ; :~f31 F, A , - C .
(1)
By inspection of the rules, it is readily seen that C must be a reduction of
Cut Elimination
XI.53]
319
A; hence by inversion, weakening, we have:
R S n ~ p F, C a.
(2)
Since Rn(C ) < Rn(A), and a < a # f l I (as 0 < fl), a cut between (1) and (2)
yields RS n f- ~ # ~ r , A . D
53.6. T H E O R E M (1-step-cut elimination)
(~
If RS nf-p+la F a n d p > O , thenRS nF-wp F.
PROOF. We may assume that the last inference ~, which concludes to F, is
a cut of rank p > 0 on a formula A; otherwise, the claim follows by IH
applied to the premises of 3. Thus we have by monotonicity lemma, for
some a o < a:
s0
sO
RS~ t- p+l F, A and RS n F- p+l F,--A with p - Rn(A ).
Hence by IH:
sO
s0
F A a n d R S nF-p
RS n ~- p
F, -,A.
By the reduction lemma, we obtain RS n ~- ~ F, where 7 -
ws O 92 <
~a.
0
53.7. T H E O R E M ("Sch~ttte's first cut elimination theorem")
If RS n ~- po~ F, then RS n ~ r1
F.
PROOF. By main induction on p and secondary induction on a. Clearly we
may assume that p > l , a > 0 .
Case 1: the last inference 3 is not a cut or is a cut of rank zero. Then we
apply secondary IH to each premise F k of F, thus obtaining RS n f- r
Fk
(where k G (0, 6), 6 is the number of premises of ~ and a k < a). Since r is
increasing in the second variable (w45), an application of 3 yields
RS n F- r
F.
Case 2: the last inference 3 is a cut on a formula A with Rn(A ) > 0, i.e.
R S n ~ pS ~
and RS nF-pa l
F,
-A.
Then by secondary IH and monotonicity:
CPao
RS~t R(A)
F, A and RSnl
Hence, by reduction lemma, RSnIr
COal
R(A)
(,)
[',--A.
# 6P~1
Rn(A
)
F. {Alternatively, we might
apply a cut to (,) and then 53.6.). Since Rn(A ) < p, we obtain by main IH,
RS
n
[ dp(Rn(A))(r162
1
F. But Rn(A ) < p and a0, a 1 <
C~,
and hence
Proof Theory of Finitely Iterated Reflective Truth
320
with the property P4 of w
[Ch.ll
RSn[ r p a r . El
53.6-53.7 can be generalized (the proof is left to the reader).
53.8. THEOREM ("Tait's second cut elimination theorem")
If RS n I 6+w
~ V F, then RS n ~ r
r.
w54. S o m e derivable sequents of RS n
We prove some useful sequents of RSn, which are needed in the proof of
the main interpretation theorem of w55.
54.1. LEMMA
(i)
Tautology:
(ii) Substitutivity:
RSnl
2"Rn(A)
1
-~A, A;
RSn]
2"Rn(A)
1
-~t = s, -~A[x := t], A[x := s].
PROOF. (i)-(ii): by induction on R n ( A ). We apply the logical axioms LOG
of 52.8.1 (ii)-(iii) and the rules corresponding to the main logical symbol of
A. For instance, let A = Ta+l(t).
Then by IH, since Rn(T%Clause(t)) < w ( a + l ) = Rn(A), we have:
RSn [- 1<~(a+l)-~Ta-Clause(t)
Ta-Clause(t).
The conclusion is immediate by use of (T ~+1) and (~T~+i). [3
54.2. LEMMA (Independence)
(i) We can find k E w such that, if C, D are substitution instances of
distinct elements of the set
Jt-
{ I n - b], [Na], [Ta], [T0a], ... ,[Tna], a A b, -~a, Va},
then RS n F- 1k - ~ t - C , - - , t - D.
(ii) We can find m E ~ such that, whenever s(a',b') = s[a := a',b := b']
and s(a, b) E A, then
RS n }__~n -~s(a, b ) - s(a', b'), a - a' A b - b'.
(iii)
We can find p E w such that, if i < k < n
RS n S ~ ~ T i T k t (or ~ F i T k t ).
The proof of (i)-(ii) above is a simple consequence of identity, pairing and
Derivable 5equents of RS n
XI.54]
321
number-theoretic axioms, like the independence lemma of 48.6; (iii) is a
consequence of (i)-(ii) with the fixed point axioms of 52.8.5 (observe that
we can derive -.Ti-Clause*(Tkt) if i < k).
At this point, the reader should recollect the notions of n-positive
Ln-formula from 50.1.3 and a-transform A a from 52.5 (for A E Ln).
54.3. LEMMA (Persistence)
(i)
Let A be an n-positive Ln-formula. Assume
RS n ~ ~ -.Tat, T~t
and RS n F- p~ -,Fat, F~t (t arbitrary).
Then for some finite k,
7+k
RSnl p
--,A~, A~.
(ii)
i < n ~ RSnl <1" ~Tit' T~t (or-~Fit , F~t).
(iii)
a < ~ ~ RSn~-,Tat,
w~
T~t ( o r - F a t ,
F~t).
P R O O F . (i): immediate by induction on A, using the definition of atransform, POSn, and the hypothesis.
(ii) We first check by induction on/3, for arbitrary t:
RS n F < w _~Tn_ l t, T~t and RS n F < w _~Fn_ l t, F~t.
(1)
If/3 = 0 (fl is a limit), we apply the initial axioms 52.8.6 (and the limit
rules LIM~3). In the successor case observe that (1) implies, for arbitrary t,
RS n F- 1< w ~Tn_l_Clause,(t), Tl3_Clause(t)
(2)
RS n ~ < w __,Fn_l_Clause,(t), Fl3_Clause(t).
From (2), the axiom F I X n _ ] (see 52.8.5) and the rules (TI3+i), (F~3+]), we
can derive"
RS n F 1< w -~Tn_lt , Tf3+lt and RS n F < w -,Fn_lt , F[3+lt (t arbitrary).
To establish the general case, we apply (1) and sufficiently m a n y cuts with
the sequents {~Tit , T n _ l t } and {-,Fit , Fn_lt}, which are RSn-derivable by
means of the persistence axioms (52.8.3).
(iii): by induction on ft. We assume that (iii) holds for every a < 7, if
7 </3. We explicitly consider the T-case, as the F-case can be
simultaneously handled with similar arguments.
If 13 - 0, the conclusion is trivial; if fl is a limit, RS n F :/~+l/a+l -'T~t, T~+lt
322
Proof Theory of Finitely Iterated Reflective Truth
[Ch.ll
and the conclusion follows by one application of ( L I M B ) . Let f l - 6+1 and
choose a < t3. By tautology and IH, we get, for arbitrary t:
RS n F w6w6_.Tat, TSt and RS n F w5
F t.
(3)
If a < 6, RS n F w(a+l
w(a+l/ ~ T a t , T a + l t again by IH; on the other hand, (3), (i)
above and the successor rules ( ~ T a + l ) , (T 6+1) imply, for some finite k"
RS n F- w6+k
...,Ta+l t, T~t;
w5
the required conclusion follows by a cut of rank w ( a + l ) < wfl.
Let a - 5. If 5 - 0, we apply (ii); if 5 - ~+1, we apply IH and (i). If 5 is a
limit, we have, for every ~ < 6:
R S n F ~ I ~ ++1
II-~T't,T'+lt~
(4)
RS n F w6w6_.T~t, TSt and RS n }- ~6
w6 _~F~t, F 6 t.
(5)
(5), (i) and the successor rules (--T ~+1) (T 6+1) imply, for some finite k
for every ~ < 5 , RS n F- w6+k ~ T ~ + I t, T6+l t.
(6)
Then we get for each ~ < 6, by use of a cut of rank w ( ~ + l ) < w6 with (4):
RSn ~ ~6wS+k+1 _~T~t, TS+lt.
(7)
w~ -~T6t, T 6 + l t by application of ( - ~ T - L I M ~) [3
Then RS n F wf~
54.4. L E M M A (Consistency)
RS n k 2+wa _~Tat, _~Fat.
P R O O F . By induction on a. If a - 0, the statement follows by cut with
length 2 from the initial axioms 52.8.6. If a is a limit, the conclusion is an
immediate consequence of IH with the limit rules for T and F, respectively.
Thus it is enough to check, for t arbitrary"
RSn }- wa
< wa _.Ta_Clause(t), -~F%Clause(t),
(1)
under the assumption
R S , F ~a ~ T a t , - ~ F a t .
(2)
(1) clearly yields the conclusion by the appropriate successor rules for T and
F. In order to verify (1), we define formulas Ai(a,b,t ) and Bi(a',b',t), where
0 _<_ i _<_ n + 6 and a, b, a', b ! are distinct parameters not occurring in t; in
short, we below omit the explicit mention of t, a, a', b, b':
A i "- (-,t - [Tia ] V - , T i a ) , if 0 < i < n;
A n "- (-.t - [Tna ] V ~Taa);
An+ 1 "- (~t - [Ta] V ~Taa);
An+ 2 "- (-~t - [Na] V ~Na);
An+ 3 "- (-,t - [a - b] V-,a - b);
(3.1)
XI.54]
Derivable Sequents ofRS n
A n + 4 "- ( - t - (~a) V - , F a a ) ;
323
A n + 5 : - (-~t - (a A b) V - ~ T a a V-~Tab);
A n + 6 "-- (~t -- (Va) V 3 u . ~ T ~ ( a u ) ) ;
(3.2)
B i "- (~t - [Tia' ] V Tia'), if 0 _< i < n;
B n " - (-~t - [ T n a ' ] V - ~ F a a ') ; B n + 1 "- (-~t -[Ta']_ V-~Faa');
B n + 2 "-- (-.t -- [Na'] V Na');
B n + 3 "- (~t - [a' - b'] V a' - b');
B n + 4 "- (-~t - (~a') V-~Taa'); B n + 5 "- ( - t - (a'A b') V (-,Faa' A ~Fab');
Bn+ 6 "-(-~t-
Let C(a, b, t ) : -
(Va') V V u . ~ F a ( a ' u ) ) .
n { A i 90 < i < n}" if we show
RS n ~_ < ~a C(a, b, t) Bi(a' b', t) for each i < n+6
(4)
(4) will imply (1) by applications of ( A ) and (V). As to (4), it is enough to
check
RS n F- ,oc~
< wa A j ( a , b , t), Bi(a',b' , t), for each i, j _< n+6.
(5)
If j # i, (5) is a consequence of Lemma 54.2(i). Let i - j .
If i < n or
i - n+2, n+3, we simply apply the second part of the independence lemma
above and LOG; else, we use 54.2 (ii), the identity lemma and (2). !']
54.5. LEMMA. Let S : - T (or F). Then we have:
(i) RS n I-- 1< w(/3+l) _~S~3_Clause(t),s~+l(t);
RSn ~ 1< w(i3-1-1) ..nSl3+l (t), S/3_ Clause(t).
(ii) RS n F- < ~ / ~ ; 2 / ~ S ~ t , s ~ + a - C l a u s e ( t ) .
PROOF. (i): immediate by tautology, ((--)T ~+1) and ((-~)Ft3+l).
(ii). By persistence lemma 54.3 and part (i), we have:
< w(~3+2) ,Ti3_Clause(t),
RS n F- w(13+l)
T[3+l-Clause(t);
(1)
RS n I- 1< ~(~+a) ~TI3 +l(t), Tr - Clause(t);
(2)
RS n F- ~(~+1
w(13+l/ ~ T ~ t , T~+I t.
(3)
The conclusion follows with two applications of cut rule. The case of F is
similar, rl
Proof Theory of Finitely Iterated Reflective Truth
324
[Ch.11
w55. Embedding I T ~~ into RS n
The time is ripe to call upon the family of infinitary systems I T S ,
which were introduced in the final section of Ch. X and where T L R can be
suitably embedded. By inspecting w50 and w52, we immediately realize that
I T ~ is just the s
of RSn+I; thus it should not come as a surprise
that the cut elimination argument of w53 works for I T ~ as well. Of course,
we must have clear that in the derivability relation I T ~ ~ p~ F , p i s a n
upper bound for the n-complexity of the cut formulas (see 50.1.4). For the
reader's sake, we recall that n-complexity simply counts the usual logical
complexity, except that it considers n-separated, subformulas as atoms;
furthermore, a formula A of s is n-separated whenever A is n-positive or
n-negative (in the sense that T n and F n occur only positively or negatively
in A, but not both).
55.1. T H E O R E M (Partial cut elimination). Assume that I T S ] c, F, where
i E {0,1}, 1 < k < w. Then:
k+~i
either i - 0 and I T ~ F- ~ k(a) F, or i -
1 and I T ~ F- 61a F.
Here w0(~ ) - ~, Wk+l(~ ) --w ~k(a) and r is the Veblen hierarchy of w
for
the proof, apply the elimination lemma and Tait's refinement of cut
elimination in 53.8. The partial cut elimination theorem ensures that cuts
can be always reduced to cuts over n-positive or n-negative conditions; and
this fact is essential to establish an interpretation of I T ~ into RSn, which is
based on the "separation" of positive and negative occurrences of Tn, Fn.
55.2. Inductive definition of A[fl, 7] (where
of I T S ) .
(i)
0 < fl < 7
and A is a formula
(Tnt)[~, 7] - T~s and (~Tnt)[~, 7] - -~T~s;
(Fnt)[~, 7] - F~t and (~Fnt)[~, 7] - -~F~t;
A[~, 7] - A, if either A - Tit , Fit , -"Tit , -.Fit and i < n or
A - ( - . ) N t , ( - . ) t - s;
(B o C)[fl, 7] - S[fl, 7] o C[fl, 7] and (QxB)[fl, 7] - Qx(B[fl, 7]);
A , V and Q - V , 3 ) .
(o(ii)
n -formulas; we set
Let F - { B 1 , . . . , B m } be a set of IT r162
F [fl, 7] - {B1 [fl, 7],..., B m [fl, 7]}.
55.3. FACT. Let A E s
and K n ( A ) - 0 : if A is n-positive, A[fl, 7 ] - A~ =
= the 7-transform of A; if A is n-negative, A[fl, 7] - A~3 - the fl-transform
(see 52.5 above).
Embedding 17~n into RS n
XI.55]
325
Verification: by induction on A.
55.4. LEMMA
(i) Let O <_ 3, <_6 < ~ < (~. Then
RS n ~ {MOf
< w(c~+i)_~(A[6, ~]) ~ A[7 ~].
(ii) Assume RS n F u F[6,~],A
with O < 7 < 6 < ~ < ~ and u,
(r < w(c~-F1). Then RS n F- w(~
< w(c~-F1) r [')', o~], A.
PROOF. (i) Induction on A. If A is an atom not of the form T n t , Fnt , we
apply the logical axioms and the N-rules; else, we use persistence lemma
54.3. If A is built up from A, V, V, 3, we apply IH and the corresponding
logical rule.
(ii) By induction on the cardinality of F, using (i) above, cuts on formulas
of n-rank < w(~+l) < wa and hypothesis on u, ~r. ['1
55.5.THEOREM. /f I T ~ F- a1 F, then RS n F- wt')'+l)
w(7+1) r[fl,'y], for every fl >_ 0,
7 >_ ~ + w~PROOF. By induction on c~.
Case 1" F is an axiom of I T S . We neglect the side formulas (this is not
restrictive by weakening lemma) and we consider a number of subcases.
1.1. F is an axiom which does not involve the level-n-predicates Tn, F n.
Then the conclusion is trivial, since r - F[/3, 7] is an axiom of RS n as well.
1.2. F is a logical axiom of the form ~A, A, where A can be assumed
n-positive. Hence by 55.3 above, (-~A)[fl, 7] --~Af3 and A[fl, 7] - A.y.
Since /3 < 7, by 54.3 (iii)
(-T~t, T~t} and
{-~F~t, F'Yt} are RSn-derivable
with length and cut rank < w(7+1). But A is n-positive and the conclusion
is immediate by application of 54.3 (i).
1.3. F - {-~Tnt,-~Fnt } or F - {-~Tit , Tnt}, { ~ F i t , Fnt } with i < n: apply
54.4 and 54.3 (ii).
1.4. F is a fixed point axiom involving T n and F n.
1.4.1. F - {-,Tn-Clause*(t),Tn t }. Then by 54.5 (i)
RSn [- 1< w(/3+l)-~T~-Clause(t), T/3+lt
If 7 -
fl+ w ~
f l + l , we are done; else, by 54.3 (iii) we also have
RS n F- ~w~
_~T~+lt ~ T-~t
,",/
9
By a cut of n-rank w(/3+l) < WT, we get RS n F- w"/
~ + a ~T/3_Clause(t), T~t.
1.4.2. r -
{ ~ T n t , Tn-Clause * (t)}. Then by 54.5 (ii)
RS n I- << w(/3+2) -~T~t, T~ + 1-Clause(t) 9
326
Proof Theory of Finitely Iterated Reflective Truth
[Ch.ll
If 3 ' - f l + l we are done; if f l + l < 7, we have by 54.3 (i)-(iii)"
RS n [_. w7
< r
-~Tl3+l-Clause(t), T%Clause(t).
The required conclusion follows by a cut with n-rank < w ( f l + 2 ) < w ( 7 + l ) .
The interpretation of the fixed point axioms for F n is similar.
Case 2: F is not an axiom and c~ > 0, fl > 0 and 7 > fl+ wa. We have to
distinguish seven subcases corresponding to the possible inference ~ - (-~N),
(N), ( A ) , ( V ) , (3), (V), (Cut)of I T S .
2.1. Let :1- (--,N) and assume, for each k E w:
c~k
I T ~ F- 1 F, - ~ t - k, where a k < c~.
Now, if we put g k - fl +wak, we have by IH for each k:
w(erk+l )
RS n ~-W(ak+l ) r[fl, trk],-~t- k.
Clearly ~rir < 7 and w(~rk+l ) < w ( 7 + l ) ; hence by lemma 55.4 (ii):
RS.
<
o.,T
By application of (-~N), we conclude
RS n I- :/")'+1
7+11 r[fl, 7 ] , - N t .
2.2. Let :1- (Cut), then for some a o < a
IT n F - 1c~0 F, A a n d l T n ~ - ac~0 F,-~A;
(1)
(by monotonicity lemma, it is not restrictive to assume that the premises
have the same bound on the length). By assumption Kn(A ) = 0; we may
also suppose that Tn, F n occur in A and that A is n-positive (hence -~A is
n-negative). Choose any 3' > fl+wa> fl+ wa~ By IH applied to the left
premise of (1), we obtain with (ro - fl+w ~~ and 55.3:
W(ao+l)
RS n ~-w(a0+l ) F[fl,o'0] ,
A%.
(2)
If we apply IH to the second premise of (1), choosing fl - (r0, (rI - (ro+W a~
we obtain, since -~A is n-negative and again with 55.3"
w(O'l+l )
RS n F-W(al+l ) F [ % , a l ] , ~ A%.
(3)
c~
0
~c~
Since w . 2 <
and fl+wa<7, we immediately have w((ro+l),
W ( g l + l ) < w(7+1) and hence by application of 55.4 (ii) to (2)-(3) we have:
< w(-'/+l)
RS n f_ < w("/'+l) F[fl, 7] Aa o and RS n k w-y
r[z,
-
A%.
An application of (Cut) with n-rank < w ( 7 + l ) yields the conclusion. The
remaining cases do not present any additional difficulty and are left as
exercises. I-i
The Upper Bound Theorem for I-I~n
XI.56]
327
w56. The upper bound theorem for I T ~
We combine the strength of cut elimination for the infinitary systems
I T S , RS n with the embedding of ITn~ into RS n. The main result shows
that, if a statement A in the language Lop is IT~-derivable for some n,
then not only A is true in the given ground model, but A is "cut-free"
derivable with an infinitary derivation tree, whose height is bounded by F 0
(whence the name of the theorem).
56.1. DEFINITION
(i) OP c~ is the subsystem of I T ~ in the language Lop ( - t h e language
of the theory OP of Ch. I). More specifically:
1) OP ~ has the same axioms of I T ~ in the language Lop;
2) the rules of OP ~ include ( A ) , ( V ) , (V), (q), (Cut), (g), (--,N);
3) the derivability relation O P ~ F - p F is inductively defined as in
50.2.3, except that now p is a strict upper bound to the usual standard
logical complezity of cut formulas, which assigns complexity 0 to e-atoms
and counts distinct occurrences of logical symbols (cf. 47.2.6).
(ii) We put f r o - O P ~ and ~ n + l -
ITS.
56 92. LEMMA (OPt-cut elimination). If O P ~ F - c~
F, either p - k + l
t9
some k e ~o and OP ~ ~ lk(
)r,
or p - ~ and OP ~176
F- r1
'
for
r.
The verification follows the standard pattern of w53.
If F is a set of formulas in RSn, let Rn(F ) = 0 stands for "every
formula of F has n-rank 0". Clearly if Rn(F ) = 0, the only formulas of F
which are not formulas of fin must have the form TOt, F~ or-,TOt,-,FOnt.
In order to get a "level lowering lemma", we define by cases a
transformation F - of F into the language of if'n, whenever Rn(F ) = 0:
(i)
n = 0:
F- is obtained from F by replacing each occurrence of TOt,
F~ by the formula ( - ~ t - t);
(ii) n > O: F- is obtained from F by replacing each occurrence of TOt
by the formula T n _ l t (Fn_lt).
(F~
{According to 52.1.1 we leave the index n for the ramified predicates T and
F implicit}
56.3. LEMMA. If RS n ~ C~lF and Rn(I' ) - 0 ,
then ~n F- < ~+~ F-.
(~n F- ~ F "- ~ n ~ ~p F, for some p < w).
PROOF. Induction on a. Let F be an axiom of RS n. If F is a logical axiom,
Proof Theory of Finitely Iterated Reflective Truth
328
[Ch.ll
say F = A , A , - , A where R n ( A ) = O , then by 51.5 and monotonicity of
ordinal assignment, ~1"n ~- 1< ~+a F-. As to non-logical axioms, observe that
the --transform sends them into corresponding axioms of ~n" Assume now
that F is derivable with a cut. Then we obtain by IH and by hypothesis on
the cut rank, for some A with R n ( A ) = 0 and some s 0 < c~,
~ n F- < ~+~0 F-, A - and ~ n F- < ~ + % F-,-~A-.
A cut on A - yields fin F- < w+a F-; but note that K n ( A - ) may be greater
than 0; this explains why the cut rank may increase up to w. The remaining
cases are immediate by IH. I"1
56.4. LEMMA (Level Lowering Lemma) 9Assume that RS n I- p~ F, where F
is a set of formulas in the language of ~I"n (so T~, F no l do not occur in F).
Then, if p > O or a > O,
~Yn ~ r
F
P R O O F . By 53.7 we have RS n ~-r
F. By assumption on F, Rn(F ) - 0
and F -- F-. The conclusion is immediate by 56.3, observing that p > 0 or
c~ > 0 implies k+r
= Cpc~, for k E w. I'l
We now appeal to the notions of Ch.IX, especially to the basic ordertheoretic properties of w
concerning the Veblen hierarchy. Recall that
an ordinal c~ is an c-number, if c~ has the form r
for some ft.
56.5. DEFINITION
(i)
C~o "- r and Cek.4.1 -- r
(ii)
H o ( f l ) - fl and H n + l ( f l ) - r
56.5.1. FACT:
H n ( a k + l ) - Hn+l(O~k).
56.6. DEFINITION. A formula A of L o ( - the language of I T S ) is T opositive iff A is inductively generated by means of V, q, A, V from atoms of
the form t - s, - ~ t - s, Nt, -~Nt, Tot.
If A is a To-positive Lo-formula , Ao6 is the formula of RS o which results
from A by replacing every atom of the form Tot with T6ot.
56.7. T H E O R E M (Upper bound)
(i)
If I T ~ F- ~ Ao, A o is a To-positive formula of s
and c~ < ~k, then
RS o ~- Z~o, for some ~ < Hn(ak).
(ii)
If I T T t - 1a A, a < a k and A E .Lop then O P ~ 1 7 6~ A, for some
6 < Hn+l(O~k).
XI.57]
Upper Bound for TI R and its Subsystems
329
PROOF. Both (i) and (ii) are verified by induction on n.
Ad (i): n - 0. By the embedding theorem 55.5, we immediately have:
RS 0 F- ~(6+
where 5 - l + w a < H o ( ~ k ) -- s k (each c~k is closed under w-exponentiation).
n - m + l . Assume
I T ~ F- 1 A o with ~ < (~k
Again by 55.5, it follows, for ~i = l+wa,
w(8+ll/ A0;
RS n F- ~(~+
hence, if we set ~ = w(~+l), we can conclude by level lowering lemma 56.4
and partial cut elimination 55.1:
r162
IT~ I-- 1
)Ao"
But c~ < c~k implies r162
< r
(1)
= c~k+1. Hence, by IH applied to (1),
RS o ~ A0~, for some ~ < Hm(C~k+l)- H n ( ~ k ) (by 56.5).
Ad(ii): n -
(2)
0. If I T ~ ~ ~ A, c~ < c~k, A E Lop and ~i- l + w " , then
w(S+l)
RS 0 F-~(a+l)
A by embedding. Hence by level lowering lemma and partial
cut elimination 55.1, again with { = r
OPC~F- )`
1 A, where ~ - r 1 6 2
< r
-- Hi(C~k).
n = r e + l : by embedding, cut elimination, level lowering, IH and 56.5.1 E!
w57. Upper bound theorems for T L R and its subsystems
We lift the upper bound theorem to the finitary formal system
TLR + T I ( l e v ) and its subsystems MF, MFp, MF c. Actually, the results are
corollaries of the elimination of variable levels, carried out in Ch. X.
In addition, we shall apply 56.7 for obtaining explicit semantical
information, which involves the intended inductive interpretation of our
basic language. To this aim, we recall that CTM is the closed term model of
OP (w
and that O(CTM), O ( C T M , ~ ) a r e the least inductive model of
MF over CTM and the &th stage of the same inductive model (in the given
order; see Ch. II, 7.6). We further recall from 37.1 of Ch.VIII that, if A E L,
the /-transform of A, denoted by Ai, is the Lv-formula , which results from
A by substituting each atom of the form T t by T i t . If A E L, A is
T-positive iff A is inductively generated by means of A, V, V, 3 from
atoms of the form t - s, N t , ~ t - s, ~ N t , T t .
Proof Theory of Finitely Iterated Reflective Truth
330
[Ch.ll
57.1. T H E O R E M
(i)
If TLR + TI(lev) F A and A is a sentence of Lop , then
OP~176
I- 1aA, for some c~ < F 0
(ii)
If A is a T-positive s
and TLR + TI(lev) F Vi. Ai, then
RS o F A~o, for some 5 < F o and hence O(CTM, Fo)I=A.
PROOF. (i): if TLR + T I ( l e v ) b A E s
we have
= v I T ~ F aq+l A, f o r s o m e a < w - 2 ,
q<w(by517);
A (by 55.1);
==~ I T ~ F Wq(a)
1
=:~ O P ~ 1 7 6~1 A with 5 - Hn+l(eO) (by 56.7 (ii)).
Ad(ii). First of all, observe that the axioms and inferences of RS o are
trivially sound for O(CTM, Fo) , as soon as we interpret T a (F a) by the set
O(CTM, a) (respectively by the set {t: t e CTM and O(CTM, cr)I=T~t)).
This remark ensures that the final part of the conclusion is true, if the first
one holds. Now, if TLR + T I ( l e v ) b Vi. A i and A is a T-positive s
we have, arguing as in (i) above, for some n and a < (o:
I T ~ b al A0
(A o being obtained from A by replacing every atom of the form T t with
Tot ). But A 0 meets the hypothesis of 56.7 (i) and we are done. VI
Theorem 57.1 (ii) immediately yields the following interesting cases:
57.2. COROLLARY
(i)
/f TLR + TI(lev) F Vi. Tit (t closed), then RS o F TSt ( - T~o(t)),
for some 5 < F 0 and hence O(G~M,to)l= Tt.
(ii)
If TLR + TI(lev) F Vi. Cli(t ) (t closed), then RS o S Clio(t), for
some 5 < F o and hence O(CTM, Fo)~CI(t ).
It is not difficult to adapt the previous machinery to the proof-theoretic
investigation of MF, MFp and MF c of Ch. II, 10.7. We begin with the
theory MF with full number-theoretic induction.
57.3. T H E O R E M
(i)
If MF S A and A is a sentence of.Lop, then OPCCF 1a A, for some
a < C~1 = r
(ii)
If MF F Tt (t closed), then RS o F Tat, for some a < Co, and
hence O(CTM, eo)]=Tt.
XI.57]
Upper Bound For TLR and its Subsystems
331
PROOF. Ad (i)-(ii). First observe that the language 2, of MF can be
translated into the language 2"0 of the system I T S , once we identify the
atom T t of s with Tot. It is convenient to leave the translation implicit; so
we use the same symbol A to denote the Z-formula A and its cognate
translation in 2"0" With this in mind, a straightforward adaptation of 51.5,
51.6 and the partial cut elimination theorem 55.1 imply:
if MF F- A, then I T ~ F- al A, for some c~ < c0.
(1)
But (1) yields (i)-(ii), as a consequence of the upper bound theorem 56.7. r'l
57.4. THEOREM. Let A be any sentence of .Lop and let t be any closed
term. Then:
(i)
(ii)
if MFp F- A, then O P m S c~
1 A, for some c~ < Cw0;
if MFp F- Tt, then RS o ~ T a t for some ~ < w w and hence
O(CTM, wW)l=Tt;
(iii)
- al A, for some c~ < %;
if MF c F- A, then OP~176
(iv)
if MF c F- Tt, then RS o F- Tkt, for some k < w and hence
O(CTM, w)~-Tt.
PROOF. We stick to the convention of identifying s with 2,0 as in the
previous proof.
Ad (i)-(ii). We exploit the fact that in these cases the infinitary system I T ~
can be replaced by its finitary version ITo(p).
ITo(P) is obtained from I T ~ by means of the following modifications:
1. we omit the N-rules ( g ) and ( ~ g ) ;
2. the operational axioms of 50.2.1 are replaced by the corresponding
axioms of A.2 and A.3 (see 47.3);
3. we add the N-induction rule in the form P - N - I N D :
infer F, ~ N t , tTla from F, 0r/a and F, Vx(xTla---, (x+l)~/a).
Clearly MFp can be embedded into IT0(p).
The derivability relation "IT0(p)~- mk F" is inductively generated by axioms
and rules of IT0(P) , as in 50.2.3; of course, since the rules are finitary, we
have that k, m < w (m being an upper bound on the 0-complexity of the cut
formulas occurring in the given derivation of F). Now, due to the restricted
form of P - N - I N D , IT0(P) enjoys partial cut elimination:
if IW0(P)~- km + l A, then IW0(p)~- n1 A for some n e w.
(1)
332
Proof Theory of Finitely Iterated Reflective Truth
[Ch.ll
Hence the embedding theorem 55.5 can be specialized to ITo(P):
if IWo(P)b k1 F, then RS o F w(~,+l)
~t "~+1) r [ z , 7], for every fl, 7,
(2)
such that 0 < fl and ~+w k < 7.
Verification of (2): by induction on k. We only consider the case of positive
N-induction and we assume:
ITo(P) k k1 o F, 0via and IWo(P) F k~
1 F, -~trla, (t+ 1)~la,
where leo < k, while the second premise is obtained by (V), (V)-inversion
property, adapted to IT0(P ) from 53.4. Fix arbitrary ~ > 0, "r > / ~ + wk and
put a n "- ~+wkO(n+l).
We check, by secondary induction on n E w:
RS o F w-),
< ~(-r+l)r[fl,7], ~rlana.
(2.1)
Indeed, by main II-I we also have, for arbitrary fl > 0 and a(fl) - fl+wk~
,-Oqa(~)a
RS 0 b :{a(~)+l
(2.2)
RS o b :/aa/~/++~/F[fl, a(fl)],-~tq~a,(t+l)r]a(~)a (t arbitrary).
(2.3)
If n - - 0 , (2.1) is simply a consequence of (2.2) with the persistence lemma
55.4 and 7 > ao. If n -- m + l , we have by secondary IH:
RS 0 F < w(~,+l)rift,,),] ' m~/ama.
(2.4)
If we choose ~ = am in (2.3) above, we get, since a(am) = an:
RS o I-- w(O-n+l
W(an+l I
a"
~qana.
(2.5)
-~rlana.
(2.6)
But 7 > an and hence, by persistence lemma 55.4,
RS 0 I-- w-'/
< ~(o'-F1) r[fl, 7] '
~rlama,
An application of (Cut) between (2.4) and (2.6) concludes the verification
of (2.1). On the other hand, if we apply the substitution lemma, we obtain,
with t arbitrary, for each n:
a.
(2 7)
r[fl, ~/]' "~t - -~, trl'la "
(2.8)
RS 0 }_ w')'
< w(-/+l) r [ z ,
~
-
Hence by 54.3 (iii) and a cut of rank wa n < wT:
RS 0 ~ toy
< w(-/+l)
A final application of (-~N) yields (2.8):
w("/+l
RS o k w(~/+l / r[fl, "y], -~Nt, tri~a. 0
Finally, if MFp b A, we have for some n > 0, by (1)-(2) with/~ - 1:
Upper Bound for TLR and its Subsystems
XI.57]
333
wn+2
RS o ~ wn+2 A[1, wn],
whence by Tait's cut elimination RS o f-r1
A[1, wn]. If we apply
the upper bound theorem and the closure properties of Cw0, we obtain
O P ~ F - a< r
A, if A E s
and O(CTM, Cw0)l=A, if A is T-positive. !"1
Ad(iii)-(iv). We introduce a variant ITo(c ) of ITo(P) such that, if MF c F- A,
then ITo(c ) ~ A. IT0(c ) is obtained from IT0(P) by replacing the rule P-WI N D with C L - N - I N D below:
B
infer r, ~Nt, trla from the premises r, Cl(a); r, 0~a; F, Vx(x~a~(x+l)~a).
k
As for ITo(P), ITo(c ) enjoys partial cut elimination, i.e. if ITo(c ) F- m+aA,
then ITo(c ) F- 1n A, for some n E w. Instead of reproving the appropriate
form of embedding for ITo(c), we establish the crucial case of the
interpretation theorem in semantical terms; we then give directions to
obtain its proof-theoretic version. First of all, we recursively define:
I=A[m, n] iff either A is an e-atom and CTMI=A; or
(3)
A - Tt (Ft) and t E O(CTM, n) ((-~t)E O(CTM, n)); or
A - - ~ T t (~Ft) and t ~ O(CTM, m) ((-~t) ~ O(CTM, m)); or
A - VxB (3xB) and ]=B(t)[m, n] for every (some) t E CTM; or
A-
B A C (B V C) and [=B[m,n] and (or)I--C[m,n].
We also write [= Tnt (Fur) for I= (Tt)[m, n] (I= (Ft)[m, n]) and I= ~Tmt
(~Fmt) for I= (-~Tt)[m, n] (]- (-~Ft)[m, n]). If F - {A1,... , Ak} , I= F[m, n] is
interpreted disjunctively as I-- (A1 V ... V An)[m,n ]. As expected, we have:
if I=F[m ', n'],A and m < m' < n' < n, then I=F[m, n], A.
(4)
Now we claim:
if IWo(c ) F- k F, then I=F[m, n], for every m > O, n >_m+2 k.
(5)
Verification of (5): by induction on k. Let us only consider the case where,
for some/c o < k, we have:
ko
~
ITo(c ) t- 1 F Cl(a)
ko
ITo(c ) F- 1 F, Or/a
and
(5.1)
ko
ITo(c) F- a -~t71a,(t+ 1)rla (t arbitrary).
Then by I n applied to (5.1), and (4), we have, for every m > 0, n > m + 2 ko
with Po - m+2k0 and remembering that t71P~ - TP~
t~P~ - FP~
9
334
Proof Theory of Finitely Iterated Reflective Truth
[Ch.ll
I=F[m, n], to po a, t~ p~ a (t arbitrary);
(5.2)
I=r[m, nl, OnPOa;
(5.3)
I=r[m,n],-~to m a, (t+l)n po a (t arbitrary).
(5.4)
Now fix m I > 0, n 1 > m 1-4-2k and set Pl - m l +2k~ We check by secondary
induction on l E w
I=F[ml, nil, I~Pl a.
(5.5)
If I - 0, apply (5.3) with m - m 1. If I - j + l , assume by secondary IH:
(5.6)
I-F[m 1, n 11, ~r/pl a.
If we apply (5.4) with m -
Pl, we get, for q - Pl+2k~
(5.7)
I=F[pl, q], ~ ~//Pl a, lT]q a.
By (4), since n 1 > m 1+2 k > Pl +2k~ - q > ml, we get
I=r[ml, n 1], -~ jOP~ a, 70q a;
hence, with a cut:
[=F[ml, nll,7~lq a.
(5.8)
By Ch. II, 7.7, we have O(CTM, n) f-! {t" t E CTM, O(CTM, n)l=T~t) - 0
and O(CTM, n) C_ O(CTM, m) for n < m, whence
[=F[ml, n l ] , ~ l~q a, --1 lr/q a.
(5.9)
I=F[ml, nl], -~ 7~Pla, 7~qa.
(5.10)
The conclusion follows by application of the cut rule to (5.S), (5.9), (5.10)
and (5.2) (in the last case choose m - ml). El
If we inspect the verification of (5), we can observe: the levels involved are
finite, by contrast with the corresponding step (2) for ITo(P); the levels
depend only on the given parameter m and on the derivation length. Hence
the interpretation theorem for ITo(c ) can be carried out within the fragment
of RS o with finite levels only. In particular, we can modify the notion of
0-rank for formulas of RSo, in such a way that the 0-rank is always finite
and cut elimination still works with respect to the new notion (Hint: choose
Ro(T k) - R o ( F k) - 3 0 k and reprove 54.1-5, 55.4). Finally the embedding
theorem 55.5 is refined for ITo(c), to the extent that:
if IT0(c ) F- ~ A, then RS o F- ~ All, 1+2 k] for some )~ < Co,
which implies (iii)-(iv). We leave a complete check of (6) as exercise. [i
(6)
The Conservation Theorems
XI.58]
335
w58. Conclusion: the conservation theorems
If we piece together the results of Ch. IX with the main theorems of w57, we
can obtain a characterization of the operational theorems of TLR-4-TI(lev),
MF, MFp, MF c. In order to state the theorem, let us agree that:
(i) a < fl stands for the Lop-formula defining the primitive recursive
well-ordering of type F 0 and lower case Greek letters a, fl, 7 . . . r a n g e over
the field of < ;
(ii) Progr( < , B ) = Vc~(Vfl(fl < c~~ B(fl))--, B(c~));
(iii) TI(c~) is the schema Progr( < , B ) ~ Vfl < c~. B(/~);
TI( <
u
<
TIop ( < c~)is TI( < ~), restricted to Lop-formulas.
58.1. THEOREM (Conservation). Let A be a formula of Lop. Then:
(i) TLR + TI(lev) F- A iff OP + Tlop ( < F0) F- A;
(ii) MF F- A iff OP + Tlop ( < r
F- A;
(iii) MFp F- A iff OP + Tlop ( < Cw0) F- A;
(iv) MFc F- A iff OP F- A .
PROOF. r
w
of Ch. IX contains the relevant facts for proving the
implications from right to left. Indeed, observe that elementary
comprehension and P W O ( ~ ) i m p l y TIop(a); then apply 46.2, 46.2.6 for (i)
and (iii). As to (ii), 46.2.3 and full N-induction imply TI( < Co) for
arbitrary formulas of s which yields the existence of the ramified hierarchy
up to any c~ < ~0 (by 44.3), and hence the conclusion by 46.2.5.
==~" we informally sketch the argument. First, it is essential to realize that
the proofs involved in chapters VIII-IX are constructive: indeed, they can be
formalized in the elementary theory OP extended by the schema TI( < (~)
on an appropriate segment c~ of F 0. The major obstacle to the formalization
is to find Lop-formulas which adequately represent in OP the infinitary
derivability relations RS n F- p F, I T ~ F- pa F, O P ~
pa F . Now it is possible
to find adequate Lop-formulas by making the derivation trees themselves
explicit, and by observing that they can be effectively encoded. Then one
shows, by means of the recursion theorem for primitive recursive functions,
that the operation of cut elimination is primitive recursive, and that also
the embedding operations are effective. These steps are non-trivial, but wellknown; details for formalizing cut elimination can be gained from Mints
(1975), Schwichtenberg (1977), Buchholz (1991) {or from the appendix}.
336
Proof Theory of Finitely Iterated Reflective Truth
[Ch.ll
In order to state the formalized results, we adopt the following stipulations.
First of all, we fix a G6del numbering of basic syntax; if E (respectively
"... F- ...") is a syntactical expression (a derivability predicate), which
belongs to one of the systems involved, let [E] ([... ~ ...1 respectively)
denote the corresponding arithmetized term (predicate) of s
Sentop(X)
defines the predicate " x is a closed formula of .Lop"; Com(x,y) stands for
"the sentence encoded by x has logical complexity < y". In addition, if SF is
a finitary formal system, let Dimk(x,y, SF ) stand for the arithmetized
predicate "x encodes a proof of y in SF, having < k lines".
If we combine formalized versions of the embedding theorems 48.8,
49.11, weak cut elimination 49.8 and the main corollary to the asymmetric
interpretation of T L R + T I ( l e v ) in STLR (see 48.8-49.19), we obtain for
each given k E ~, provably in OP, if ~ := TLR + TI(lev):
VdV[Al(Sentop([A1) ADimk(d, WAl,~)-~([STLRr
F- AI)),
(1)
for a suitable term r
representing an a(k)-recursive function, and for
some a(k)< co (cf. 49.20).
Now define a 0 = Co, an+ 1 = Can0. Then by formalization of 51.6, 55.1
and the upper bound theorem 56.7 (ii), we have for each given m, provably
in OP + Tlop(am+l):
V[A](Sentop([A]) n [STLR~ ~- A]-~ 3a(a < a~+i ^ [ Oe~ ~ ~ A])).
(2)
At this stage, we can use a well-known "reflection" technique. If OP ~162
~ ~A,
then each formula occurring in the given OP~176
~ must have
logical complexity _< co(A)( = the logical complexity of A), because the only
possible cut formulas within ~, by assumption, either have the form t = s
or Nt. Hence the correctness of a "cut-free" derivation (in the above sense)
can be checked by reference to a truth predicate Tr n for .Lop-sentences of
logical complexity _< n, for some given n; and Tr n can be defined within
OP. Hence we obtain, for each finite n and k, provably in OP + Tlop(ak)"
a < a k A Com([A],-~) A Sentop([A1) A OP ~176
F- ~ A -~ Trn([A]);
(3)
o n ~ Trn([A])-~ A (where A E s
(4)
and has logical complexity _< n).
The verification of (i) is now straightforward, as soon as we note that
TLR + TI(lev) F- A with A E s
implies OP F- [TLR + TI(lev) F- A] and
we apply (1)-(4) above. The other results follow by the formalized versions
of the corresponding upper bound theorems for MF, MFp and MF c. V!
The previous theorem, coupled with the general conservation theorem
15.5 concerning PWc, yields an exhaustive proof-theoretic classification of
the systems introduced up till now (with the exception of the impredicative
extensions of w167
41-42).
The Conservation Theorems
XI.58]
337
58.2. Final remarks
(i) The techniques of this chapter adapt those applied in Cantini
(1985a) to certain predicatively reducible theories ID* of iterated inductive
definitions. J~iger(1984, 1986) develops an elegant and uniform approach to
reductive proof theory, based on admissible set theories; the exact
relationship of TLR and its variants with iterated admissibility is yet to be
investigated in some detail.
(ii) The results of chapters X-XI can be strengthened with the addition
of local weak generalized induction principles. To be more specific, let us
consider the term I ( W , a , i ) of lemma 41.11, which defines the collection of
well-founded trees recursive in the /-class a. We already know that it is
consistent to assume the C_-minimality of such collection. Formally let
LGI( - local generalized induction) be the axiom:
Cli(a ) A Vx(Wi(x, a, b)--,xriib) -~I(W, a, i) C_ b,
where Wi(x,a,b ) formalizes the operator, which inductively generates the
collection of trees recursive in the set represented by the /-class a. It turns
out that TLR + L G I is not stronger than TLR (the arguments parallel
those of w
but the Veblen function Ac~Afl.Oc~fl is required for the
asymmetric interpretation of LGI).
(iii) A further refinement concerns the direct proof-theoretical analysis
of PW c + GID; this is already in the literature for an equivalent system
(Cantini 1992) and entirely analogous to the method of w
Thus we only
sketch the basic idea. The starting point is to axiomatize, in a natural
ordinal theory PWO, the features of the inductive model of Ch. II, w that
make PW c + GID true. One can easily find a system PWO, where:
1. the inductive generation of T in ordinal stages is made explicit via
certain local predicative closure conditions (extending those of 52.9 so that
r-axioms become true);
2. the full transfinite induction schema T I on ordinals is accepted, together
with a form T R of ordinal reflection, granting the existence of transfinite
ordinals;
3. number-theoretic induction is needed only in a restricted, local form, i.e.
for properties at a given ordinal stage.
Then PWO is reduced to OP. First of all, the schema T I is eliminated:
PWO is interpreted in a system P W O ~ where ordinal variables are forced
to range over finite ordinals (via ~-rule). P W O ~ is consistent and it admits
an asymmetric interpretation in a ramified system with finite stages PWR.
The final step shows that P W R is proof-theoretically reducible to OP using
a standard cut elimination argument.
Proof Theory of Finitely Iterated Reflective Truth
338
[Ch.ll
Appendix: primitive recursive cut elimination for RS n
Preliminaries
We outline a primitive recursive cut elimination algorithm for the
predicative systems RS n. We combine the use of repetition rule, due to
Mints(1975), with the encoding technique of Schwichtenberg (1977). For an
elegant unified treatment of primitive recursive and continuous cut
elimination, the reader is urged to consult Buchholz (1991). All the cited
references only deal with Peano arithmetic with w-rule.
Roughly speaking, RSn-derivations are (possibly) infinite trees of
inferential figures, inductively generated according to the clauses DER.1DER.2, which define the derivability relation RS n ~- p F of 52.10 Now the
problem is that we need a finitary description of the associated infinitary
trees and their properties: thus we shall only consider those derivations that
can be effectively described by finite sequences of data, the so-called codes.
In particular, we apply an infinitary inference, only if we have a primitive
recursive control over the immediate subderivations of the premises. As a
preliminary step, we fix the following data:
(i) a primitive recursive G6del numbering [ ] of the language of RSn;
(ii) a primitive recursive indexing {[e]:e C w} for primitive recursive
functions (PR in short); Uni is the recursive universal function such that
Vni(e,x)=[e](x); P R I is the set of PR-indices ( P R I is primitive
recursive);
(iii) a PR-injection Inj" NT---,N, with PR-projections; we assume that
the image of In j, Inj[N 7] is disjoint from the set P R I of PR-indices; if
x C Inj[NT], x i (for 0 < i < 6) denotes the i-th coordinate of x.
A sequent F = {A1,... , An} is encoded as a finite set of G6del numbers:
if A1,... , A , are distinct, [ F ] - 2rAil + ... + 2 [An].
For notational simplicity, we systematically identify syntactical entities
with their number-theoretic encodings; in general, this will cause no
ambiguity. We also use the formal expression A X for referring to any
possible axiom of RS n. R E P stands for the repetition rule: infer RS n ~ po( F
from RS n F- ~ F, whenever ~ < a. R E P is semantically trivial, but it makes
sense as a geometric operation, which adds a new node to a given derivation
tree of length fl and increases the height of the given tree. If a PR-function
F operates on F and on an inference label, say (CUT) for the cut rule, we
simply write F(F, CUT) instead of F(FFI, FCUT]). Similarly, if S o , . . . , S 6
are syntactical entities, we write (So,... , $6) instead of Inj([So],... , [$6] ).
The basic data structures we deal with for representing derivations are
Append&
XI.A]
called labels and they are suitable vectors f = ( f 0 , ' " , f 6 )
Formally, we are led to introduce the following
339
of dimension 7.
1. D E F I N I T I O N
f e LABEL iff either f = 0 (0 stands for the code of the e m p t y set,
too) or f = ( f 0 , ' " , f6) and:
1. fo = RF(f) is the name of an inference ~ of RSn; from f0 we can
read off
(i)
1.1. the eigenvariable of ~, if ~ requires it; else
1.2. an index a E [0, Fo), which gives the "position" of the premise
of ~ (see the case of the repetition rule below), or identifies the
minor formula of ~, if there is any ambiguity;
2. f l - E N D ( f ) encodes the disjoint union of the set of side formulas
LAT(f) and the set of active formulas AF(f);
3. f2 = L(f)( = the length of f), f3 = R ( f ) ( = the rank of f ) are
elements of OT;
4. f4 = P A R ( f ) is a finite set of parameters;
5. if f5 = SOl(f), f6 = SD2(f), then:
5.1. if R F ( f ) = AX, each SDi(f)is empty (i = 1,2);
5.2. if R F ( f ) = REP and its position index is 5 E OT, or R F ( f ) is
infinitary, then S D 2 ( f ) = 0 and S O l ( f ) E P R I (i.e. a PR-index);
5.3. if RF(f) # REP and RF(f) is 2-ary (1-ary), then
SDi(f ) :/: O, for i = 1, 2 (SOl(f) # 0 and SD2(f) = 0, or
SD2( f ) # 0 and SOl(f) = 0, respectively).
(ii)
We inductively define the operation DEP as follows:
DEP( f ) - O, i f f - 0 o r f ~ L A B E L ; e l s e
D E P ( f ) = max(DEP(SDl(f)), DEP(SD2(f)))+l.
Clearly LABEL and DEP are primitive recursive.
If f E LABEL, we can build up a formal figure, which is to be regarded
as a finite stump of a (possible) derivation; DEP(f) is the depth of f.
Informally, if f is a label, RF(f) names the final inference of a derivation,
whose conclusion is E N D ( f ) and whose immediate subderivations are
identified respectively by SOl(f), SD2(f). L(f), R(f) play the role of the
length and the rank, while P A R ( f ) lists the parameters occurring in the
derivation; it can be assumed that we can effectively identify the subset
E I G E N ( f ) of eigenparameters. We set PAR(f, g) = P A R ( f ) U PAR(g), if
f, g E LABEL. By PAR(f)(b/b +) we denote the set which is obtained
Proof Theory of Finitely Iterated Reflective Truth
340
[Ch.ll
from P A R ( f ) by replacing b with b+; the "plus"-sign is used to declare that
b occurs as eigenparameter. Below we generally skip places corresponding to
empty coordinates of f, unless some ambiguity arises.
We now proceed to the definition of a collection CODE C_LABEL,
whose elements represent true derivations.
2. Inductive Definition of CODE
CODE is the smallest set of labels, which satisfies the initial clause below
and is closed under clauses, corresponding to the rules of RS n"
(i) If F, A is an axiom of RSn, (AX; F, A; a; p; PAR(F,A)I E CODE,
for every .. ; e O T (here 1 6 - f ~ - 0);
(ii) if f i E CODE, E g D ( f i) - {F.A~}. a > L ( f ~). p > R ( f ~) (i - 1.2).
F -- {F, A 1 A A2} , then
((A); F, A 1 A A2; a; p; PAR(f1, f2); fl; f2)E CODE;
(iii) if fi E CODE, E N D ( f i ) - {F, A 1 V A2, Ai} ( i - 1, 2), a > L(f2) ,
p > R(fi) , then
((( V ),i); F, A 1 V A2; a; p; PAR(fi); f i ) E CODE
{it is understood that S D I ( f ) -
0 (SD2(f) - O ) whenever i - 2 ( i - 1)};
(iv) if g E CODE, END(g) -- {F, A[x "- a]}, a q~PAR(F, VxA),
a > i(g), R(g) <_p, then
((V,a); F, VxA; a; p; PAR(g)(a/a+); g)E CODE;
(v) if g E CODE, END(g) - {F,A[x "- t]}, a > L(g), p >_R(g), then
((B, t); F, SxA; a; p; PAR(g); g)E CODE;
(vi) if h, g E CODE, E N D ( h ) - {F,A), E N D ( g ) - {F, -~A}, L(h),
L(g) < a, max(Rn(A)+l , R(g), R(h)) < p, then
((CUT, A), F; a; p; PAR(g, h); h; g ) E CODE;
(vii)
if e E P R I and END([e](~))- F, L([e](~)) < a, R([e](~)) < p,
V - PAR([e](~i)), [e](~) E CODE and [e](~') - q) for ~' ~- ~, then
(PEPs; F; a; p; V; e)E CODE;
(viii)
if g E CODE, END(g) - {F, Nt, t - k} for some k E ~, a > L(g),
p > R(g), then
(((N),k); F, Nt; a; p; PAR(g); g)E CODE;
(ix) let e E PRI, fn - [e](n) E CODE and E N D ( f n ) - {F, --,t - ~}, for
XI.A]
Appendix
341
every n E w; assume a > L(fn) , p >__R(fn) for every n E w, and
let V = U {PAR(fn): n E w} be finite. Then
((-~N); F,~Nt; a; p; V; e ) E CODE;
(x)
if g E CODE and END(g)= {F, (-~)T%Clause(s)}, a > L(g),
p >_R(g), then
(((--)Ta+l); F, (--)Ta+ls; a; p; PAR(g); g)E CODE;
(xi)
the clauses corresponding to the introduction of F a + l are obtained
from (x) by replacing T with F;
(xii)
let e E P R I and f a - [ e ] ( a ) E CODE with END(fa) - { F , - , T a t } ,
for every a < 5, 5 limit; assume/3 > L(fa), p >_R(f~) for every
a < 5, and let V - U { P A R ( f a ) ' a < 5} be finite. Then
((-~T-LIMh); F, ~Tht; c~; p; V; e ) E CODE;
(xiii)
let g E CODE and END(g) - {F, Tat}, where a < 5 and 5 limit;
if /3 > L(g), p > R(g), then
(((T-LIMh),a); F, Tht; /3; p; PAR(g); g)E CODE;
(xiv)
the clauses corresponding to the introduction of F 6, for 5 limit, are
obtained from (xiii) by replacing T with F.
NB" that only derivations with a finite set of parameters are encoded, as
implied by (ix), is not restrictive for our aim of embedding finitary systems
into infinitary ones.
By generalized recursion theory, CODE is the fixed point of a positive
arithmetical operator (indeed a boolean combination of II O- and E ~
conditions) and it is generally not elementarily definable. However, we can
effectively associate to each derivation label f a well-founded tree of formal
figures, which is locally correct (hence a derivation in the true sense),
exactly when f E CODE. Since local correctness is an elementary condition,
we can get an elementary representation of CODE.
3. D E F I N I T I O N . Let 9 denote concatenation on OT-elements.
(i) OT* is the smallest set X such that" ( ) E X; if s E X, s , a E X, for
aEOT.
(iN) Length(( ) ) - 0; Length(s,a)- Length(s)+l.
(iii) If s, s' E OT*, s C_1s' iff s - s' or s ' - s , a , for some a E OT. C_ is
the transitive closure of C_ 1; s C_ s' iff s is a subsequence of s'.
(iv) S COT* is an OT*-tree if ( ) E S and S is closed under the
subsequence relation C (s' C s E S implies s' E S).
Proof Theory of Finitely Iterated Reflective Truth
342
[Ch.ll
An OT*-tree S is well-founded if every S-path is finite; by S-path we
mean a maximal C-chain.
4. D E F I N I T I O N (by course of value recursion on Length(s), s E OT*).
1. D e r ( f , s) - 0, if f ~ L A B E L or s ~ OT*;
2. else, assume f E L A B E L , s E OT*: then
2.1. Der( f , ( ) ) 2.2. let s -
f;
s',fl, for some fl E OT; assume ~1- RF(Der(f,s'));
2.2.1. ~ - AX: D e r ( f , s ) -
0;
2.2.2. ~ - REP~: then Der(f, s) - 0 if fl 5/: t~;
else, Der(f,s) - Uni(SDl(Der(f,s')),fl);
2.2.3. ~ has ~ premises ~ > w, 6 limit: then Der(f, s) - 0 if fl > 5; else,
D e r ( f , s ) - Uni(SDl(Der(f,s')),fl ) (here Uni is a fixed
recursive universal function for the unary PR-functions);
2.2.4. ~ I - ( V ,i), i - 1 or 2: then D e r ( f , s ) fl - i - 1; else Der(f, s) - 0;
SDi(Der(f,s'))if
2.2.5. ~ is not as in 2.2.4, but is k-ary (1 _< k _< 2). Then D e r ( f , s ) - 0
if fl > k; else n e r ( f , s) - SD~+ l ( n e r ( f , s')).
By inspection of 4, we obtain:
5. L E M M A
(i) The operation )~x)~y.Der(x,y) is primitive recursive in Uni ( = the
given universal function for 1-ary PR-functions).
(ii) We can find a PR-operation D E R such that D E R ( f ) is an index
for )~x. Der( f , x).
Clearly (ii) follows by the s-m-n theorem. Henceforth we write s E f for
D e r ( f ,s) # O and we say that s i s a n o d e o f f .
T ( f ) - {s E OT*" s E f } is
obviously an OT*-tree.
6. D E F I N I T I O N . L C ( D E R ( f ) ) holds iff it satisfies the following conditions:
1. f E L A B E L , ( ) E f; D e r ( f , s ) E L A B E L , if s E f;
2. if s E f, ~ = R F ( D e r ( f , s ) ) , then
2.1. either ~ = A X , E g D ( D e r ( f ,s)) is an axiom and s.~ ~ f
for every ~ E OT; or
2.2. ~ ~= A X , E N D ( D e r ( f , s)) follows from the set
Appendix
XI.A]
343
{ E N D ( D e r ( f , s , f l ) ) : s,fl E f} by means of 3;
3. in 2.1-2.2 the conditions on the length and the rank are met; this
means in particular that if s E f, s,/3 E f, then
L(Der(f, s)) > L(Der(f, s,j3)).
{ L C ( D E R ( f ) ) is read as " D E R ( f ) encodes a locally correct derivation"}.
7. LEMMA. Let f E L A B E L .
(i)
The predicate L C ( D E R ( - ) ) is definable in s
(ii)
L C ( D E R ( f ) ) implies that T ( f ) is well-founded;
(iii) f E C O D E iff L C ( D E R ( f ) ) holds.
PROOF. (i): by a straightforward formalization of 6.
(ii): by condition 6.3, any descending infinite sequence in the tree ordering
would produce a descending infinite sequence in OT*.
(iii): ~ : by generalized induction on the definition of CODE. ~ : by (ii),
T ( f ) is well-founded and we can apply transfinite induction to verify
f E CODE. D
As consequence of 7 (iii), there is an s
A(f), which represents
the condition f E CODE; for convenience, we write f E D E R instead of
A(f), and we simply say that f is a derivation. Lemma 7 makes possible to
define elementary operations on CODE; more properly, when we talk about
operations on DER, we mean PR-operations
F : LABEL---, LABEL,
which preserve the property CODE, i.e. if f E CODE, F ( f ) E CODE. As a
rule, we only rely on the surface structure of labels, without appealing to
transfinite induction; instead, we apply course-of-value recursion (on the
depth of the derivation labels), PR-distinction of cases and the second
recursion theorem for PR-indices (after Kleene 1958):
8. THEOREM. There is a Kalmar elementary operation Fix such that, if e
is a PR-index for a k+l-ary PR-function, then Fix(e)E P R I and
[Fix(e)](nl, . . . , n k ) = [e](Fix(e), ha,... , nk).
Operations on DER and cut elimination
We essentially exploit the finitary presentation of RSn-derivations; the basic
operations involved in cut elimination are shown to be primitive recursive
and they naturally work on arbitrary derivation labels. Transfinite
induction is needed only for checking that the basic operators work properly
(correctness proofs). First of all, we see that it is possible to deform
Proof Theory of Finitely Iterated Reflective Truth
344
[Ch.ll
monotonically the ordinal assignment (of lengths and ranks), and to rename
parameters.
If f E L A B E L , f(a) abbreviates the condition "a E P A R ( f ) " ; f ( a / t )
denotes the result of replacing everywhere a by t. D E R is not closed under
the operation t~-. f ( a / t ) , because substitution may spoil the eigenparameter
restrictions. In order to reconcile the present notations with those of w52, we
introduce the following
9. DEFINITION. f F- aP F "- L C ( D E R ( f ) ) A f E L A B E L A
A E N D ( f ) = F A L ( f ) = a A R ( f ) = p.
Hence, as a consequence of lemma 7, we have:
f F - pa F is definable in the language s
9.1
Now we state a few technical lemmata, which are essential for
manipulating derivations. Proofs are straightforward: however the formal
definitions of the primitive recursive operations, which are claimed to exist,
require lengthy PR-distinction by cases, course-of-value recursion on D E P
and a final application of the second recursion theorem. The technique will
be illustrated below in the case of the inversion lemma and the subsequent
theorems.
10. LEMMA
(i)
We can find a PR-operation M O N such that, if f F- pc~
F~ o~ _
< C~~,
Ol !
p <_ p', then M O N ( f , a', p') - f ' E D E R and f'F-
, F.
P
(ii)
Renaming: we can find a PR-operation R E N A M E
oz
f F- p F, c ~ P A R ( f ) , a E P A R ( f ) , then
a, c ) - f (a/c) E D E R , ](a/c) F- Ol
p
RENAME(f,
PAR(f(a/c))(iii)
such
P A R ( f ) ( a / c ) and D E P ( f ) =
such that, if
r[a--
c],
DEP(f(a/c)).
Renaming of eigenparameters: we can find a PR-operation E G
that, if f F- Ol
p F, a E E I G E N ( f ) ,
c ~ PAR(f),
then
E G ( f , c, a) - f ' E D E R , f ' F- p F, D E P ( f') - D E P ( f ) and
EIGEN(f')
= (EIGEN(f)
U {c})-{a}.
By lemma 10(iii), it is not restrictive to assume that, if a parameter b
occurs as an eigenparameter at a certain node of a derivation f, then b does
not occur below that node.
Append&
XI.A]
345
11. LEMMA (Substitution). We can find a PR-derivation S O S T such that,
if a ~ P A R ( r ) and f ~ ~ r,
th~n S O S T ( I , a , t ) - f ( a / t ) ~- "p r[a--t] (t
arbitrary term). Moreover D E P ( f ( a / t ) ) = D E P ( f ) and
P A R ( f ( a / t ) ) = P A R ( f ) U PAR(t)t.J P A R ( C o ) - ( V o t_J{a}),
where Y 0 = { a l , . . . , a n } = E I G E g ( f ) f-I PAR(t), C 0 = { e l , . . . , Cn} is the
set of the first n parameters, which do not belong to P A R ( f ) .
The proof is a combination of eigenparameter-renaming and of simple
inductive considerations; we also need the finiteness of P A R ( f ) .
12. LEMMA (Weakening). We can find a PR-operation I N D E B ,
that, if PAR(A) fl E I G E N ( f ) - 0, and f ~- pOl F, then
(i)
(ii)
such
I N D E B ( f ) ~ - p or F,A;
DEP(INDEB(f))=
PAR(INDEB(f))
D E P ( f ) and
= P A R ( f ) t3 P A R ( A ) .
The condition on P A R ( A ) is not restrictive by lemma 11 above.
We now assume in toto the notions of w
in particular, the notion of
reducible formula of RS n is primitive recursive.
13. LEMMA (Inversion). We can find a PR-operation I N V such that, if
f ~- Olp F, C, Rn(C ) > O, C is reducible to A, and
P A R ( A ) N E I G E N ( f ) = 0, then
I N V ( f , A ) - f' C D E R and f ' b pa F A, ;
DEP(INV(f,A))
< DEP(f).
P R O O F . We sketch the formal definition of INV; but the reader should
keep in mind the informal argument of 53.4 (which runs by transfinite
induction on L(f)). Recall that ~ stands for an inference name. We set:
1. F ( a , f , A ) = O if f ~ L A B E L ,
C C E N D ( f ) reduces to A;
or
EIGEN(f) VIPAR(A)#O
else, if f has the form (~; F, C; a; p; V; f l ;
V' = ( V - P A R ( C ) ) t 2 P A R ( A ) , we define:
or no
f2), C reduces to A and
2. F ( a , f , A ) = (AX; F,A; a; p; V ' ) p r o v i d e d ~ = A X (hence f l = f2 = 0);
else:
3. F(a, f, A) = (:1; r, A; a; p; V'; F(a, f l , A); F(a, f2, A)), if ~ is finitary, C
is not active; else:
346
Proof Theory of Finitely Iterated Reflective Truth
[Ch.ll
4. F(a, f, A) -- (:l; F, A; a; p; V'; j(a, e, A)), provided 3 is infinitary (hence
f l = e E P R I , f2 = 0), C is not active, and j is a primitive recursive
function such that j(a, e,A) is a P R - i n d e x for )~x. [a]([e](x),A), if a E PRI;
else:
5.
F ( a , f ,A) - (REP1; F, A; a; p; V'; comp(sost, a, f l, b,t)) ,
provided
~= ((V),b), A = A[b := t], (A(b)= minor formula of ~). Here comp is a
PR-function, which produces a primitive recursive index for the function
)~x.[1]([sost]([a](fl, A), b, t), x), if a E PRI; while sost is a P R - i n d e x for
S O S T (see lemma 10), and in general, if a E OT, [a](h,x)= h, if x = a and
[tr](h, x) = 0 otherwise; else:
6. F ( a , f , A ) = ( R E P i ;
F,A; a; p; Y'; g(i,a, fi, A)), provided ~ is finitary,
C is active, A is the minor formula of the i-th premise of ] or the minor
formula of ] is located at the i-th place (e.g. ] = (( V ),i)), while g is a PRfunction such that, if a E P R I , g(i,a, fi, A) is a PR-index for the function
Ax.[t~([a](fi , A), x) (for Ax.[t'](f, x), see item 5); else:
7. F ( a , f , A ) = (REP~; F,A; a; p; V'; h(6, a, fa, A)), provided the rule :l is
infinitary, A is the reduction of C determined by the &h-premise of ~, C is
active, h is a PR-function such that, if a E P R I,
h(6, a, f l, A ) is a P R - i n d e x for )~x.[b]([a]([fl](6),A),x ).
By theorem 8, we choose a P R - i n d e x inv such that"
F(inv, f, A) = [inv](f , A);
it is straightforward to check that CODE is closed under )~x.[inv](x,A) and
that DEP([inv](f,A))<_ D E P ( f ) (by induction on L(f) and D E P ( f )
respectively). V!
R E M A R K . (i) In the previous proof, we introduced a PR-function j,
yielding a P R - i n d e x j(a,e,A) for )~x.[a]([e](x),A). It is tacitly understood
that, if a q:. P R I or e ~ P R I (i.e. they are not PR-indices), then j(a,e,A)
assumes an arbitrary fixed value (say 0). Similar assumptions are made for
h, g, comp and will be presupposed in the definitions of RD C and CF (to
be given below).
(ii) R E P 6 is essential to the primitive reeursive character of I N V .
Consider, for instance, the code f - ( ( - " T 6 ) ;
F,-~T6t; a; p; V; e), where
/3 < ~ and 6 is a limit. Then we could have simply defined
I N V ( f , [--TOt]) - INV([e](fl), [--T~t]).
(,)
The point is that (,) makes sense only by effective transfinite recursion on
L(f) and by the application operation for PR-indices. Hence I N V would
not have been primitive recursive. On the contrary, R E P allows to delay
the application of I N V , and it is exactly this device which makes possible
Appendix
XI.A]
347
to consider only the "name" of the procedure enumerating the infinitely
many subderivations of f.
14. LEMMA (Reduction). We can define a PR-operation RDC, such that,
F , ~ A and f l ~ ~PlA,-~A, R(A) _>1, R(A) -> Po, Pl, then
if f o ~ p0
(i)
RDC(fo, f l ) :
f E DER;
n
(ii)
fl
a p@ /~ F, A and p _< R(A);
(iii)
D E P ( R D C ( f o, f l)) <- D E P ( f o)+DEP(f l)"
(# is the natural sum, which is well-defined on OT).
PROOF. We only write down the specific PR-equations, whose fixed point
solution is RDC. The verification proceeds by transfinite induction on a#fl.
By course of value recursion on D E P ( f o ) + D E P ( f l ) , we can find a
PR-operation F(a, f0, f l ) such that:
F(a, fo, fl) = 0 , if it is false that fo, f l are labels, where END(fo) ,
END(f1) have the form F, A and A , - - A respectively, Rn(A ) >_R(f0) ,
Rn(A ) > n ( f l ) , nn(A ) > 0;
else,
suppose
that
Y = P A R ( f o) U P A R ( f 1), E N D ( f o) = {F, A},
END(f1) = {A,-~A}, L(fo)= a, L ( f l ) = fl, R ( f o ) = Po, R ( f l ) = Pl; then
we define:
1. F(a, fo, f l ) : (AX; F, A; c~ # fl; p; V) if the final inferences of fo, f l
are axioms; else, let ~l be the last inference of fo; then:
2. F(a, fo, f l ) = (:1; F, A; a#/3; p; V; F(a, SDl(fo),fl),F(a, SD2(fo),fl)) ,
whenever ~ is finitary, A is not active; else"
3. F(a, fo, f l) =
r,/x;
p; v; cp(a,e, f l)), if ~ is infinitary, A is not
active, cp(a, e, f l ) is the PR-index of Az. [a]([e](x), fl) and e = SD l(fo);
else:
4. F(a, fo, f l ) :
-((CUT,~MA(fl)); r, zx;
p; v; F(a, fo, SDl(fl)); G(fo, fl)) ,
if G(fo,fl ) = I N D E B ( I N V ( f o , ~ M A ( f l)),A)) , and the formulas A, -~A
are both active, A is reducible but not disjunctive, ~ M A ( f l ) - "negation of
the minor formula of the last inference of fl""
The symmetric cases are left to the reader. RDC is the fixed point of
F(a, fo, fl)" I']
15. THEOREM. We can find a PR-operation CF such that, if f F-p F,
then CF(f) E D E R and CF(f) F r pa F.
PROOF.
As usual, we define C(a,f)
by distinction of cases on PR-
Proof Theory of Finitely Iterated Reflective Truth
348
conditions and by course-of-value recursion on
[Ch.ll
DEP(f).
1. C(a, f) - 0 if f ~ LABEL; else"
2. C(a, f ) - f, if R ( f ) - 1 or a - 0 (in this last case replace p - R(f) by
1, if p > 1); else, assume p - R ( f ) >
1, a - L ( f ) > 0 ,
R F ( f ) - ~ , and
f - (~; F; ~; p; V; f0; fl)" We have three subcases.
3.1. ~ is finitary, but it is not a cut of rank > 1:
C(a,f)-(:1; F; Cpc~; 1; V; C(a, fo); C(a, fl) )
(this is well-defined since
D E P ( f i ) < DEP(f), i C {0,1}).
3.2. ~ is infinitary, so f0 is a PR-index: we let
C ( a , f ) - (:t; F; Cpc~; 1; V; comb(a, fo)),
where
comb(a, fo) is a P R - i n d e x for Sx. [a]([fo](X)).
3.3. ~ is a cut of rank v - tJ(f) > 1, v < p and L(fi) - ai, for i - 0,1. By
lemma 11, there is a PR-function #i such that #i(a) satisfies
[#i(a)](f)- MOg([a](fi),r
v), whenever a E PRI. By lemma 14, there
is a PR-function h such that, for a E PRI:
[h(a)](f) - [a](RDC([Po(a)](f), [Pl(a)](f))).
Hence, recalling lemma 13 (item 5) and s-m-n theorem, there is a PRfunction r such that, if a E PRI:
[r
f)](x) --
[O]([h(a)](f),x).
Finally, we put:
C ( a , f ) - (REPo; F; Cpc~; 1; r
If we choose a PR-index CF by theorem 8, we can verify, by main
induction on p and secondary induction on a, that CF(f)F- r
F. [:]
W i t h similar arguments, we can prove that there is a P R - o p e r a t i o n
satisfying Tait's second cut elimination theorem (and hence 1-step cut
elimination). By inspecting the construction of theorem 15, we can conclude
that OP+TIop(Fo) proves the cut elimination theorem for RS n.
PART E
ALTERNATIVE VIEWS
"On a signal~ beaucoup d'antinomies, et le d~saccord a subsistS, personne
n'a ~t~ convaincu; d' une conlradiction, on peul toujours se tirer par un
coup de pouce ! Je veux dire par un distinguo." (H. Poincar4, 1913).
This Page Intentionally Left Blank
CHAPTER 12
NON-REDUCTIVE SYSTEMS FOR TYPE-FREE
ABSTRACTION AND TRUTH
w
w
w
w
w
~64.
The core system V F - and transfinite induction
Supervaluation models of V F An abstract sequent calculus and truth
Cut elimination and related properties
A provability interpretation and the upper bound theorem
Reconciling supervaluation models with provability interpretation
In this chapter we critically reconsider the basic truth axioms of w7 and
their semantics. An essential feature of these principles is that they are
reductive: they (roughly) presuppose a "compositional theory of meaning",
in that the truth conditions of logically complex sentences are reduced to
corresponding conditions for logically simpler sentences. In particular, the
basic idea is predicativistic in spirit: a statement is justified only if its truth
can be ultimately grounded upon elementary truths (see Kripke' s
classification in w34).
A major consequence of this general attitude is that even a tautology
may not be accepted, whenever it involves ungrounded sentences. It is
therefore natural to investigate the consistency of alternative views, which
are well-behaved with respect to logical consequence. In particular, it seems
reasonable to accept every classical tautology A, irrespective of its
complexity and its specific content (e.g. A might have the form r~lr ~ r~r,
r being the Russell property).
In w we present a non-reductive system VF-, which has non-trivial
mathematical content (indeed, it proves a generalized induction principle).
The main theorem tells us that VF-, even if number-theoretic induction for
classes is assumed, is proof-theoretically equivalent to OP (and hence to
PA); furthermore, the same equivalence holds between VFp "- "VF- plus
internal number-theoretic induction axiom" and the impredicative theory
ID 1 of elementary inductive definition. Thus the non-reductive approach
overcomes the deductive limits of the reductive notion of reflective truth.
The rest of the chapter is devoted to illustrate two types of semantics
for VF-: supervaluation models (SV-models, in short) and a provability
interpretation. SV-models are introduced in w and they take inspiration
Non-Reductive Systems for Truth and Abstraction
352
[Ch.12
from the supervaluation method (van Fraassen 1968, 1970). Indeed, we show
that there is a simple monotone operator, whose fixed points provide models
for VF-. w167
describe a proof-theoretic semantics for VF-: the truth
predicate T is interpreted as provability in an abstract infinitary system,
which enjoys cut elimination (w
We underline that the proof-theoretic
investigation is worked out in a restricted metatheory, i.e. the theory
P W - + GID of w16 with approximation operator and generalized induction.
As a byproduct, we shall obtain an upper bound for V F - and, at the same
time, insights on new principles for truth (w
In addition, w shows that
for countable ground structures, SV-least fixed point models and provability
models coincide.
w59. The core system V F - and tran.,fflnite induction
For the sake of simplicity, we restrict our consideration to a variant of .5,
which assumes ---~, V as primitive logical symbols; we let _L " - ( K - S ) and
-~A "-(A---, _L ), while V, A, 3 are defined as usual in classical logic. We
write T A and FA for T[A] and T[-~A], [A] being the term__ of .5op, which
represents A (cf.w we assume that I M P L Y "-AxAy.(31,(x,y)) encodes
---,). In order to simplify a few arguments below, it is convenient to fix an
axiomatization of classical logic with modus ponens as the only inference
rule (Tarski 1965).
59.1. DEFINITION
(i) V F - is the elementary theory (in the language s as modified
above), which includes classical first-order logic with equality, the axioms of
O P - and the five schemata below:
T-out:
T A ~ A (A arbitrary sentence);
T-elem"
(A---, TA), where A has the form t = s, -~t = s, Nt, ~Nt;
T-imp:
T(A~B)~(TA~TB);
T-univ:
VxTA~TVxA;
T-log:
TA, provided A is a logical axiom.
(ii) VF c " - V F -
plus the class number-theoretic induction axiom CL-
N I N D , i.e. the formula
Cl(a) A Orla A Vx(xrla ~ (x+l)r/a) ~ Vx(Nx ~ xrla);
VFp "- V F - + P - N I N D "- Orla A Vx(xTla ---, ( x + l ) r / a ) ~ Vx(Nx --, x~la);
P - N I N D is the number-theoretic induction for properties.
XII.59]
The Core System and Transfinite Induction
353
59.1.1. REMARK. The T-schemata above are theorems of NMF-; what
really makes the difference, is the closure of T under logical deduction.
59.2. LEMMA
(i) The LOG-rule: if A is a formula of 2. and A is provable in pure
logic, then V F - F TA.
(ii)
The internal abstraction schema:
VF- F T(Vu(u~{x: A} ~ TA[x := u]));
(iii) VF- F T A A T B ~ T ( A A B);
(iv) VF- F -~(TA A FA);
(v) VF- F (TA V F A ) ~ ((A ---,T B ) ~ T(A ---,B));
(vi) VF- t- T A V T B ---,T ( A V B);
(vii) VF- F T V x A ~ VxTA;
(viii) VF- F 3 x T A ~ T3xA;
(ix) VF- F T A ~ F-~A;
(x)
if A is a formula of s
(A does sol contain occurrences of T),
VF- F ( T A V FA) A (TA ~ A).
PROOF. (i) By induction on the derivation of A in pure logic. If A is a
logical axiom, we are done by T-log. If A is obtained from B and B--, A,
we get T B and T(B--, A) by IH, whence TB--~ T A by T-imp, and finally
TA.
(ii) By identity logic and (i) above, we have (u fresh variable):
T((Ax.[A])u = [A[x := u]]---,.(ur]{x : A} ~ TA[x := u])).
We then obtain, by T-imp and T-elem,
(~x.[A])u = [A[x := u]] ~ T ( u ~ { x : A} ~ TA[x := u]),
whence T(uy{x: A } ~ T A [ x := u]) by /%conversion. The conclusion follows
by logic and T-univ.
(iii) AAB---,A, A A B ~ B
hold by logic, whence by LOG-rule
T(AAB)~TAATB;
in the other direction we apply the tautology
A ~ (B -~ (A A B)) and T-imp.
(iv): trivial consequence of T-out.
(v) Assume T A V F A , A - , T B . If T A holds, then A holds by T-out,
whence TB; but T ( B - ~ ( A - - , B)) by LOG-rule; hence TB--, T ( A ~ B) with
T-imp, i.e. T ( A - ~ B ) . If F A is assumed, we have T ( ~ A - - , ( A - ~ B ) ) by
LOG-rule; the conclusion again follows by T-imp. The reverse direction
354
Non-Reductive
Systems for Truth and Abstraction
[Ch.12
follows by observing that T A V FA implies A ~ T A (use T-imp, T-out).
(vi)-(ix): by LOG-rule and T-imp.
(x): by induction on A, applying T-elem, LOG-rule and the previous
facts. 0
59.2.1. REMARK. (i) T ( T A ~ A ) i s
inconsistent with T-out, T-imp and
T-log. If we choose A :--"TL, where L - [-"TL], we have the following
chain of implications:
T ( T L ~ L) ~ T(-"L ~ -"TL):=V T(-"L ~ L ) ~ TL ~ T-"TL ~ -"TL.
(ii) Assume the schema T ( 3 x A ) ~ 3xTA. Then we have in pure logic, with
-"K=S, 3x(x=g~A),
hence T ( 3 x ( x = g ~ A ) ) ,
i.e. for some c,
T(c=K~A),
which implies by 59.2
either ( c = K A T A ) o r
(-"c -- K A T-,A), i.e. T A V T-"A: absurd.
To sum up, V F - + { T 3 x A ~ 3 x T A } is inconsistent. But a special case of
the schema is consistent with V F - (see 63.9).
Fix an enumeration {Ai} i ~ ~ of s
which have exactly two
distinct free variables. Let x 9 y stand for any formula Ai(x , y) in the given
list.
59.3. DEFINITION
(i) Progr( 9 ,B) := Vx(Yy 9 x.B(y)---, B(x)) ( = the property defined
by B is 9 progressive);
(ii) W ( 9
:= Y z ( P r o g r ( 9 ~xrlz);
(Progr( 9
stands
Progr( 9 B) with B(x):= xrlz and we simply say that z is 9
for
(iii) WF( 9 ):= {u: W( 9
WF( 9 ) is clearly suggested by the set-theoretic definition of the largest
well-founded part of a relation; remarkably, VF-justifies the corresponding
transfinite induction schema.
59.4. T H E O R E M (Transfinite induction). VF-proves:
(i) Progr ( 9 WF( 9 ));
(ii) Progr( -~ , B)-~ Vx(x~WF( -~ )-~ B(x)),
where B is an arbitrary s
PROOF. (i) Progr( 9 ,W( 9 , - ) ) i s clearly derivable in pure logic;
by LOG-rule we can infer
T(Vx(Vy 9 x.W( 9 , y) ~ W( 9 , x))).
Since
9 is defined by a formula of s
hence
(1)
we have T(x 9 y)V F(x 9 y) by
XII.59]
The Core System and Transfinite Induction
355
59.2 (x). If we apply 59.2 (vii), 59.3 and 59.2 (v), (1)implies
Vx(Vy -~ x. T W ( -~ , y) ~ T W ( ~ , x)),
(2)
which yields Progr( -~, W F ( -~ )) by means of T-out and 59.2 (ii).
(ii) It is enough to check
Vx(x~lWF( -~ )---* B'(x)),
where B ' ( x ) " - P r o g r ( -~ , B ) ~ B(x). In pure logic, we have
Progr( -~ , B').
Then we can repeat the argument for (2), thus obtaining
Vx(Vy(y -~ x ~ T B ' ( y ) ) ~ TB'(x)),
whence by abstraction,
Progr( ~ , {u" B'(u)}).
(3)
If we assume x~IWF( -~ ) and we choose z "- {u" B'(u)}, we get
Progr( -z,, {u" B ' ( u ) } ) ~ X~l{U" B'(u)}.
(4)
From (3)-(4)and abstraction, it follows TB'(x), hence B'(x) with T-out. F1
59.4.1. REMARK. The previous argument only requires that { ( x , y ) ' x -~ y}
is a class, and not the stronger s
To appreciate the strength of VF-, the reader with a "logicistic"
inclination may be willing to verify the following theorem. Let s be 2.
without the predicate N and with the combinators K and S as the only
individual symbols; let VF 0 be the subsystem of VF-, formalized in the
fragment s Then we have:
59.5. THEOREM. Peano arithmetic is interpretable in VF 0.
PROOF (Hint). Simply define (x-~-l)"- {x}, 0 " - 0 and
--
Vy(Clo
N(y)-
where ClosN(Y ) "--O~ly A Yu(u71y ~ (u+l)~/y). Plus and times are introduced
s la Dedekind as the least relations satisfying the obvious recursive clauses.
By adapting 59.4, we can verify the appropriate induction schemata (for
details, see Cantini 1991). F1
It will follow from the main result of w63 that VF 0 is not stronger than
OP. In contrast to MF, VFp ( - V F - + p r o p e r t y N-induction axiom) goes
beyond the limits of predicative mathematics. This is most easily seen by
appealing to the theory IDl(acc ) of accessibility elementary inductive
definitions, which proves the 1-consistency of Predicative Analysis (see
356
Non-Reductive Systems for Truth and Abstraction
[Ch.12
Buchholz et al., 1981).
For the reader's sake, we outline IDl(acc ). If L ( P A ) i s the language of
Peano arithmetic, fix an effective enumeration {Ai: i E w} of all L(PA)formulas, containing two distinct free variables. The language of IDl(acc ) is
L(PA), expanded by a countable sequence of unary predicate symbols, say
{IN: i C w}. As above, let -~ stand for any Ai(x,y); we use WF( ~ )(x) as a
more suggestive notation for Ii(x ) (whenever -~ is any Ai). Formulas are
inductively generated as usual; atoms obviously have the form t - s,
WF( -~ )(t). The axioms of IDl(acc ) contain: (i) Peano axioms; (iN) numbertheoretic induction for the full language; (iii) for each -~ and arbitrary
B(x), the axioms:
WF( -~ ).1
Progr( ~ , WF( -~ ));
WF( -~ ).2
Progr( -~ , B)-~ Vx(WF( ~ )(x)-~ B(x)).
59.6. THEOREM. IDl(acc ) is interpretable in VF p"
PROOF. By the theorem 59.4, it only remains to check that VF p proves
the number-theoretic induction schema for arbitrary formulas of 2.. Set
ClosN(A ) "- A(O) A Vx(A(x)--, A ( x + l ) ) and A'(x) "- ClOsN(A ) ---,A(x),
where A is a given arbitrary formula of .5; then ClosN(A' ) is trivially
derivable in pure logic.
Thus V F - ~ TClosN(A'), which implies VF-~-ClosN({X: A'(x)}), whence
by property N-induction Vu(iu--,uTl{x: A'(x)}); T-out and exchange of
premises imply VFp F- ClosN(A)---~ Vu(Nu--~ A(u)). [-!
59.6.1. REMARK. Conversely, VFp has a model in a set theory, which is
proof-theoretically equivalent to ID1; the basic steps are similar to those of
the main theorem of w and the result is essentially contained in Cantini
(1990). Thus we concentrate on VFc, in accord with our choice of stressing
systems not stronger than PA.
To conclude, the reader may naturally wonder whether the strength increase
sensibly affects the structure of classes in VF-. For instance, does any of the
properties WF(-~ ) define a class? The answer is negative and it can be
readily obtained with a recursion-theoretic investigation of the inductive
models of w
By the way, the fundamental closure properties of
CL := {x: Cl(x)} in V F - a r e best summarized by the non-surprising
59.7. THEOREM. V F - p r o v e s that CL is closed under the join principle
and the elementary comprehension schema (see Ch. II, 9.7-9.9).
The proofs are straightforward and make use of the elementary facts of
59.2.
5upervaluation Models
XII.60]
357
w60. Supervaluation models of VFWe keep using the conventions and notations of w7 and w30; we fix a
standard operational structure 31~I=OP-; Lop(Mr,) and .5(title) are the usual
languages expanded with distinct constants for elements of M, M being the
universe of 31,. If tin a closed term of s
,~t~(t)is the value of t i n dtl~.
60.1. DEFINITION
(i) Once 3l~ is given, X C_ M and A is an arbitrary sentence of s
XI=A stands for "A holds true in the structure ( ~ , X ) " , whenever Lop
receives its usual interpretation in Ml~ and T is assigned the subset X, i.e.
(all,, X}I= Tt iff 31~(t) E X.
(ii) Xll-A
iff for every Y C_ M, if X C_ Y, then
YI-A (where
A is an
arbitrary s
(iii) If X C_ M, we let (I)o(X) : - {3t,([A])'XIJ-A, A L(Ml~)-sentence}.
(iv) X is (~o-dense iff X C_ (~0(X); X is (~o-closed iff (~0(X)C_ X; X is
consistent (complete) iff for no b E M, b E X and (-,b) E X (for every b E M,
b E X or (-~b) E X). CONS(drip)"- {X C_M" X consistent}.
(v) SENT(Jft~)"- {~([AI)" A s
Since ~ is fixed, we generally omit the explicit indication of 31~ and we
simply speak of "sentence" and "consistent" tout court.
60.1.1. REMARK. Variants of the relation X I [ - A are obtained by imposing
additional constraints on the possible extensions of X (cf.w67). For instance,
if we define the relation X I ] - A by quantifying over all consistent and
complete extensions of X, we obtain van Fraassen's notion of supervaluation
for s
For this reason and because of theorem 60.3 below, the fixed
points of the operator (I)0 are also called supervaluation models (in short
SV-models).
60.2. LEMMA
(i) X[[- A implies X[=A (A s
(ii) Xll- A--, B and XI[- A imply X][- B;
(iii) If X[[-A(a) for every a E M, then Xl[-VxA;
(iv) (I)0 is monotone: X C_Y =~(~o(X) C_(~o(Y) (X, Y C_M).
Hence FIXo(31~ ) - {X C_M" X -
(~0(X)} is non-empty.
(v) If X is (~o-dense, then X C_SENT(Jf[~) and X]= TA--, A; hence
FIXo(Jf~ ) C_CONS(J~).
358
Non-Reductive
Systems for Truth and Abstraction
[Ch.12
PROOF" (i)-(iv) are trivial by definition of II- and the Knaster-Tarski
theorem. As to (v), if X C_Go(X), T-out holds by (i) and trivially
X C_SENT(J~). If NI~([A])E X and ~ ( [ ~ A ] ) E X , then X [ [ - A and
X I I - ~ A , which yield a contradiction by (i). E!
60.3. THEOREM
(i) If X E F I X o ( ~ ) and ~ I - O P - ,
then XI=VF-. In addition we
have, for arbitrary a E M:
T-rep
XI= Ta ~ TTa;
(ii) If ~ is an w-model of OP and X E FIXo(Jtl~ ), then XI= VFp (cf.
59.1). In particular, if A is an arbitrary instance of N-induction in the
language L, XI= TA.
PROOF. (i). T-out, T-imp, T-univ: apply 60.2 (v), (ii), (iii).
T-elem: if A has the form Nt, t - s or the negation thereof, X[= A implies
MI~[=A, whence YI=A, for every Y _DX , i.e. XI[-A, i.e. X[= T A as X is
(~o-closed. The converse is similar. T-log: if A is a logical axiom, X[= A, for
arbitrary X C M, i.e. X[[-A. Hence, if X E FIXo(Jf6 ), NI~([A]) E X by (I)oclosure, i.e. X]= TA. T-rep: if XI= Ta, also a E Y and Y]= Ta, for every
Y 2 X, whence X I I - T a , i.e. X]= T T a ((I)o-closure).
(ii) If A is an L(.At~)-instance of N-induction, Y[= A, for every Y C_M. The
conclusion follows by Co-closure of X. Vl
60.4. COROLLARY. V F - + T - r e p + { T A " A is a logical axiom or an
axiom of OP-, or an arbitrary instance of N-induction in the language s
is consistent.
At this stage we shall not undertake the investigation of the latticetheoretic structure of the fixed point models of (I)0: suffice it to say that also
in the present situation the encoding techniques of Ch. VII can be profitably
applied. For instance, the reader can verify:
60.5. THEOREM. Card(FIXo(alg))- 2 card(M).
However, we warn against mechanical repetitions of the old arguments.
w61. An abstract sequent calculus and truth
We consider the problem of giving a more constructive semantics for VF-;
in particular, we show how to avoid universal quantification over arbitrary
subsets in the definition 60.1. To this aim, we shall define a generalized
sequent calculus %, in such a way that provability in % yields an
interpretation of the predicate T of VF-. This step is rewarding in two
XII.61]
An Abstract Sequent Calculus
359
respects. First of all, provability semantics validates new schemata and
hence we shall obtain a stronger consistency result. Furthermore, the
definition of % and the derivation of its main properties can be easily
carried out in the system P W c + GID with N-induction for classes, which is
conservative over OP (and hence PA; see Ch. III, 15.5).
Since the %-provability interpretation is the identity over s
it
will follow that VF-, VF c are conservative extensions of OP; in addition,
VFp turns out to be interpretable in P W + G I D , the system with full
induction schema, which can be shown proof-theoretically equivalent to the
theory IDl(acc ) of w
The crux of the construction lies in devising a sequent calculus %, which
enjoys cut elimination and hence is consistent, provably in P W c + GID.
Clearly % has to be infinitary (by axiom T-univ). However, the problem
with the usual cut elimination proofs is that they require induction on cut
formulas (of maximal complexity), i.e. forms of number-theoretic induction
that may not be available in weak systems like P W c + GID. In essence, the
solution w e present here is simply to replace the usual finilary sentences
with natural abstract counterparts, which are introduced by generalized
induction. It follows that N-induction can be avoided by means of ordinal
transfinite induction, which is available in unrestricted form using GID (see
w
m
Henceforth, we use the abbreviations: (Va)"-<6, a), (a---,b)'-<31, (a,b>>,
and ( ~ a ) ' - ( a - - - ~ [ K - S ] ) ; Vx.t stands for V(Ax.t) The other logical
operations are introduced by mimicking the definitions of the corresponding
logical operators"
(a A b ) " - (-~(a ---,-~b));
(a V b ) " - (-~a ---, b);
(3a) "- -~(V()~x(-~(ax)))).
Also the map A H [A] on atomic formulas is defined as in w7. We put"
A t e ( a ) :- 3x3y(a - [x - y] V [Nx] V [Tx]);
Eatoo(a) "- Atc~(a ) A ~ 3 x ( a - [Tx]).
We usually omit outer square brackets to distinguish an element of
{x: A t e ( x ) } from the corresponding formula, whenever the distinction is
clear from the context. Thus T a - a, (Ta---, a ) - a abbreviate [ T a ] - a and
([Ta] ---, a) -- a (respectively).
61.1. LEMMA. We can find a formula A S T ( x , v) operative in v, such that,
if Sentc~ := I x v . A S T ( x , v) ( = the fixed point of A S T ( x , v), cf. 10.1), then
we have, provably in P W - + GID:
1.1
Atc~(a ) ~ a~iSentcr
1.2.
(a --~b)~Sentc~--~a~Sent ~ A b~iSent~;
Non-Reductive Systems for Truth and Abstraction
360
[Ch.12
1.3.
(Va)ySent~ +-~Vx((ax)rlSentc~);
1.4.
V x ( A S T ( x , B ) ~ B(x))~Vx(xySentoo---~ B(x)), B arbitrary.
PROOF: clearly A S T ( x , v) formalizes the clauses corresponding to 1.1-1.3,
which are positive in Sentcr Then we apply the fixed point theorem for
predicates ahd GID to get 1.4. I"1
The elements of S e n t ~ are naturally viewed as abstract sentences and
they suffice for the main result below (cut elimination).
61.2. LEMMA. If A ( Z l , . . . , x n )
is a formula of s with the free variable
shown, then
P W - + GID F- Vxl . . . Vxn([A(Xl, . . . , xn)]71Sent~).
PROOF: by metamathematical induction on A, using 61.1. [3
An adequate notion of ordinal in the sense of von Neumann
in P W - + G I D
(see w167
in particular, we can define
Ord(x), two binary relations < , - ,
such that if the lower
letters a, fl, p, ... range over O N = {x: Ord(x)}, then P W - +
(see 23.4 for notations):
is available
a predicate
case Greek
GID proves
61.3. LEMMA
(i) x r l O N ~ O r d ( x ) ; V x ( O r d ( x ) ~ x r l A n ) (every ordinal is a set);
(ii) < is linear, i.e. ( a < f l ) V ( a - - f l ) V ( f l < a ) ;
(iii) T I ( O N , B ) : = Vc~(Vfl(fl < c~---, B ( f l ) ) ~ B(c~))~ Vc~B(c~).
Since the operator f a ' - { x " A S T ( x , a ) } is C-monotone, we exploit the
ordinal analysis of inductively defined predicates and rephrase 23.6-23.10 in
the special case of S e n t i .
61.4. LEMMA.
We can find a term Ax.Sentoc(x ) such that P W - + G I D
prove8:
(i)
c~ < fl ~ Sent~(c~) C Sentc~(fl ) (where a < fl "- (c~ - fl
(ii)
Yx(x~TSentc~(c~ ) ~-~AST(x, {y" 3/?(/5 < c~ A yrlSentc~(fl))}));
(iii)
Vx(xrlSentoo ~+ 3c~(xrISentoo(tx) ) );
(iv)
Vx(Atc~(x ) ~ xrlSent~(O)) (where 0 - O);
(v)
(vi)
Vz((Vz)~S~%(~) ~ w3#(# < ~ A (~)0s~t~(#))).
V c~ <
fl));
61.4.1. REMARK. If arlSentoo(a ) is read as "a has complexity a", then the
An Abstract Sequent Calculus
XII.61]
361
associated notion of complexity has the expected properties: the atomic
elements of Sentcr have complexity 0 (by 61.4(iv)); the immediate
constituents of a non-atomic element b of Sentcr have strictly lower
complexity than b itself (61.4 (v)-(vi)).
We proceed to describe the announced generalized sequent calculus %.
In order to enhance readability, we content ourselves with an informal
presentation; however, we underline with appropriate remarks the steps that
require non-trivial considerations for the formalization in P W - + GID.
Notation. F , A := F UA; F,a := F,{a}; F and A will stand for classes of
abstract sentences.
61.5. DEFINITION. ~- pc~ F ::~ A is the least relation such that
(i) c~, p are ordinals and F, A are classes C_ Sentcr
(ii) it is closed under the axioms and rules below.
61.5.1. Axioms
AX.I:
if b~/F f3 A and Ate(b), i.e. b has the form Ix - y], [Tx], [Nx],
then f- ~P P =:~ A;
AX.2.1-if A - (a - b), (Na), A is true and [A]r/A, then F- p F =:v A;
AX.2.2" if A - (a -- b) , (Na) , A is false and [ / ] u r , then F- p~ F = v A ;
AX.3:
if {[Ta] , [T-~a]} C F, then ~ ap
F::~A.
61.5.2. Rules
(R--~):
if ( F- ~ F, a =~ A, b), (a---~b)rlA and fl < a, then F-,o~ F ::~ A;
(L--~):
i f ( ~ ~ F =~ A,a), F- ~ F,b ::~ A and (a---~b)~F with fl < a,
7 < a , then ~ pa F = ~ A ;
(RV):
assume that for every x and some/3 < a F- ~ F =~ A, ax and
let (Va)~A; then ~- pa F ::~ A;
(LV)-
if e
F, ax ::~A for some x, fl < c~,
(Va)uF, then
}- ap F =:VA;
o~
(RT):
if ( F- ~ ==~b) and 13 < a, 5 < p, [Tb]r/A, then ~ p F :=VA;
(LT)"
if ( ~ 2 b ::v) and fl < a, 5 < p, [Tb]uF, then ~ p F ::V A.
(~
61.6. LEMMA. We can find a formula D(z, w), such that
(i)
(ii)
D is operative in w;
D formalizes the inductive clauses corresponding to the axioms and
the rules of 61.5;
362
Non-Reductive Systems for Truth and Abstraction
[Ch.12
(iii) P W - ~ TD(z, w)+-+ D(z, w).
PROOF. By straightforward formalization of 61.5 and application of
lemmas 61.3-61.4; note that, in order to get (iii), it is essential to have
condition 61.5 (i). F1
GID and the fixed point theorem for predicates imply:
61.7. LEMMA. We can find a term Dercr such that P W - + GID proves
(i) V z ( z ~ l n e r ~
D(z, nercr
(ii) zTIDercr --, 3x3y3u3v(z - (x, y, u, v) A
^ ~Oi
^ yuON ^ Cl(u) ^ Cl(v) ^ u c_ S ~ t ~ ^ ~ c_ S~nt~).
(iii) If B is arbitrary,
V z ( n ( z , B ) - - , B ( z ) ) - ~ Vz(z~lnercc-~B(z)).
(iv)
The formula (~, p, u, v)~IDercr is extensional in ~, p, i.e.
a -- a' A p -- p'--+ ((a, p, u, v)~lnerc~ ~ (a', p', u, v)~lnerc~ ).
61.8. CONVENTION
(i) We henceforth write ~ p r ~ A, in place of (a, p , r , A ) ~ D e r ~ .
Accordingly, t- pa F, a ==~A, b is an abbreviation for the formula
(a, p, F U {a}, A U {b})~lDeroo.
(,)
By lemma 61.7(ii), ( . ) i m p l i e s CI(F), C/(A), F C Sentoo , A C S e n t i ,
arlSentoo , brlSentoo , arlON , prlON. So we keep using capital Greek letters
F, A, etc., as variables for classes C Sentoo and
v r ( . . . ) - - W(V/(x) A W(u~x -~u~Sentoo)...);
Vc~(... ) ' - V x ( x , O N ~ . . .
).
(ii) We finally let:
~- pF:=V A " - 3 a ( ~ p
~- F ~ A
"- 3p( F p r ~ A).
We repeatedly use the notational conventions of 61.8 in stating the
propositions below.
61.9. LEMMA ( P W - + GID).
(i) Monotonicity:
( t- a F :=VA) A c~ </~ A p < a.---+ ~ ~ F ==~A.
An Abstract Sequent Calculus
XlI.61]
(ii)
Weakening:
( F- pa F ~ A )
(iii)
Consistency:
( ~- F =V A ) -,-~(F U A - ~q)).
363
AF C
_ F' A A C
_ A'.---, ~ ap F ' ~ A ' .
PROOF: by transfinite induction on c~ (61.3 (iii)) using the definitions and
the elementary properties of ordinals. [-l
61.10. L E M M A (Inversion; P W - +
(i)
( F - ap F , a - - . b = v A ) ~
GID)
~- p~ F , b ~ A ;
( ~- ~p F , a - - , b = v A ) - - ~- pa F ~ A, a;
(ii)
(F- ~p r ~ A, a -o b) - - ~- pa F, a ::~ A,b;
(iii)
( F- ~p F ~ A, Va) ~ Vx( F- ~p F =V A, ax);
(iv)
o~
(I- pa :2zTa)--,3cr(a < pA t- a = ~ a ) ;
arlSentoo A ( F- p T a ~ ) . ~ 3 o ' ( ~
(v)
< p A ~- o" a =:~);
Eatc~(a ) A T a A ( }- pa F,a=:~A).---, ~ pc' F::~ A;
Eatc~(a ) A Fa A ( F- p F=:vA , a).---~ [- ap F==~A;
(vi)
( F- ~p F,T-~a=~ Ta, A ) ~
~- p F,T-~a=~ A
P R O O F . Standard induction on the length a. We only check (vi).
Case 1: F, T ~ a =~ Ta, A is an axiom. Then we m a y assume that either T a is
active or T ~ a (else the conclusion is trivial). If T a is in F, then F, T-~a ~ A
is already an instance of the axiom AX.3; if T-~a is in A, F, T-~a =~ A is an
instance of AX.1.
Case 2: F,T-~a=~Ta, A is not an axiom. If T-~a, T a are not active, we
simply apply IH to the premises of F,T-~a=~ Ta, A. If T-~a is active, we
simply erase T a from the conclusion of (LT). If T a is active, F ~ =~ a, with
~r < p, fl < c~; hence by ( L ~ ) and AX.2.2 we have F- ~r-~a =~. An application
of (LT) yields ~ p F, T-~a :ez A. [:]
R E M A R K . 61.10.1. If arlSentoo, the equation a = - ~ a has no solution (see
62.4 below); this is essential for the soundness of the second part of (iv).
61.11. D E F I N I T I O N . Sent-do C_ Sentoo is the least property, which contains
{x "Eatoo(x)} and is closed under ---, and V. Clearly, the elements of Sent-do
are "abstract" counterparts of T-flee sentences.
61.12. L E M M A ( P W - +
GID)
(i)
Tautology:
F- o F, a ==~A, a.
(ii)
Lop-Completeness:
arlSent-oo --. ( F- o :ez a) V ( ~- o a ==~).
364
IVon-Reductive Systems for Truth and Abstraction
[Ch.12
PROOF. (i)-(ii): standard induction on a with the assumption aqSentoo(a),
AX.1, AX.2 and logical rules. F1
w62. Cut elimination and r d a t e d properties
:~ is closed under cut rule:
62.1. T H E O R E M (Cut elimination). P W - + GID proves:
(,)
,,S ntoo( ) ^ (
r A,a) ^ (
PROOF. Let C(p, ~, ~, Z, a,r, A) stand for (,); let
C'(p, ~, a, fl) . - VaVFVA C(p, ~, a,/?, a, F, A). We apply transfinite ordinal
induction to check
VpWVoNI3C'(p,5, ~, ,8).
(**)
As usual, we can rephrase the proof of (**) as a multiple induction on the
well-ordering defined by: (p', 5', a', t3') < (p, 5, a,/?) iff (p' < p) or (p' - p and
6' < 6) or ( p ' - p, ~ - 5', c~' < a) or ( p ' - p, ~ ' - ~, a ' - c~ and fl' < fl).
The induction hypothesis is:
'~' b, F' ~ A ' ,
if F o~', F' = ~ A ' , b a n d Fp,
then Fp, F' ::~A',
(IH)
P
whenever F', A' C_ Sentoo , brlSentoo(6'), (p', 6', a', fl') < (p, 6, a, fl) and F', A'
are classes. Then we assume that
arlSentoo(6 ) A ( I- po~ F ==~A, a) A ( ICase 1: Eatoo(a ) holds, i.e. a by 61.10 (v).
I v - d] or a -
r, a :=~A).
(Ass)
[Nc]. Then F p F ==~A follows
Case 2: a - Tc. We distinguish the following subcases.
2.1.1" Tc is not active and either F :=~ A, Tc is an axiom, or F, Tc =~ A is an
axiom. Then F ==~A is trivially an axiom.
2.1.2: Tc is active and F ~ A, Tc is an axiom. Then F - eF', Tc, for some F'
and hence F p F ==VA follows from the other assumption.
2.1.3: Tc is active and F, Tc =V A is an axiom. Then either A - - e A', Tc or
F - e F', T - c . In the first case, we get F p F ~ A from the first assumption,
while 61.10 (vi) applies in the second case.
2.1.4: a - Tc and Tc is active in either premise. Then we have applied
(RT) and (LT). Hence, for some 7, crlSentoo(7) and
c~'
~'
( F p, ::~ c) and ( F p, c:=v), for some
p, <
p.
As (p', 7, a', ) < (p, 0, a,/?), the empty sequent =~ would be derivable in %
by IH, against 61.9 (iii). Thus, either Tc is not active in F orp F =v A, Tc, or
XII.62]
Cut Elimination
365
Tc is not active in F- p~ F, Tc =~ A. Since the argument is symmetric, let us
consider only the first case. If F =:~ A, Tc follows with (RT) or (LT) applied
to some d ~ Tc, we simply erase Tc in the conclusion and we get F ==~A.
Else, the premises of F :=~ A, Tc follow with a rule % ~: (RT),(LT). For
instance, let ~ = (RV) and assume that A = ~A'U {Vd} and
Vx37(7 < c~ A I- ~ r ~ A, Tc, dx).
By weakening (61.9 (ii)), we also get k ,,q F, Tc::~ A, dx, for each x. By IH,
if 7 < a and x is arbitrary, we get ~- ~pF ~ A, dx, where ~ - max(a, ~)+1
(as (p, 0, 7,/~) < (P, 0, c~,/3) holds). Hence F- p F =V A by (RV). The extant
cases are similar.
Case 3: -~Atcv(a ). We again consider a few subcases.
3.1. either F ~ A,a is an axiom or F , a ~ A is an axiom. Then F ::V A is
already an axiom, as a cannot be active by assumption on a and 61.5.1.
3.2. F =v A, a and F, a ::V A are not axioms and a is active in either premise.
3.2.1. Let a -
Vb. Then we must have t-p
F, Vb, bc=vA, for some c and
/~'< fl; by weakening (61.9(iii)) , also ~ ~
p F , bc ~ A, Vb. Then we obtain
F- ~ F, bc ~ A, for some ~ (we use IH because (p, 6, a,/3') < (p, 5, a, ~)), and
by inversion t- p F =V A, bc. But (Vb)E Senti(6) yields (bc)TiSent~(6'), for
some 6' < 6 by lemma 61.4 (vi); hence (p, 6', a, ~) < (p, 6, a, ~) and by IH
F- P F ~ A .
!
3.2.2. Let a - (b---~c): then we must have ~ p F, b ::v (b--~ c), c,A for some
c~'<c~; hence, by weakening, F - p ~ F , b ~ c , b = v c , A and by IH, since
(p, 6, c / , ~ ) < ( p , 6, c~,/~), k~F,b=vA, c, for some ~. Now 6 1 . 1 0 ( i ) a n d
weakening yield F- p~F, c, b ==~A and F- ~ F :=~ b, A. But b, crlSentcr
for
some 6 ' < 6 by Lemma 61.4(v): then ~p, ~', ~, ~) < (p ~, a, ~), whence IH
implies k p~F, b ::> A, for some 7. But (p, 6', fl, 7) < (P, 6, a, fl) and finally
pF~A.
3.3. F ::> A , a and F,a::> A are not axioms, a is not active in one premise.
We apply IH to the premises of the final inference % and then we use %. D
Not surprisingly, theorem 62.1 leads to a form of Herbrand's theorem
for 9(;, which is required by subsequent consistency results. However, we
have to establish that certain equations have no solutions in Sent co; but
this project encounters difficulties, derived from the clause that introduces
(V f ) in Sentcr In contrast to the case where f = Ax[A] and A is a usual
sentence, we are not entitled to infer that f x and f y have the same logical
form (e.g. if f x is (bx~cx), then also f y = by~cy). This suggests a
suitable notion HSentcv(f ) of hereditary senlenlial funclion.
Non-Reductive Systems for Truth and Abstraction
366
[Ch.12
HSentoo is inductively generated by the following clauses:
(i)
(ii)
(iii)
~x[T(ax)], ~x[ax = bx], )~x[N(ax)] are in HSent~;
if a, b are in HSentcr , then )~x(ax ~ bx) is in HSentcr
if ($y. azy)is in HSent~,
for every x, then ~z.V($y. a y x ) i s in
HSent~.
Obviously, HSentcr is formally representable; if we further assume
extensionality for operations, we notice that we can associate elements to
every formula of HSentcr
We can find a formula ASF(x,u), operative in u, such
that, if HSentcr := Ixu.ASF(x, u), then we have provably, in P W - + GID:
62.2. LEMMA.
(i)
f~lHSent~
3h3gVx((fx = [T(hx)]) V ( f x = [g(hx)]) V
V ( f x = [hA = gx]) V ( f x = (hA ---,gx) A hzlHSent ~ A gTIHSentcr ) V
V ( f x = V(Ay. hyx)A Vz(Ax. (hzz)rlHSentoo))).
(ii)
V x ( A S F ( x , B ) ~ B ( x ) ) ~ V x ( x T I H S e n t ~ B(x)), (B(x) arbitrary).
Moreover=.
GID F frIHSentc~-~ Vx((fx)ySentoo ).
(iii)
PW-+
(iv)
If A is a formula of 2. with free variables in the list x, Wl,... , w n
then
P W - + GID + Extop F VWl ... Vwn(~X [A(x,
( E x t o p - extensionality for operations, see w
point given by 10.1).
Wl,...
,
wn)])~iHSent~).
I x u . A S F ( x , u ) is the fixed
PROOF. (i)-(ii): by 61.1 and GID.
(iii): straightforward induction on HSent~, using the closure properties of
Sentcr
(iv): m e t a m a t h e m a t i c a l induction on A with repeated applications of {conversion (2.2) for operations. For simplicity, let w also denote a finite list
wl, ... , w n of variables. If A is atomic and A(x, w) = (t(x, w) = s(x, w)),
choose h := ~x. t(x, w), g := ~x. s(x, w), f := )~x. [t(x, w) = s(x, w)]. Then we
get f u - [hu - gu] by (/~)-conversion, hence f - ~x. [ h A - gx]TiHSent ~.
The other atomic cases are similar. Assume A(x, w) = (B(x, w) ~ C(x, w)).
Then by In, h = $x.[B(x,w)] and g = ~x.[C(z,w)] are in HSent~; if we
choose f = ($x.[B(x,w)~C(z,w)]), then fT1HSentcr (as f u = ( h u ~ g u ) ) .
Assume A = VyB(y,z, w); by IH, $z[B(y,x, w)]~lHSentcr , for every y. Then
by (i), $x. [VyB(y,x, w)]rlHSentcr 0
Cut Elimination
XII.62]
367
62.3. LEMMA ("No solution lemma")
(i) P W - + GID F- frlHSentoo ~ V z V x V y ( ~ ( f x - ( f y ~ z))).
(ii) P W - + GID proves
^
y)
-
^
-
(iii) P W - + GID F- V x ( x r l S e n t o ~ -~((Tx ~ x) - x)).
(iv) If A is an s
P W - + GID F- --([A(u, Vl,... , vn) ] - [VxA(x, Wl,... , wn)]).
PROOF. (i) If fyHSentoo , (fx)rlSentoo , for every x by 62.3 (iii). Then by
61.3-61.4, we show by transfinite induction on a:
VvVz(f # (fv--, z))).
(,)
If Ato~(fx ) holds or f x - (Vg), (,) is clear, by the independence of ~ , N,
= , T, V. If f x - ( b - - - ~ c ) , there exist h, g in HSentoo such that
f x - (hx ~ gx). Were f x - ( f y ~ z), we ought to have, by pairing and by
definition of the operation )~x)~y.(x ~ y)"
hx-(hy--,gy),
(**)
where (hx)rlSentoo(fl), for some/3 < a; but (**) contradicts IH.
(ii)" similar to (i), arguing by induction on Sent oo with the independence
properties of---~, M and the pairing axioms.
(iii)-immediate by (ii).
(iv) We verify, by induction on A, that the assumption
[A(u, Vl,... , vn) ] - [VxA(x,
Wl,...
,
Wn)]
leads to a contradiction; in the case A(u, Vl, ... , vn) - VyB(y, u, Wl,... , wn)
we apply IH. VI.
62.4. THEOREM. P W - + GID proves:
(i) Eatoo(a ) ~ ( F- Ta ==~a) A ( F- a =>Ta);
(ii) arlSentcr A brlSentoo A (a r -~b) A (b 9s -,a)
--~(( ~ Ta, Tb==~)---~( F- a=> ) V ( ~ b=> ));
(iii) arlSentoo A brlSentcr A a r b ~ (( ~- Ta =>Tb) ---,. ( F a :=~) V ( F- :=~ b));
(iv)
Herbrand: f~lUSentcr ~ (( ~ V x T ( f x) =>) ~ 3x( ~ f x :=~)).
(v) If A is an arbitrary formula of.L, P W - + GID + Extop proves
(( F- V x T A ( x ) =V T V x A ( x ) ) - - , .( F- ~ VxA(x)) V 3x( F A(x)=V).
Non-Reductive Systems for Truth and Abstraction
368
[Ch.12
P R O O F . (i) If a encodes a true atom of the form t = s, Nt, F- Ta ==~a is an
axiom (by AX.2.1) and F a=:~Ta is derivable from F- =:~a by 61.5.2 (RT).
A dual argument works if the a t o m in question is false.
(ii) By hypothesis, the sequent cannot be an axiom; hence, either Ta is
active and F a =:~, or Tb is active and hence F b =:~.
(iii): similar to (ii).
(iv) First of all, let us say that F is a class of T f-instances (in symbols
Tf-Instan(F)) iff F satisfies the following condition:
c l ( r ) A Vu(
,r
=
(1)
Then observe that, if f is in HSentoo , and F is a class of T f-instances,
for no x, both T ( f x ) and T ~ ( f x ) a r e in F.
(2)
Indeed, if there is some y such that T ( f y ) and T-~(fy) are in F, then for
some z, T ( f z ) = T-~(fy)= T ( f y - - , _L ), whence f z = fy--. _L, against
62.3 (i). We now claim"
f~lHSentoo A Tf-Instan(F) A ( F r U {V(,~xT(fx))}=V)--, 3x( F f x ~ ) .
(3)
Clearly (iv) is an immediate consequence of (3); (3) is verified by induction
on a such that ( F a F U {V(Ax.T(fx))} =~). If the given sequent is inferred
by (LV), we have (t- ~ r U{V()~x.T(fx))}U{T(fc)}=~), for some c and
fl < a: the conclusion follows by IH. If F U {V()~x.T(fx))}::~ is inferred by
(LT), then for some c, t- ::~ fc, and we are done.
(v) If F is a class of T f-instances and f =)~x[A], then the sequent
{VxTA(x)} U F ::~ TVxA(x)) cannot be an instance of an axiom (use 62.2
(iv), (2) above and 62.3 (iv)). Hence we obtain, by induction on a:
Tf-Instan(F) A ( S F U {Vx. TA(x)} :=VTVxA)---,
--, 3x( t- A(x) =v ) V ( F =v VyA).
The cut elimination theorem implies that those predicates, which are
decidable in the sense of %, are equiextensional with classes; more precisely"
62.5. LEMMA. We can find a term As. t(u) such that:
P W - + GID F Vx(( F a x :2z) V ( F =~ ax))--~ (Cl(t(a)) A t(a) = e{x: F =~ ax}).
P R O O F . a ( 1 ) = {x: F=~ax} and a ( 0 ) = {x: F ax=~} are disjoint by cut
elimination; by dual representation (theorem 16.11), we can choose
t(a) = ER(a(1),a(O))= e l ( l ) s u c h t h a t - t ( a ) = el(0). By assumption on a,
t(a) is a class. ['1
62.6. T H E O R E M (Internal N-induction)
(i)
% derives N-induction for arbitrary conditions: if A(x) is an
XII.63]
arbitrary s
A Provability Interpretation
369
(with the free variable shown only),
PWp + GID F ( F A(0), V x ( A ( x ) ~ A ( x + I ) ) ~ Vx(Ux ---+A(x))).
(ii) % derives the N-induction principle for %-decidable predicates,
provably in PW c + GID:
F- Vx(( ~ :::~ax) V ( F ax:::~))----~( F- a-O, Vx(ax---.a(x-4-a)):::~Vx(Nx---.ax)).
PROOF. (i)Set C l o s ( A ) " - {A(0), V x ( A ( x ) ~ A(x+l))} and
Ind(A) " - { x " F : ~ C l o s ( A ) - ~ A ( x ) } . By AX.2 and P - N I N D (see 10.7)it
suffices to check that N C_Ind(A). But the tautology lemma yields
F C l o s ( A ) ~ A(0); F C l o s ( A ) ~ A(a) implies b Clos(A) =:~A(a+l) by
tautology, inversion and cut.
(ii) For every x, assume:
b ~ ax or F ax ::~.
(1)
Then we claim:
either F =:~a0 and F ::~ Vx(ax---,a(x+l)), or
F aO, Vx(ax----,a(xA-1)):::~.
(2.1)
(2.2)
If (2.1) does not hold, either not F :=~a0 or not F-::~Vx(ax---.a(x+l)). In
the first case, (1) implies F a0 :=~, and (2.2) follows by weakening. As to the
second case, we have by inversion 61.10 (ii)-(iii)"
for some b, not b ab ::~ a(b+l).
By (1), b ::~ab and b a(b+l)=~, for some b, hence (2.2) by (L---~), (LV) and
weakening. We can prove for any x, if Clos(a)"- aO A Vx(ax ~ a(x+l)),
( F :=~Clos(a) ---. ax) or ( F Clos(a) ---. ax :::~).
(3)
(3) is trivial i f ( F :=~ax) or ( F Clos(a)::~). Else, we must have ( F ax:::~)
and ( F =:~Clos(a)) by (1)-(2); hence ( F Clos(a)---.ax ::~). Now consider
Ind(a) "- {x "( F ~ Clos(a) ---. ax)}, Ind(a)- "- {x "( F Clos(a)---. ax :=~)}:
by lemma 62.5 and (3), there exists a class c such that c - eInd(a) and
- c - eInd(a)-. So we can apply class induction to check g C c - eInd(a)
and we finally get with AX.2, ( F ~ Clos(A)---, v x ( g x ~ ax)). D
w63. A provability interpretation and the upper bound theorem
Provability in the sequent calculus % provides an interpretation, which
validates VF--axioms plus certain additional T-schemata, directly inspired
by closure properties of %. The pay-off of the main interpretation theorem
Non-Reductive Systems for Truth and Abstraction
370
[Ch.12
is that it gives precise information on the proof-theoretic strength of the
object theories.
First of all, we define the announced provability interpretation.
63.1. DEFINITION (Induction on the definition of formula).
A~=A,
ifA=(Nt),
(t=s);
(A ~ B)o o = Aoo ~ Boo;
(Tt)o o=(F-::>t);
(VxA)o o = Vx(Aoo ).
If k is a natural number, it is understood that in the context Sentoo(k), k
stands for the (closed term representing the) corresponding finite ordinal.
63.2. LEMMA (Local truth). For each finite k, we can define a predicate
T R k ( x ) such that P W - + GID proves:
(i) T R o ( [ T a ] ) ~ ( F- :=Va);
TRo([X = y ] ) ~ x = y;
TRo([Nx])+-~ g x ;
(ii) TRk+l(a---+b)+--~a~Sentoo(k ) A brlSentoo(k ) A (TRk(a)---, TRk(b));
TRk+I(Va ) ~-~Vx((ax)rlSentoo(k ) A TRk(ax));
(iii) Atoo(a ) ---+(TRk+l(a) ~ TRk(a));
if k < n, arlSent~(k ) ---, (TRk(a) ~-+TRn(a));
(iv) Yx(xrlSentc~(k ) ~ - ~ ( T R k ( x ) A TRk(-,x));
(v) TRk([A(Xl, . . . , Xn] ) +--,A~(Xl, . . . , Xn) ,
for every formula of logical complexity <_ k.
The proof is well-known; (iv) requires outer induction on k; as to (v), we
proceed by induction on A, using 61.10 (v), whenever A "- Tt.
63.3. NOTATION. Let F, A be classes of abstract sentences; we set:
(i) T R k ( A F ) : = Vx(xrIF ~ TRk(x));
(ii) T R k ( V A ) : = 3x(xrlA A TRk(x)).
63.4. THEOREM (Formalized %-soundness). For each finite k, P W - + GID
proves:
(i) v r v A ( r u A c S nto (k) ^ (
r
(TRk( A 17')---,T R k ( V A))).
(ii) If A is an arbitrary formula of complexity < k,
P W - + GID I- ( I- ::~ A ( x l , . . . ,Xn) ) --> Aoo(Xl,...,xn).
A ProvabilityInterpretation
XII.63]
371
PROOF. (ii)is immediate from (i) and 63.2 (v).
(i) is checked by induction on the definition of t-(see 61.7 (iii)). We rely
without explicit mention upon the truth conditions of 63.2. and the fact
that TR k is monotone in k (i.e. TRk(a ) implies TRm(a), whenever k < m).
1. If F =:~ A is an instance of AX.1, the conclusion is trivial. If F =:~ A is an
instance of AX.2, either TRk( A F) is trivially false or TRk( V A) is trivially
true. If F =:~ A is an instance of AX.3, F = eF'U {Ta, T--,a}, for some a. It
suffices to show that TRk(AF) leads to contradiction. If TRk(A F) is
assumed, then (F-==~a) and ( F - a s ) ( a p p l y
61.10 and 63.2(i)). Hence by
cut elimination, we get ( F- ~ ) , against the consistency lemma 61.9 (iii) !
2.1. F=:~A is inferred by (RT): then for some a, such that [Ta]~/A,
(F- ::~ a), which implies TRo([Ta]) (63.2(i)) and hence TRk( V A). If F ==~A
is inferred by (LT), then (~-a=:~), which implies, by 62.1, not F- =:~ a , i.e.
~TRk([Ta]) , whence -,TRk( A F)since [Ta]r/F.
2.2. F =:~ A is inferred from F', a---, b, b =:~ A and F', a ~ b =:~ A, a where
F eF'U {a---, b}. We suppose
-
TRk(
A F), i.e.
TRk( A F')
and
TRk(a ~
(,)
b);
in addition we can assume by IH:
(**)
***)
TRk( A ( r U { b } ) ) ~ TRk( V A);
TRk( A F ) ~ TRk( V (A U {a}).
either TRk( V A) and we are done,
By (.) and (***),
or TRk(a), which
implies TRk(b), by the second part of (.); hence, again by (.),
TRk( A (F U {b})), and finally TRk( V A) with (**). The cases where F =:~ A
is the conclusion of an application of (R---+),(RV),(LV) are easily handled
and left to the reader. El
63.5. LEMMA
(%-Reflection) P W - +
GID ~- Vx( ~ =:~ f x ) ~
( F-=~Vf).
PROOF. The argument is not new (see 23.8); we apply CL-compactness
16.3. First consider R ( f , x ) - {c~" F- ~ ::~ f z } and assume (~- =:~ fx). Then
for every x, ~R(f,x) is a non-empty class of ordinals C_ R ( f , x ) ; hence
d - { y " 3x(yrl~R(f,x))} is a non-empty set of ordinals, such that if
7 - U d, then
Vx3p ( [
(fx)).
Finally, we find a bound a such that Vz(
(Rv). u
and we conclude by
We are now ready to state the promised strengthened interpretation result.
Non-Reductive Systems for Truth and Abstraction
372
[Ch. 12
63.6. DEFINITION
(i) VFH- is the elementary theory, based on classical predicate
calculus, which includes the schema T-out of 59.1, and, in addition:
1. the axioms corresponding to T-imp and T-univ of 59.1"
T-imPax: T(x ---, y)---, (Tx ~ Ty);
T-uniVax: VxT( f x) ---, TV f ;
2. the new principles:
T+-elem: T ( A ~ TA), where A - (x - y),-,(x - y), Nx,-,Nx;
T+-log:
TA, if A is any axiom of OP-;
T-rep:
Tx---, TTx;
T+-cons: T(T-,x ~ ~Tx);
T(-,T A ): ([A] r [~B]) A ([B] r [-,A])~. T ( ~ ( T A A TB))--, ( T ~ A V T-,B).
T(T-,):
([1] r [B])--+. T ( T A ~ TB)--, (T-~A V TB).
T-Herb:
T(VxTA(x)--, T V x A ( x ) ) - , . 3 y T ( - , A ( y ) ) V T(VxA(x)).
(T-Herb " - t h e Herbrand schema; the acronym is justified by 63.9(viii)
below).
(ii) VFH c "- VFH- plus the axiom of internal class-N-induction
I-CL-NIND
C l ( a ) - , T[aO A Vx(ax--~ a(x+l))--, V x ( N x - , ax)];
(iii) VFHp "- VFH- plus the axiom of internal induction I-NIND
T[aO A Vx(ax---, a(x+l))---, Vx(Nx---,ax)].
Of course VF- C VFH-.
63.7. MAIN THEOREM (Provability Interpretation + Upper Bound).
Let ~Y - VFH-(VFHc, VFHp), and ~f - P W - + GID + Extop
(respectively PW c + GID + Extop , PWp + GID + Extop ).
(i)
If ~ ~- A, then Y F- Aoo.
(ii) If A is a formula of Lop, ~ - VFH- (VFHc) , Y - OP- + Extop
(OP + Extop), then ~Y F- A implies Y F- A.
PROOF. (i) First assume ~ - VFH-. It suffices to derive the translation
Aoo of each VFH--axiom in the metatheory P W - + GID.
XII.63]
A Provability Interpretation
373
(T-out)~: apply the %-soundness theorem and 63.4 (ii).
(T+-elem)oo: by theorem 62.4 (i), 61.2.
(T-imPax)c~: by 61.10 (ii) and the cut elimination theorem 62.1.
(T-uniVax)~: by the %-reflection lemma 63.5.
(T+-log)~: apply 61.12.
(T-rep)~: by closure of k- under (RT).
(T+-cons)c~: by axiom AX.3 for F .
(T(~T A))~, (T(T~))oo, (T-Herb)~: by 62.4(ii),(iii),(v) (in the given
order).
If ~ - VFHc, VFHp, we apply the N-induction theorem of 62.6.
(ii): notice that A o o - A, if A E s
15.5. [:]
and use the conservation theorem of
63.7.1. REMARK. (i) A provability interpretation was already exploited by
Cantini(1990) in the context of a proof-theoretic investigation of truth
theories over Peano arithmetic, proposed by Friedman and Sheard (1987).
The idea was suggested by W. Buchholz, in order to simplify a formalized
model-theoietic construction of the present author.
(ii) The statement of 63.7 (ii) leaves the case of VFHp open; but it can
easily be settled by building a model of P W p + G I D + E x t 0 p in the
admissible set theory of J~iger (1982), which is proof-theoretically equivalent
to ID 1(acc).
We conclude with a few consequences of the extended systems.
63.8. DEFINITION. W V F - is the (weak) subsystem of VF-, whose only Taxioms are T ~ a - - ~ T a
and TA~-,A, where A is an e-atom;
WVFc "- W V F - + I-C1-NIND, WVFp
W V F - -~-I-NIND (see 63.6 above
for notations).
" - -
63.9. PROPOSITION. The following schemata are provable in VFH-:
(i) T ( T A ~ - ~ T B ) - - . T ( A ~ - ~ B ) ;
(ii) T ( T A ~ T B ) ~ T(A ~ B);
(iii) T-~TA ~ T-~A;
(iv) T A ~ T T A ~ FFA;
(v) FA ~ F T A ~ TFA;
374
Non-Reductive
Systems for Truth and Abstraction
[Ch.12
(vi) the Meta-Lhb principle:
T(T(TA~A)~TA)~TA;
(vii) T(T(A ~ T A ) ~ T A ) ~ TA;
(viii) T(-~VxTA)-, 3xT-,A.
(ix) If A is a sentence of s and W V F - F A, then VFH- F TA.
The same holds if we replace W V F - with WVF c (WVFp) and V F with VFc (VFp).
PROOF. (i) If [A] ~: [--B] and [B] ~: I--A], T(~T A) implies T~A or T-B:
in either case T(A ~-~B) is derivable by using the LOG-rule (see 59.2) and
T-imp. Otherwise, T(A ~ - , B ) has the form T(C ~ C ) or T(C ~ - , C ) and
we are done.
(ii) Assume T(TA--,TB); since T ( T B ~ T - ~ B ) ( T + - c o n s ) , we get
T(TA--,-~T~B), whence by the previous step T ( A - - , ~ B ) and finally
T(A~B).
(iii) Assume T-,TA; by LOG-rule T ( - , T A ~ ( T A ~ - , T A ) ) , whence
T(TA ~ T A ) , i.e. by (i) above and LOG-rule, T~A. Conversely, apply
T-rep, T-imp, T+-cons.
(iv)-(v): use (iii), T-rep, T-out.
(vi) Assume T(T(TA
(ii). Hence by T ( T ~ )
we get T ( T A A ~A),
direction easily follows
~ A ) ~ TA). Then [TA ~ A] r [A] by lemma 62.3
we get T(-,(TA ~ A)) or TA. If the first case holds,
which finally yields a contradiction. The reverse
in VF-.
(vii) Let T(T(A ~ T A ) ~ TA). Then T((A ~ T A ) ~ A) by (ii)above. But
Peirce's law and LOG-rule yield T(((A ~ T A ) ~ A ) ~ A), whence TA by
T-imp. For the converse, apply T(A ~ (B ~ A)).
(viii) If T-,VxTA, then T(VxTA ~ TVxA) (by T-imp and T-log), whence
TVxA or 3yT-,A(y) (T-Herb). But TVxA and T-~VxTA together imply a
contradiction (again by T-imp, T-log and T-out). Hence 3yFA(y).
(ix): apply T-univ, T-imp, T+-log, T+-elem, T+-cons. 0
63.9.1. REMARK. (i) By 63.9(vi) V F H - s y s t e m is incompatible with Lhb
schema
T(T(TA~A)~TA).
(ii) VFH- refutes the schema T+-univ "- T(VxTA ~ TVxA)" apply T-Herb
with A(x):= (x = x A ~TL), where L is the Liar (i.e. L = [--TL]).
XII.64]
w64. Reconciling supervaluation models with
The
375
5upervaluation Models and Provability Interpretation
reader
will certainly
notice
that
provability interpretation
the definition
of the
operator
X ~ o ( X ) of w is highly impredicative: we quantify over all possible
interpretations of T, and we do not have a set of constructive rules to
produce (~0(X) from X. This situation is in sharp contrast with the case of
the F-operator of Ch. II. Thus it is natural to wonder whether there is any
constructive access to the least fixed point of (I)0. In particular, is it possible
to reconcile the provability interpretation with the supervaluation models of
w
We show that the answer is positive, at least in the case where we deal
with ground countable structures. Most work has been carried out in w167
60; the essential step is to relativize the derivability relation F F=vA of w
to finite sets of usual s
and to observe that the new
derivability relation characterizes the least fixed point model. Actually, we
shall consider the relation ]]-, restricted to consistent subsets of Jtl~.
64.1. D E F I N I T I O N
(i) X I ] - 1 A iff " Y I - A , for every Y E C O N S ( R ) such that Y _~ X"
(recall that X E CONS(JfI~)"- X C_M and for every a E M, either a ~ X or
x).
(it)
(1)1 is the operator defined by ( ~ I ( X ) " - {J~([A])'X]]-IA ). Let
FIXl(Jft~ ) "- {X C_M" (I)I(X) - X}.
Clearly (~1 is monotone and we set V o o ( ~ ) - the least fixed point of (I)1.
(iii) F-*M F ::~ A is inductively defined as F F =~ A in 61 5, except that
we require"
1.
F, A are finite subsets of L(.)l~)-sentences;
2.
the axioms AX.2.1-AX.2.2 of 61.5 are replaced by
~-AX.2.1"
F- ~ r =V A, provided A E F and ~ ] = - ~ A , A -
~-AX.2.2:
F- ~ F =V A, provided A E A and Nt~]= A, A -
Nt, t - s;
Nt, t - s.
(iv) If X C_ M, the relation X F - ~ / F = ~ A (to be read as "F:=~A is
M-derivable with X-axioms") is inductively defined by modifying the
definition of F- ~ r ~ A as follows:
1. we omit the T-rules (RT) and (LT);
2. we add the X-axioms:
X F- ~V/F ==~A, provided for some t, (Tt) E A and .)tl,(t) E X;
Non-Reductive Systems for Truth and Abstraction
376
x ~
r
•
[Ch.12
provided for some t, (Tt) E r and .Al~(--t) E X;
3. we add the cut rule"
infer X F- ~ F:=~A from X ~- ~F==~ A,A and X F- ~ A , F = : ~ A .
Clearly, X ~ M
* F :=~ A defines a certain (non-effective) theory in M-logic
( --first order logic extended with the infinitary M-rule: if A(a) is derivable
for each a E M, so is VxA). If we specialize 62.1 and 61.9-61.12 to ~ M,* we
immediately have:
64.2. PROPOSITION
(i)
F-*M
* F~A
F =:~A, A and F- M
* A, F ~ A imply F- M
(ii) If A is a sentence of Lop(.Al~), either F- *M ~ A or ~- M* A ~ .
(iii)
The set DER(alg)"- {[A]" A is an L(.Ag)-sentence with ~ *M~ A}
is consistent (i.e. DER(JII~) E CONS(J~)).
64.3. LEMMA. If ~
is countable, XII-1A iff XF-*M ~ A
(A arbitrary
PROOF. If not X F-~/~A, we have, by the Henkin-Orey w-completeness
theorem (Shoenfield 1967), a subset Y of M such that YI=~A, Y satisfies
the X-axioms and the consistency axiom AX.3; hence Y E CONS(.Ab) and
Y _~ X, which implies not xiI-1A. Conversely, it is straightforward to
check x i I - l ( A F)---+( V A) by induction on the definition of X F- ~F=:~A
(here A F "--conjunction over F; V A .--disjunction over A). rl
64.4. THEOREM (Characterization)
If ~1~ is countable, Vcr
) - DER(alg).
PROOF. (i) Closure of DER(alg). By lemma 64.3, it suffices to check"
DER(.At~) F- M
* F~A
implies ~ M F ~ A .
(a)
Inductive verification of (1). If F ~ A is an instance of AX.1, AX.3,
*
IfA'-A'
Tt
al~-AX.2, we are done by the corresponding axioms of ~- M"
with .Ale(t)E DER(.A~), then .A~I=t - [ B ] , for some sentence B E s
that F - ~ ~ B .
Hence F - ~ F ~ A
by (RT). The case of the extant
DER(.Al~)-axiom is disposed of by means of (LT). In the induction step, we
apply 64.2 (cut elimination theorem) and IH.
(ii) If ~ I ( X ) C_ X, then DER(att~) C_ X. It suffices to check
*
M
* F~A
F ~ A implies X F- M
(2)
which implies XI[- 1/~ r ~ V A (by 64.3); we then apply ffl-Closure of X.
XII.64]
5upervaluation Models and Provability Interpretation
377
(2) is inductively checked. We only consider the case where F ~/ F :=~ A is
obtained via (RT) and (LT). Assume F - ~ =VA and let A - A ' , T t with
r i b ( t ) - 3t~([A]). Then, by IH, X F ~==~A, i.e. by 64.3 X]]-IA, whence
rib(t) C X ((I)l-Closure of X) and X F ~v/F =:~Tt, A' by (X)-axiom. Assume
~ A ==Vand let r - F', Tt (t as above). Then by IH, 64.3 and (~l-Closure,
dtl~([~A]) e X. Hence X F ~ F', T A =:~A is an X-axiom. V1
This Page Intentionally Left Blank
C H A P T E R 13
THE VARIETY OF NON-REDUCTIVE APPROACHES
{}65.
w
w
w
w
An inconsistency
On a truth theory of Friedman and Sheard
Fitch's models
Introducing semi-inductive definitions
Semi-inductive models for reflective truth
We address the problem of strengthening the internal logic of T with
truth principles suggested by modal logic, according to the natural idea of
reading T as a necessity operator. Of course, according to Montague (1963)
and subsequent work up to recent classification results by Friedman and
Sheard (1987), there are severe limitations to this move.
Be as it may, we know from Ch. XII that there is a consistent notion T
of self-referential truth that "believes" in its own consistency and is closed
under logical consequence. Can we coherently maintain that, in addition to
its consistency, T believes in its own closure under infinitary logical
consequence? Formally: can we add to T+-cons "- T(T--,a---+~Ta) the
schemata:
T+-imp := T ( T ( A --+B) ~ (TA ~ TB));
T+-univ := T(VxTA(x)---+ T(VxA))?
Of course, we already know by 63.9.1 that T+-imp and T+-univ force us to
give up the provability interpretation of w63. But, more surprisingly, as we
shall see in w
there is no chance to find a model for theories extended
with T+-imp and T+-univ, insofar as we stick to the core system VFp of
Ch. XII with internal number-theoretic induction. The point is that the
number-theoretic induction of VFp is strong enough to exclude certain
simple w-inconsistencies, which are, on the other hand, called into existence
by the above mentioned axioms.
Nevertheless, as w shows, the internal logic of truth can be greatly
expanded, up to include T+-univ, T+-imp, T+-cons and even internal
completeness T+-comp := T ( T A V T-~A), once we give up the soundness
schema T-out "- T A ~ A together with T-rep "- T x ~ T T x . The resulting
The Variety of Non-Reductive Approaches
380
[Ch. 13
theory, which appears in Friedman and Sheard (1987), turns out to be
w-inconsistent, and yet consistent by means of a semi-inductive procedure,
to be developed in w
In w
we refine the supervaluation models,
according to an idea of Fitch (1963), in order to obtain a system which
extends VFp and is compatible with T+-imp, T+-cons, but not with
T+-univ. The final sections outline a radically new solution of the problem.
We introduce models for VF-, which use non-monotone semantic valuation
schemata and are based on the notion of truth revision.
In particular, we first define in w
Herzberger's semi-inductive
definitions with their basic properties (stabilization and periodicity
theorems). The new machinery produces strengthened internal logics (w
which were first investigated (as far as we know) by Aczel and Turner.
These systems differ from those of Ch. XII and w67, and they are subsumed
under the Friedman-Sheard theory. However, we are quite far from having a
satisfactory, complete axiomatic treatment of the two approaches.
w65. An inconsistency
We observe that a theorem of Mc Gee(1985), concerning the
w-inconsistency of certain theories of truth over Peano arithmetic, extends
to VFp. As immediate consequence, we obtain the announced inconsistency
result. Below we stick to the notations of the previous chapter. Henceforth,
we presuppose an axiomatization of predicate calculus with modus ponens
as only inference rule (cf. p.352).
65.1. DEFINITION
(i) IL (=internal logic) is the least set s of L-formulas such that:
s is closed under modus ponens, and the rule
T-intro:
infer T A from A;
s contains the axioms of classical predicate calculus, the axioms of
O P - a n d the following schemata:
T-elem:
A ~ T A , where A has the form t -
T-imp:
T ( A ---, B)---, ( T A ---, TB);
T-univ"
V x T A ---, TVxA;
T-cons:
T-~x ---, -~Tx;
NIND:
A(0) A V x ( A ( x ) - ~ A ( x + l ) ) ~ V x ( N x -~ A(x)).
.
s, ~ t -
s, Nt, ~Nt;
_
The acronyms T-elem, T-imp, T-cons correspond to the schemata of w
and w
T-elem is the adequacy schema for elementary atoms. We write
IL F- A, instead of A E IL.
An Inconsistency
XIII.65]
381
(ii) VF + is the elementary theory (in the language s of w
which
includes: classical first-order logic with equality; the axioms of the theory of
operations OP-; the T-axioms"
T+-elem:
T ( A ~ TA), where A -
(t - s), ~ t -
s, Nt, ~Nt;
T+-imp:
T(T(A~B)~(TA~TB));
T-rep:
Tx~TTx;
T+-cons:
T(T-~x --+ ~Tx);
T+-univ:
T ( V x T A -~ TVxA);
I-NIND:
T[A(0) A V x ( A ( x ) - , A ( x + I ) ) ~ Vx(Nx -~ A(x))];
and the T-rules;
T+-log:
T A , provided A is a logical axiom, an axiom of OP-;
T-elim:
from T A infer A (A arbitrary sentence).
VF + is closed under a strengthened LOG-rule (see 59.2 (i)), namely:
65.2. LEMMA. If I L F A , then VF + F T A
(and henceVF + F A ) .
PROOF (Induction on IL-definition). If A is an axiom of IL, by inspection
V F + F TA. If A - T B
has been obtained by T-intro from I L F B,
VF + F T B by induction hypothesis and hence VF + F T A by means of
T-rep. The remaining cases only require the obvious provability of T-imp,
T-univ (via T-elim) and the application of induction hypothesis. D
The following remark is essential to the inconsistency argument:
65.3. LEMMA
IL F Vx(Nx - , T A ( x ) ) ~ V x T ( N x ~ A ( x ) ) ~ T ( V x ( N x ~ A(x))).
PROOF. ::~: apply T-intro, T-imp, T-elem in the form -~Nx--~T[-~Nx],
and T-univ.
r apply T-imp, T-intro and Nx--~ T[Nx]. 0
65.4. THEOREM. IL is w-inconsistent, i.e. we can find a formula H(x)
such that:
(i) IL F -~Vx(ix -~ H(x));
(ii) IL F H(~), for each n 6 w.
PROOF. We lift Mc Gee's trick to the present framework. By recursion on
N (Ch. I, 3.2), we can find a term G such that,
382
The Variety of Non-Reductive Approaches
[Ch.13
IL F- GOy -- y and G(k+l)y -- [T(Gky)].
(1)
By fixed point theorem, there is a solution M to the equation
IL F- M -
[--,Yx(Nx---, T(GxM))].
(2)
As a consequence of T-cons,
---,T(GxM))).
(3)
IL I- T ( M ) ~ - - , g x ( g x --+T(G(x+I)M)),
(4)
IL b T ( M ) ~ T ( V x ( N x
By lemma 65.3 and (1),
whence, with the trivial V x ( N x ~ T ( G x M ) ) ~ V x ( N x ~ T ( G ( x + I ) M ) )
(which requires the axiom Vx(Nx ~ N ( x + I ) ) ) ,
IL F- T ( M ) ~ V x ( N x
~ T(GxM)).
(5)
But (1) again yields, using the axiom N(0),
IL ~- Vx(Nx ~ T(GxM))--. T(M), whence IL P ~Vx(Nx ~ T(GxM)),
i.e. IL F - T ( M ) (by T-rule), or equivalently, IL F-T(GOM). By iterating
T-intro, we inductively get IL F- T(GkM), for each k E w. Hence the choice
H ( x ) - T(GxM) proves the theorem. F1
We are now in the position to refute VF+:
65.5. COROLLARY. VF + is inconsistent.
P R O O F . If we choose M and G as in Theorem 65.4, from L e m m a 65.2 we
havp VF + b T(M) and hence VF + b ~Vx(Nx--, T(GxM)) by T-elim. By
definition of G,
VF + b T(G-OM).
But VF + b T ( G x M ) ~ TT(GxM) (by T-rep) and hence, by choice of G:
VF + t- Vx(T(GxM)---, T(G(x+I)M)).
We conclude by internal N-induction that
VF + F- Vx(Nx ~ T(GxM)))" contradiction !f'l
Of course, it may be asked whether IL is nevertheless consistent: a positive
answer is offered in the next section. We do not know whether the argument
above works with number-theoretic induction for classes.
On a Truth Theory
XIII.66]
383
w66. On a truth theory of Friedman and Sheard
If we omit T-rep from VF +, we can produce a theory of truth FSL _DIL,
which is consistent but w-inconsistent. The theory was (essentially)
introduced as an extension of Peano arithmetic by Friedman and Sheard
(1987). The construction below shows that Mc Gee's construction is sharp:
in the theorem 65.4 "w-inconsistent" cannot be simply replaced by
"inconsistent". Moreover, as we shall see, there are negative applications to
the case of the logic of truth revision.
66.1. DEFINITION
(i) Fix any structure .AI~I-OP-; M is the domain of .Ate. If X C_ M, A
is a sentence of .L(.At~), X I - A means "A holds true in the structure
(JII~,X)", i.e. A is true whenever s
receives its usual interpretation in
and (.AI~,X)I-Tt iff Jit~(t)E X (.A~(t) being the value of t i n Jtt~). As
usual, we systematically rely on abuse of notation: a, b, c, d, e... stand both
for elements of MI~ and the corresponding constants. Once ~ is fixed and
b, c E M, we shall write be, ~b, Vb, b--+c, instead of the proper ~(Ap(b, c)),
~(NEGb), ~t~(ALLb), J~(IiPLYbc) (in the given order; for IMPLY, see
w
(ii) If
(iii)
X C_M, SENT(.At~)'- { ~ ( [ A ] ) - A s
J(X) "- {.AI,([A])" ~ ( [ A ] ) E SENT(J~lt,) and X I- A}.
(iv) We set:
X(0)-
O and X ( k + l ) -
J(X(k));
Thoo(dtl~) : - {.Ate([A]) -.At,([A]) E SENT(.31t,) and, for some k,
X(m)]= A, for every m > k}.
If MI~([A]) E Thcc(Mt, ),
every n > m.
(v)
ko(A)"-least number m such that X(n)[= A, for
Th(.At,)- {.Ate([A])" A is an Lop(.At,)-sentence such that .AI~[=A}.
(vi) We say that Thcc(Jtt~) is closed under an inference rule %, if
Thcc(Jll~) contains the .Al, term encoding the conclusion of %, whenever
Thcc(.At,) contains the .Ab-terms encoding the premises of % (e.g. "Th~(.AI~)
is closed under modus ponens" means b E Th~(MI~), whenever a E Thcr
)
and (a-,b) E Thcc(Ml~)).
Now observe that we cannot have both ~([A])EThcc(.AI~ ) and
~([-~A])EThcc(J?6 ) (otherwise X(m)I=A and X(m)I=-A , for every
m > max(ko(A),ko(~A))" contradiction). A similar argument shows that
Thcc(Jtl~) is closed under modus ponens. If A is any valid formula of s
The Variety of" Non-Reductive Approaches
384
[Ch. 13
or dl~ I- A and A does not contain T, X(m)l--A for arbitrary m. Hence we
have:
66.2. LEMMA
(i)
Thoo(.Al~) is consistent and closed under (first-order)logical
consequence.
(ii)
Th(J~) C Thoo(Jtl~).
66.3. DEFINITION
(i)
T-Comp: TA V T-~A; T-exist: T3xA--,3xTA (A s
(ii)
Let us consider the following finitary inferences:
(iii)
(T-intro): from A infer TA;
(T-dim): from TA infer A;
(~T-intro): from -~A infer -~TA;
(~T-elim): from ~TA infer -~A.
F S L ( - " t h e Friedman-Sheard logic") is the least set of s
such that:
FSL contains the axioms of classical predicate logic with - , the
non-logical axioms of OP-, T-elem, T-cons, T-comp, T-imp,
T-univ, T-exist and I-NIND (see 65.1 (ii));
FSL is closed under modus ponens and the rules T-intro, T-dim.
We write FSL ~ A for A E FSL.
66.3.1. REMARK. FSL is closed under ~T-intro and ~T-elim.
(Verification: assume FSL~--~A; then F S L b T-~A (by T-intro). But
FSL ~- T-.A---+-,TA (by the axiom T-cons of IL), whence FSL ~ -.TA.
Closure under-~T-elim: assume FSL ~--,TA: then FSL ~-T-.A (by T-comp)
and FSL F--~A by T-dim.)
66.4. T H E O R E M (Soundness). Assume that ~1~ is an w-standard model of
OP. Then
FSL F- A implies J~([A])E Thoo(Jf~).
PROOF. We argue by induction on the definition of FSL. Thus it suffices
to check: (i) every axiom of FSL belongs to Thoo(Jtl~); (ii) Thoo(Jtl~)is
closed under logical consequence, T-intro and T-dim.
(i). By lemma 66.2, all valid sentences are in Thoo(Jl~). Observe that if
A Es
then for every m, X(m+l)~-m
iff X ( m ) I - A
iff
X ( m + l ) I = TA, and hence 3t,([m ~ TA]) E Th~(J~), which takes care of Te/e///
.
Ad T-imp. If X ( m + 2 ) I = T ( A ~ B ) and X(m+2)I=TA, then X(m+I)I=A
On a Truth Theory
XIII.66]
385
and X ( m + l ) l = A ---+B; hence X ( m + l ) l - B, i.e. X ( m + 2 ) ] = TB. Therefore we
obtain, for each n > 2, X(n)]= T(A ~ B ) ~ (TA ~ TB).
Xd T-univ. We assume X(m+2)I=VxTA(x), i.e. for each c E M ,
X ( m + l ) l = A(c), which implies ,At,([VxA])E X ( m + 2 ) , i.e. X ( m + 2 ) ] = TVxA.
Thus for each n > 2, X(n)I= VxTA--+ TVxA.
Xd T-cons. If X(m+I)I=T-~A holds, then not X(m)I=A , which implies
Ni,([A])~ X ( m + l ) , i.e. X(m+I)I=--,TA. Therefore for each n > 0 , we
obtain X(n)]= T-,A ~ ~TA.
I-NIND" note that, since ~ is w-standard, we have, for every m,
X ( m ) l = A(0) A V x ( A ( x ) ~ A ( x + I ) ) ~ Vx(Nx ~ A(x)).
Ad T-comp. We have, for each m, either X(m)I=A or X(m)I=~A , i.e.
~ ( [ A ] ) E X ( m + l ) or .AI~([-~A])E X ( m + l ) . Then it follows, for each n > 0,
X(n)I= T A V T-,A, i.e. Jfl~([TA V T~A]) E Thoo(J~ ). The case of T-exist is
trivial as well.
(ii). By lemma 66.2, it suffices to check closure under T-intro and T-elim.
Assume 3b([A]) E Thor
). Pick any m > ko(A)+l: then X ( m - 1)]= A and
hence X(m)I=TA; so Mt,([TA])E Thoo(J~ ). Thus Thcv(J~ ) is closed under
T-intro. Assume J~([TA])EThoo(./tl~) and let m > k o ( T A ) . Were
X(m)i=--,A , then
.AI,([A])it X ( m + l )
by
definition
of
J,
i.e.
X(m+I)I=--,TA. But m + l > ko(TA): hence X ( m + I ) I = T A , absurd.
It follows that NI,([A])E X(m), for every m > ko(TA), whence
.At~([A]) E Thor
).
Thus Thoo(J~ ) is closed under T-elim. [3
66.5. COROLLARY. FSL is consistent, but w-inconsistent.
PROOF. FSL is consistent by 66.4 and 66.2 (i); but FSL contains IL and
hence it is w-inconsistent by 65.4. [3
66.5.1. REMARK. (i) It is immediate to see that FSL proves the
consistency of arithmetical analysis (see 40.3.1). By a result of Halbach
(1994), FSL is proof-theoretically equivalent to ramified analysis up to any
level < w.
(ii) The
unrestricted
inconsistent,
Lhb axiom
Tx---~ TTx.
addition of T T z ~ T x
would clearly imply in FSL the
Tarski schema T A ~ A (A arbitrary). However, FSL becomes
even if we add either T T x ~ Tx or Tx ~ TTx; in addition the
T(Tx~x)~Tx
is inconsistent with FSL, since it implies
The Variety of Non-Reductive Approaches
386
[Ch. 13
w67. Fitch's models
Despite the negative result of w
we can adapt the supervaluation
semantics of w
to validate the assumption that T recognizes its
consistency and its closure under logical consequences, plus the adequacy
schema with respect to T-free atomic sentences.
The result is implicit in Fitch (1963), and so we speak of Fitch's
models. We prove that V F - c a n be consistently enlarged by accepting, as
new schemata, T ( T ~ A ~ - ~ T A ) , T ( T ( A ~ B ) ~ ( T A ~ T B ) ) ,
and closure
under a stronger T-introduction inference. In particular, we can infer TA,
whenever the formula obtained from A by replacing T with the necessity
operator Vl, is derivable in a quantified extension of deontic logic (see
Bull-Segerberg 1983).
Technically, we first refine the basic relation ]]-o of w
67.1. DEFINITION. Fix JI~I=OP-; recall that an e-atom has the form
t - s, N t or the negation thereof.
(i) Diag(.At~)"- {J~([A])" Jlt~I- A, A e-atom of s
(ii) Recall that S E N T ( J I I , ) - {.AI,([A])" A s
X C_ SENT(Jfb) is Jfb-normal iff the following closure conditions are met"
1) X contains Diag(Jl~);
2) X E CONS(JI~);
3) X is closed under logical consequence:
3.1) if A is a logical truth in the language L(.A~), then .A~([A]) E X;
3.2) X is closed under modus ponens: a E X and (a---~b) E X imply
bEX.
(iii)
.AI, N O R "- { X C M" X is ~t-normal};
.)~-NOR(Y) "- { X C_ M" Y C_ X and X E .Ate-NOR}.
If Y E .Ate-NOR(X), we say that Y is a normal extension of X.
(iv) Once ~ is fixed, recall that
J ( X ) "- {MI~([A])" A is an L(Ml~)-sentence with (~t~,X)I-A}.
We observe a number of useful facts on 3b-normal sets. The verification is
an easy exercise.
67.2. LEMMA
(i)
Existence of Jft,-normal sets: if X C_ M, then J ( X ) is J~-normal.
(ii)
If Y C_ Jfl,-NOR and Y is non-empty, then M Y is Jl~-normal.
XIII.67]
387
Pitch's Models
(iii) If Y C MI~-NOR and Y is C -directed (i.e. for every X, Y E Y, there
is some Z E Y with X C_ Z and Y C_ Z), then U Y is .A~-normal.
67.3. DEFINITION. Let X C_ M, A s
(i) XI]- 2 A iff for every Y E .Ag-NOR(X), then Y]-A;
(ii) (I)2(X) "- {~([A])" XII-2A}; FIX2(atg ) :- { X C_ M" X - (I)2(X)}.
Clearly (I)2 is monotone and hence FIX2(.Ag ) is non-empty.
(iii) We set Nc~(a?l~) "- the C_-least element of FIX2(.AI, ). As usual
X C_ M is (I)2-dense iff X C_ (I)2(X).
67.4. LEMMA
(i) If tf is a C -directed family of .A~-normal r
then U Y is Jtl~-normal and r
subsets of M,
(ii) if X is .~-normal, ff2(X) is J~-normal;
(iii) ff2(0) is Jll~-normal.
PROOF. (i): immediate from 67.2 (iii) and ff2-monotonicity.
(ii): consider the family Y ( X ) - {J(Y)" Y E Jtl~-NOR(X)}. Then Y(X) is a
non-empty family of Jll~-normal subsets by 67.2(i). But f f 2 ( X ) - A Y(X)
and the conclusion follows by 67.2 (ii).
(iii): (I)2(q)) - A 3'(0) and Y(q)) is a non-empty family of R - n o r m a l subsets
(apply 67.2 (i)-(ii)). E!
67.5. LEMMA. Let X E Jtl~-NOR; then
(i) XII-2A implies XI= A (A arbitrary sentence of L(MI~));
(ii) XI= T ( A ~ B) ~ ( T A ---, TB);
(iii) if A is a sentence of Lop(.A~ ) of the form t - s, ~ t -
s, Nt, -~Nt,
XI---A~TA;
(iv) if X is r
, XI= T A ~ A;
(v) if X E FIX2(alI~ ), XI= V x T A ---+T V x A .
PROOF. Assume that X is alg-normal. (i)-(ii) are trivial by definition and
assumption, while (iv)is a consequence of (i) with (I)2-density.
(iii). Let A = -~(a = b). If XI=-,(a = b), then .Agl=-~(a = b) and hence
Jll~([--,(a = b)]) E Diag(Jfg) C_ X
by normality of X, i.e. XI=T--,(a = b). Conversely, if XI=T--,(a = b) and
The Variety of Non-Reductive Approaches
388
[Ch. 13
Xl= (a - b), .A~([a - b]) E Diag(J~) C_X, against the consistency of X. The
other atomic cases are similar.
(v). Assume XI=VxTA; by (I)2-density , XII-2A(a), for each a E M, whence
XII-2VxA, which finally implies, by (I)2-closure , XI= TVxA. [:]
67.6. LEMMA. Ncc(atg ) is ~-normal. Hence FIX2(atg ) n atg-NOR is non-
empty.
PROOF. Observe that N~(Mg) - U {N(c~)" c~ E ON and c~ > 0}, where
N(0)-0,
N(/~+l)-O2(g(/~))and
N(A)-U{N(6)'6<A},
A limit.
Noo(Jt~) is .Al~-normal by 67.4. !"1
67.7. DEFINITION
(i) FT ( - "Fitch's theory") is the theory which contains classical
predicate logic with - , the non-logical axioms of OP, T-out, T+-elem,
T+-cons, T+-imp, T-rep, T-univ, T+-log (cf. 65.1 and 59.1) and the
schema T-negT: T ( - ~ T A ) ~ T ~ A
(ii) F T - is FT minus number-theoretic induction.
67.7.1. REMARK. Clearly VF-C_ F T - a n d
w59 can be lifted to the new systems.
VFp C_ FT; thus the results of
67.8. THEOREM. /f ~l=OP-, x ~ FIX2(J~ ) and X is Mg-normal, then
X is a model of FT-. If atg is w-standard, X I = F T and X falsifies the
schema T(VxTA ~ TVxA).
PROOF. We only check T-negT; the remaining principles are consequences
of 67.5 or use arguments already applied in w
Assume X]=-~T-~B, then
aIg([--,B] ~ X. Hence X'-XU{.AI~([B])} is consistent, by closure of X
under logical consequence.
Let X" be the closure of X' under logical consequence: clearly X" is a
normal extension of X such that X"I=TB. Hence we cannot have
XII-2-~TB; therefore XI=-~T-~TB. Finally, if atg is w-standard, observe:
XI]- 2 A(0) A V x ( A ( x ) ~ A(x+I))---, Vx(Nx --, A), for arbitrary A.
The final claim is immediate from 65.5. [-1
67.9. DEFINITION. I F T ( - Fitch's internal theory) is the least set of s
formulas satisfying the conditions:
(i) IFT is closed under modus ponens, and the rules T-intro, T-elim,
~T-intro, -,T-elim;
(ii) IFT contains logical axioms for classical predicate calculus, the
Fitch's Models
XIII.67]
389
axioms of OP-, the schema of N-induction and T-elem, T-imp, T-cons.
As usual, we write IFT F A instead of A E IFT. I F T - F A means that
the induction schema has not been applied in deriving A.
67.10. PROPOSITION. F T - is closed under the following rule: if I F T - F A,
then F T - F T A (the same holds for FT and IFT).
PROOF (of. lemma 65.2). We use T-negT, if A follows by -"T-dim, while
T-out, T+-cons, T+-imp, T-rep apply in the case of -"T-intro. [3
67.11. PROPOSITION. F T - p r o v e s lhe schemata:
(i) T ( T A ~ T B ) ~ T(A ~ B);
(ii) T-"VxTA---, T-"VxA;
(iii) (-"TA ~ T - " T A ) ~ TA V T-"A (T-Shcomp).
PROOF. (i) T-cons implies:
I F T - F (TA ~ T B ) ~ -"T(A A -"B).
Hence F T - F T ( T A ~ T B ) ~ T-"T(A A -"B)) by 67.10 and T-imp. Then
T-negT implies:
F T - F T ( T A --, T B ) - , T(-"(A A -"B)),
whence the required conclusion with T(-"(A A -"B))-, T(A--, B).
(ii) I F T - F TVxA-~ VxTA, whence F T - F T(-"VxTA -~-"TVxA) by 67.10.
Thus we conclude that F T - F T-"VxTA --, T-"TVxA, which immediately
yields with T-negT the provability of (ii).
(iii): trivial with T-negT. [3
67.12. REMARK
(i) Let T + - n e g T : = T ( T - . T A - , T - " A ) .
Then FT-+T+-negT is
inconsistent (note that 67.11(i) and T-out immediately yield -"TA--~-"A,
whence TA~--~A, for arbitrary A). A similar argument shows that
F T - + T ( T A --, T T A ) is inconsistent.
(ii) The following schemata are not provable in FT-:
T-Rcomp := T ( T A -, A)--, T A V T-.A;
T-S4comp := T ( T A --, T T A ) - , TA V T-"A.
Verification: consider a term S such that S = [TS], provably in OP-. Then
trivially Noo(M1,)]=T(TS--,S), but Noo(MI~)]=-"TSA-"T-"S (inductive
argument, using the fact that each Nc~(a ) is ~ - n o r m a l , for a > 0). The
proof for T-S4comp is similar.
The Variety of Non-Reductive Approaches
390
[Ch. 13
w68. Introducing semi-inductive definitions
There is a natural extension of the theory of inductive definitions, which
involves arbitrary non-monotone operators. Below we state and prove the
basic results of semi-inductive definitions, which concern the existence of
stabilization, ordinals and the periodicity phenomenon. These results can be
found in Herzberger (1982), Burgess (1986), Visser (1989).
We keep assuming that M is an infinite set (for more general results, see
Visser, cit.).
68.1. DEFINITION. Let A. ~ ( M ) - ~ ~ ( M ) be an arbitrary operator on the
subsets of M. We call the pair (A,X) A-process based on X.
(i) If {G(fl): fl < A} is a family of subsets of M indexed by an ordinal A,
liminf{G(fl)" fl < A} - {a
(ii)
E M" 37 < A. Vfl(7 _< fl < A ~ a E G(fl))}.
A-iteration based on X C_M (by transfinite reeursion):
It(
x,x, o) - x ;
I t ( A , X , a + l ) - A(It(A,X,a));
I t ( A , X , A ) - liminS{It(A,X, fl)" fl < A}, if A is a limit.
(iii)
In(A,X):= {a E M : 37Vfl(7 _< 13 ~ a E It(A,X, fl)));
In(A,X) is the set of those a E M that are stably inside the family of
c~-iterations (modulo A and X);
Out(A,X) := {a E M : 37Vfl( 7 _< f l ~ a ~ It(A,X, fl)));
Out(A, X) is the set of those a E M that are stably outside the a-iterations
(mod A and X).
(iv)
Stab(A, X) := In(A, X) U Out(A, X) ( = set of stable elements
rood A, X);
:=
M-Stab( X,X)=
set of ..
tabl
(moa
For instance, if we consider the complement operator A ( P ) have:
It(A,X, 2n) = X, It(A,X, 2n+l)= m - x
M - P , we
and I t ( A , X , w ) = 0;
hence Stab(A, X) = 0.
ff A is monotone, In(A, X) is the least fixed point of A, and liminf and U
coincide.
68.2. CONVENTION: once A is fixed, we set
Introducing Semi-Inductive Definitions
XIII.68]
X ( a ) := I t ( A , X , a ) ;
391
I n ( X , a ) := liminf{X(~): ~ < a};
Out(X, a):= {a E M : V~(a _< ~ ~ a ~ X(~))}.
We keep using ON for the class of ordinals (in set-theoretic sense). Clearly,
if a is a limit ordinal, X ( a ) = I n ( X , a ) = I t ( A , X , a ) . We shall verify that
In(X) = In(X, a) and O u t ( X ) = Out(X, a), for some a < R ( M ) = the least
cardinal > card(M).
68.3. DEFINITION
In(X) C_X(7)
and
(ii) A limit ordinal 6 stabilizes (A,X) iff 6 covers (A,X), I n ( X ) and Out(X) - Out(X, 6).
X(6)
(i) A limit ordinal 6 covers
Out(X) M X(7 ) - 0 , for every "/> 6.
Clearly, if 6 covers (A,X)
moreover, Out(X) C_Out(X, 6).
and
(A,X)
6<7,
iff
x(7) c_ I~(x) u u~t.b(x);
68.4. LEMMA (Covering). Let (A,X) be a A-process on X C_M: then for
every a < R(M), there exists a limit ordinal 6 > a with ti < R(M), which
covers (A, X).
PROOF. If a E In(X) (a E Out(X)), we set:
Height(a) "- the least c~ such that a E X(fl)(a ~ X(fl)), for all/3 _> c~.
Then card({Height(a)'a C Stab(X)})<: R(M) and hence we can choose the
minimum limit 6 < R(M) such that
6 > c~ and 6 > sup(Height(a)" a E Stab(X)).
By choice, 6 covers (A, X). [3
68.5. T H E O R E M (Stabilization). Let (A,X) be a process on M. Then for
every a < R(M), there exists an ordinal 6 with a < ti < R(M), which
stabilizes (A, X).
PROOF. We show that it is possible to filter out all the unstable elements
which possibly enter X(6) (6 being an ordinal given by covering). To this
aim, we choose a limit ~ < R(M) and an enumeration {a(fl): fl < )~) of
Unstab(X), where each element occurs infinitely often (for every a = a(~)
with ~ < ~, there always exists u < )~ with a(~)= a(u) and ~ < u).
We recursively define a strictly increasing ordinal sequence {f(~): ~ < A} of
length A, whose terms are < R(M):
f ( O ) - min{7"~ < 7 and 7 covers (A,X)};
The Variety of Non-Reductive Approaches
392
[Ch. 13
f ( # + l ) -- the least 7 > f ( # ) such that a(#) E X(7) (a(#) ~ X(7))
if a(#) it X ( f ( # ) ) (if a(#) E X ( f ( # ) ) , respectively);
f(~) - least 7 > sup{f(fl)" fl < ~}, if ~ is a limit.
The sequence is well-defined by covering lemma and the choice of the
enumeration of unstable elements; moreover, by definition, if fl < ~ < A,
f(fl) < f(~). Hence 5 - sup{f(~)" ~ < A} is a limit < R(M) and it trivially
covers (A,X). It is enough to check that for every a E X(5), a ~ Vnstab(X).
By contradiction, assume a E Unstab(X) and a E f3 {X(fl)" a < fl < 5}, for
some ~r < 5. Since f is increasing, there is some ~ < A such that a < f(~)
and hence:
Vfl(f(~) _< fl < 5----~a E X(fl)).
(1)
Using the enumeration of unstable elements with infinitely many
occurrences of each term, we must have a - a(~) for some ~ with ~ < 71 < A,
whence ~r < f(~) < f(~) < 5. But (1) implies a E X(f(~7)) and by
construction of f, a ~ X ( f ( ~ + l ) ) , against a E M {X(fl)" ~r _ fl < 6}. U!
68.6. DEFINITION. If (A,X) is a process on M,
(r(A,X) "-- the least stabilizing ordinal or the closure ordinal of ( A , X ) .
68.7. LEMMA
(i) ( X ( a ) ) ( f l ) - X(a+fl);
(ii) If X ( a ) -
X(fl), then X ( a + 7 ) -
X(fl+7).
PROOF. (i): induction on ft. (ii)" immediate by (i). [1
68.8. T H E O R E M (Periodicity). Let a be the closure ordinal of ( A , X ) on
M. Then there exists exactly one ordinal r - r
X) < R(M), the so-called
"period", such that:
(i) X ( ~ ) (ii)
X(~+r
for every ordinal 6;
if a < a, there is an ordinal v < r with X ( c ~ ) - X ( a + v ) .
PROOF. Let r be the least ordinal > ~, which stabilizes (A,X) and let
r
We check (i) by induction on 6. Let 5 - f l + l :
then, since
X ( ~ ) - X ( ~ ) - I n ( X ) , we have by 68.7 (i) and IH"
=
Let 5 - A be a limit and assume that for all v < A, X ( ~ ) - X ( c r + r
X(~r) C_ X ( a + r
holds, since r stabilizes (A,X).
XIII.68]
Introducing Semi-Inductive Definitions
393
Conversely, if a E X(a+r
there is some ~ < or+CA, such that a E X(~)
for every /3 satisfying ~ </3 < a+r
Since ~ < a+r
for some 7/< A,
a E X(~r+@/) and by IH a E X(a). As to (ii), if a < c~, we apply ordinal
division and we can find u < r and ~ such that c~ = a + r
Hence, with
(i) above and 68.7 (ii), we get X(c~)= X ( a + r
X ( a + u ) . !-1
68.9. DEFINITION. Set
process (A,X); we define:
I m ( A , X ) ' - - {X(c~)" c~ E ON} - image
of
the
C o n f ( X ) "- {Y E I m ( A , X ) " V/337(/3 _< 7 A Y - X(7))};
Cycle(X) "- {X(c~)" Cr _< C~< ~r+r
(r period);
Init(X) "- {Y E I m ( A , X ) " 37V/3(7 <_/3-. r ~ X(/3))}.
68.10. LEMMA. Let (A,X) be any process on M.
(i) If X(a) E I n i t ( X ) and/3 < ~, then X(t3) E Init(X);
(ii) If X(a) E C o n f ( X ) and ~ < ~, then X(j3) E C o n f ( X ) .
PROOF. (i) Let r/ be such that for every 6, if 7/__ 6, then X(6):/: X(a).
Were ~ < c ~ and X ( ~ ) ~ I n i t ( X ) ,
there would exist ~ > r /
with
X(~) = X(~). But ~ < c~ implies c~ = ~+u, for some u; hence by 68.7 (ii),
X(a) = X(~+u), which is absurd by assumption on c~ and since ~+u > T/.
(ii)" similar argument. El
68.11. T H E O R E M (Decomposition). Let (A,X) be a process with closure
ordinal ~r and period r
(i) Conf(X) - {X(c
(ii) 1nit(X) C
(iii) I m ( A , X ) -
Then:
).tr <
< cr+r
<
< o'+r
-
Cycle(X).
PROOF (i): by stabilization theorem, we have X ( a ) E Conf(X), and hence
Cycle(X) C_C o n f ( X ) by 68.10(ii). Conversely, if Y E C o n f ( X ) , there is
a > ~r+r such that Y = X(a). By periodicity, Y = X(c~)= X(cr+T/), for
some 7/< r Hence Y E Cycle(X).
(ii): if X ( a ) E Init(X), a ~- or; were a < a, for some u < r X ( a ) = X ( a + u )
(by periodicity) and X(c~) E Cycle(X); (i)implies a contradiction.
(iii): from ( i ) a n d (ii). l-1
To sum up, the behaviour of I m ( A , X ) is already determined below the
ordinal ~+r I m ( A , X ) splits into: 1) an initial piece below a; 2) a cycle
consisting exactly of the cofinal sets of Conf(X), I n ( X ) being among them.
Clearly, if A is monotone and X is A-dense (i.e. X C_ A(X)), the period is
zero and the cycle is empty. As to the elements of M, which are unstable
with respect to (A, X), they can be characterized as follows:
The Variety of Non-Reductive Approaches
394
[Ch.13
68.12. THEOREM (Characterization)
(i)
M C y c l e ( X ) - I n ( X ) - X(cr);
(ii)
U Cycle(X) - In(X) U Unstab(X);
(iii) Out(X) - M - U Cycle(X) and Unstab(X) - U Cycle(X)-X(a).
PROOF. (i) I n ( X ) - X(r
MCycle(X)is an immediate consequence of
stabilization theorem and 68.11 (i). The periodicity theorem guarantees that
if a ~ X(~), there is fl such that ~r _< fl < ~ + r with a ~ X(fl); hence
a ~ f3 Cycle(X) (for arbitrary a E M) and f3 Cycle(X) C X(~r).
(ii). From left to right, the inclusion is immediate by the properties of ~r.
If a ~ Vnstab(X) and a ~ In(X), then a E Out(X), whence there is some
7 > ~ such that a ~ X(fl) for all fl > 7- By periodicity, X(7) - X(~+~/), for
some ~/< r and hence a ~ X(~r+r/), i.e. a ~ U Cycle(X). Conversely, let
a ~ U Cycle(X), then a ~ In(X); were a E Unstab(X), then a E X(~r+r/), for
some ~/< r (r period, apply 68.8). Hence a E U Cycle(X)" contradiction!
(ii) immediately yields (iii). I"1
w69. Semi-inductive models for reflective truth.
The application of semi-inductive definitions to the semantics of selfreferential systems is due independently to Herzberger (1982) and Gupta
(1982). Applications to the modelling of axiomatic systems for truth and
property theory can be found in Turner (1987) and Friedman-Sheard
(1987). Turner (1990) observes that the internal logic of truth, which is
sound with respect to semi-inductive interpretations, is rather rich.
However, in view of the inconsistency theorem of w
the logic of stable
truth cannot in general contain the schema T+-univ, as claimed by Turner
(1990) (1990a). As far as we know, there is at present no completeness
result, which fully characterizes the logic of truth revision (possibly
involving some form of infinitary logic).
The aim of this section is quite modest" we apply the new tools of w68
to make clear that there are a few principles, which separate the logics of
truth (sound for the supervaluation models and the provability
interpretation of Ch. XII), from the logics of truth based on semi-inductive
models.
69.1. DEFINITION
(i) As in w
if .AtI-OP-, M is the domain of ~ , X C_ M, A is a
sentence of s
X ] - A means "A holds true in the structure (MI~,X/" , i.e.
A is true, whenever s
receives its usual interpretation in .A~ and
Semi-Inductive Models for Truth
XIII.69]
395
(dill, X)I= Tt iff 31~(t) E X (36(t) being the value of t i n 31~).
(ii) If 31~ is fixed and X C_ M,
J(X) "- {dtl~([A])" A sentence of L(MI,)such that XI= A).
69.1.1. CONVENTION. We restrict our investigation to processes of the
form (g,x), where J" @ ( M ) ~ ( M ) i s
defined as above, Mt~i=OP-,
X C M. The notions of stable, unstable, stabilization ordinal, etc. are
referred to the process (J,X).
Typically, In(X) represents the set of those sentences of s
which
are stably true, insofar as we choose X as initial value for the truth
predicate T. For simplicity, we identify the elements of In(X) with the
corresponding sentences; the elements of In(X) are simply called X-stably
true sentences.
We shall proceed to check that I n ( X ) i s a model of V F - (see w
plus
suitable additional T-schemata.
69.2. LEMMA
(i) Soundness of stable truth: for every X C M and every
L(.~)-sentence, I n ( X ) ~ TA --, A.
(ii)
Consistency: for every X C M, c~ > O, X(a) is consistent.
(iii) I n ( X ) l - T-~A implies A E Out(X).
(iv) In(X)I- T-~TA ~ T~A.
(v) O u t ( X ) - {MI~([-~A])" In(X)I--T-~A).
(vi) S t a b ( X ) - {~([A])" A L(Jft~)-sentence with In(X)I= T A V T~A).
PROOF. (i) If cr is the closure ordinal of (J,X) on M, I n ( X ) - X(a); so
X(~r)I-TA implies that A is X-stably true, whence .Ate([A])E X(cr+l), i.e.
x ( ~ ) l - A.
(ii): immediate by induction on a.
(iii) If X ( ~ r ) - In(X)l= T-~A, then ~t~([-~A]) E X(6), for every 6 _> ~r. Hence
by (ii) ~ ( [ A ] ) ~ X(ti), for every 6 >_ cr, i.e. by definition ~t~([A]) E Out(X).
(iv) By assumption, there exists some 6 such that for every f l > 6,
~([-~TA]) E X(fl). We inductively verify that .AI~([-~A])E X ( f l ) f o r all
>_ 6+1. Let fl - 7+1 >_ 6+1. By assumption Jtl~([-~TA]) E X(fl+l), whence
by definition of the process, X(~)I=-~TA and X(7)I=~A, which implies
~ ( [ - A ] ) E X(fl). If fl is a limit > 6+1, we have by IH dtl~([-A])E X ( 7 ) f o r
every 7 with 6+1 <_ 7 < fl, i.e. JI~([--A]) E X(fl).
The Variety of Non-Reductive Approaches
396
[Ch. 13
(v) The second set is contained in the first one by (iii) above. Let us
inductively check that 31~([A]) E Out(X) implies .~([--TA]) E X(a) for
every a__>(r§
Indeed, if J~([A])EOut(X), then, by definition of
stabilization ordinal 3t~([A]) ~ X(a) for every a _ a (here X ( a ) - In(X))
and hence X(a)I=-~TA, i.e. dlt([-,TA])EX(a+I). If a--)~ is a
limit >_ (r-i-l, by IH dtl~([--~TA])E X(~) for every 13, q + l </~ < )~; by
definition of X(A), this is sufficient for .)~([--TA])E X(A). In conclusion,
.)tt~([--TA]) E In(X) and In(X)]= T-~TA, whence In(X)l= T--~A by (iv).
(vi)" by (v) and definition of X-stable element (with respect to J). V!
We now come to the problem of characterizing the set
I L S T - {A" dil~([A])E In(X), for every .)~]= O P - a n d every X _C M}
for comparison). ILST can be regarded as the internal logic of
It is not known to the present author
whether ILST is recursively axiomatizable. The best we can offer is to show
that ILST has non-trivial closure properties, which are embodied in the
system below.
(recall w
stable truth for the language s
69.3. DEFINITION
(i) LIS- is the least set of Z-formulas with the following properties:
1) LIS- is closed under modus ponens, and the rules
T-intro, T-elim, -~T-intro, -~T-elim;
2) LIS- contains the axioms for classical predicate logic for s
OP-, T-elem, T-cons~ T-imp;
3) LIS- contains Turner's schemata (see also 67.11)"
(ii)
T-Rcomp
T(TA ---,A)---, TA V T~A;
T-S4comp
T(TA --+T T A ) ~ TA V T-~A;
T-S5comp
(-~TA --, T-~TA)---, TA V T~A;
LIS is LIS- plus the number-theoretic induction schema NIND.
LIS F- A "- A E LIS;
LIS- F- A : - A E LIS-.
69.3.1. REMARK. (i) LIS c_ FSL (where FSL is the logic of 66.3): indeed,
T-Rcomp, T-S4comp, T-Shcomp are trivial consequences of T-comp.
Turner's schemata imply that the characteristic principles of the familiar
modal logics T, $4, $5 cannot be stably true.
(ii) LIS- ~ (TA V T - , A ) ~ T(TA-~ A). (Verification: apply T-Rcomp
from right to left; the opposite direction is already derivable in Fitch's
Semi-Inductive Models for Truth
XIII.69]
397
internal logic of 67.9).
69.4. T H E O R E M . If L I S - ~ - A , then A E ILST (i.e. In(X)I=TA, for every
..~1= Op -, x c_ M).
P R O O F . Let - ~ I - O P - and assume that X C_ M. As above, X(cr) - In(X).
We first check the following claim:
In(X) contains the axioms of LIS.
(1)
If A is a logical truth of L or an axiom of OP-, then X ( a ) I = A for all a,
and hence X((r)l= TA; also note that, if A is an atom or the negation of an
e-atom, then for every a, X(a+I)I=A~-~TA. These remarks show that
logical axioms and OP--axioms, together with T-elem, are stably true.
T-cons. By induction on a > 2, we check (T~A---+~TA)E X ( a ) . Indeed,
(T-~A---,-TA) E X ( 5 + I ) , as X(5)I= T-~A---,-~TA by 69.2 (ii). If A is a limit,
(T-~A---+-~TA) E X(fl) by IH for 2 < fl < A; so (T-~A---+~TA)E X(A) by
definition.
T-imp. By induction on a > 1, (T(A---+B)---+(TA---+TB)) E X(a).
As to the case a - 2 ,
if X(1)I=T(A---,B ) and X(1)I=TA , it follows
X(O)I=(A~B) AA, whence X ( 0 ) I = B and X(1)I=TB. So
X(1)I-T(A~B)~(TA~TB),
i.e. ( T ( A ~ B ) ~ ( T A - - - , T B ) ) E X(2).
The limit case is trivial by IH. Let a - t~+l, with 5 > 2: it suffices to see
that A---, B E X(5) and A E X(8) imply B E X(5). We can assume ~ > 2; if
is a successor, the conclusion follows from the definition of X(~). If 8 is a
limit, there exists 3' < ~ with A ~ B, A E X(fl), for all/3 with 7 </3 < ~; by
IH and definition of J, B E X ( f l ) , for all fl with 7 < f l < 5 ,
whence
B e x(6).
T-Rcomp. We check, by induction on fl > 1
T((TA ~ A)---+(TA V T~A)) E X(fl).
As usual, the limit case is trivial by Itt. As to the successor, we consider
two cases.
Case 1: f l - 3'+2. Then X(7)I= A or X(7)I--~A yield
X ( 7 + l ) [ = T(TA ---,A ) ~ (TA Y T-,A),
whence the required conclusion for'ft.
Case 2" f l - A + I
with A limit: assume X(A)I=T(TA--,A), i.e. there is
some 3' < A such that TA ~ A E X(fl') for all fl' such that 7 _< fl' < A.
Subcase 2.1" X(5)[= A for some 5 such that 3' _< ~ < A. Then we can check,
by secondary induction on fl', that A E X(fl'), for all fl' with ~+1 < fl' < A,
and hence X(A)I= TA: indeed, A E X ( 5 + I ) by assumption; if f l ' - ~ + 1 and
398
The Variety of Non-Reductive Approaches
[Ch.13
A E X(~) by IH, X ( ~ ) I = T A ~ A and X(~)I=TA , whence A E X(~+I); the
limit case is trivial by IH. Hence X()~)I= T A V T-,A.
Subcase 2.2: assume that, for all fl' such that 7+1 _< f l ' < A, (--A)E X(fl').
By definition X(A)]= T-~A and hence X()~)]= T A V T-,A.
T-S4comp: first observe that
if )~ is a limit, X(A)]= T ( T A ~ T T A ) implies X(A)]= T ( A ~ TA).
Verification of (,). Let 6 such that ( T A ~ T T A ) E
satisfying 6 < fl < )~. We check by induction:
(,)
X(fl), for every fl
(A--, T A ) E X(fl), for every fl such that 5+1 _< fl < A.
If fl is a limit in the prescribed interval, the claim follows by definition of
l i m i n f and IH.
Let f l - - 7 + l and 5 + l < f l < A .
Then (TA---~TTA) E X ( 7 + 2 ) , whence
X(7+1)1= T A --. T T A , which in turn yields X(7)I= A ~ TA, i.e.
(A ~ T A ) E X ( 7 + l ) .
Let us check that, if )~ is a limit,
X(A) I= T ( A - . T A ) ---, . T A V T-~A.
(**)
Let ~ be such that, for every ~ <_fl < A, (A---, T A ) E X(fl). If X(fl)]= A for
every fl with 6 _< fl < A, we immediately have X()~)]= T A and hence the
required conclusion. Assume 6 _< 7 < .k and X(7)[= --A. Then we can
inductively see that if 7 _< fl < A, X(fl)]=-~A. (The only non-trivial case is
the successor case f l - ~+1" by IH X(~)]=--A; were X ( ~ + I ) I = A , we should
get, with X ( ~ + I ) I = A - , TA, that X(~+I)]= TA, i.e. X(~)]= A: absurd!).
Hence, Vfl(7+l _ fl < A-,(-~A) E X(fl)), i.e. X(A)[=T~A and
finally
X()~)]= T A V T--,A. (.)-(**), together with the fact that T-S4comp holds in
every X ( a + l ) , imply that T-S4comp is stably true.
T-Shcomp: as in the previous cases, the schema is true in X ( a + l ) .
We verify that it holds in X()~), for A limit. Assume X ( A ) ] = - T A - - . T ~ T A
and X(A)I=-~TA. Then there is some 6 < A such that ( - , T A ) E X(fl) for
every fl with ti _< fl < )~, which also implies Vfl(5 _ fl < )~~ X ( f l ) I = - T A ) .
If there were some fl such that 6 < fl < A and X(fl)I=A, then also
A E X ( f l + l ) , i.e. X(fl+l)]= TA: absurd! Hence
Vfl(6 _< fl < ~ ~
X(fl)l=-~A),
which implies Vfl(5+l _< fl < A ~ (--A) E X(fl)), i.e. X(A)]= T ~ A .
This completes the verification of the opening claim (1).
We now proceed to verify:
XIII.69]
Semi-Inductive Models for Truth
399
I n ( X ) is closed under logical consequence, T-intro, T-elim, -~T-intro,
-,T-elim and the M-rule: infer VxA from A(a), for each a E M.
(2)
Modus ponens: apply 69.2 (i) to the fact that
In(X)l= T ( T ( A ---, B)----, .TA ~ TB).
It suffices to check inductively that X(o')I=VxTA implies
X(fl)I=TVxA , for all fl >_ ~r+l, whence it will follow by choice of ~r,
X(~)l= T'dxA. The case of fl limit is trivial by IH. Let fl - 7+1 > ~r+l. By
assumption, for all a E M, A(a)E X(a); as ~r is the stabilization ordinal,
A(a) E X ( 7 + l ) for every a E M, whence X(7)I= VxA, i.e. VxA E X ( 7 + l ) .
M-rule.
T-intro: it suffices to check X(~r)I=TA--.TTA. Assume X(~r)I=TA; we
inductively prove that T A E X(fl), for every fl > cr+l. The limit case is
trivial by IH. If f l - ~+1, observe that the assumption A E X(cr) implies
A E X(8), if ~r < ~, i.e. X(~)I= TA, whence T A E X(8+1).
T-elim: by 69.2 (i) X(~r)l= T T a ~ Ta.
~T-intro: apply T-intro, closure of X(~r) under modus ponens and the fact
that T-cons E X(cr).
-~T-elim" if X(~r)I=T-~TB, we inductively verify Vfl > c~+l.(-~B)E X(fl),
where a is an ordinal such that Vfl > a.(-~TB)E X(fl). The limit case is
trivial by In. If f l - 6+1 > a + l , ( - T B ) E X(8+2), hence X(6+I)I=~TB ,
i.e. X(6)I=-~B, and finally (-~B)E X(6+I).
As a consequence of claims (1)-(2), we have that LIS- C_ILST. E!
69.5. COROLLARY. Let ~1~ be an w-standard model of O P - and let
X C M. Then In(X)I=LIS, but I n ( X ) I = - - , T ( V x T A ~ T V x A ) , for some
sentence A.
PROOF. The first part is obvious by 69.4 and assumption. As to the
negative claim, observe that if I n ( X ) I = T ( V z T A ~ T V z A ) , then I n ( X )
would be ;v-inconsistent by 65.4. But I n ( X ) is w-consistent, by closure
under M-rule and the fact that the extension of the predicate N is a class !
D
We now turn to the question of characterizing the external logic of
stable truth, namely the set
E L S T - - { A - A L-sentence such that In(X)l= A, for every .AI~I=O P - and
XC_M}.
A natural (finitary) approximation to ELST is suggested by the previous
results.
400
The Variety of IVon-Reductive Approaches
[Ch. 13
69.6. DEFINITION
(i) Let T+-Rcomp (T+-S4comp, T+-S5comp) be the schema which is
obtained by prefixing T to the schema T-Rcomp (T-S4comp, T-S5comp
respectively) of 69.3. For instance, T+-Rcomp has the form:
T[T(TA ---,A)--, (TA V T-,A)].
(ii) LES- is the theory which contains classical predicate logic with - ,
the non-logical axioms of OP-, T-out, T+-elem, T+-cons, T+-imp, T-rep,
T-univ, T-negT, T+-log, T+-Rcomp, T+-S4comp, T+-S5comp (see 65.1,
59.1, 69.3).
(iii) LES "- LES-+ the internal N-induction schema
T[(A(0) A V x ( A ( x ) ~ A(x+I)) ~ Vx(Nx ~ A(x))].
Notice that LES extends the system FT of 67.7. By a straightforward
argument, we obtain:
69.7. LEMMA. LES- (LES) is closed under the rule: if L I S - F A, then
L E S - F TA (LES F TA).
~l= OP-, x c_ M, then In(X)l= LES-.
In addition, In(X)]= LES, if ~1~ is w-standard.
69.8. THEOREM. If
PROOF. All the relevant work has been done above: we simply apply 69.2
and 69.4. D
69.9. REMARK. F T - is strictly contained in LES- by 67.12. In particular,
if S is a term such that S - ITS], then L E S - F TS V T-~S, while we know
that TS V T-~S is unprovable in FT-.
69.10. Problems. Is T+-S5comp independent from FT-? What is the
proof-theoretic strength of FT-, LES- ? We conjecture that they are
equivalent to VF-.
CHAPTER 14
E P I L O G U E : A P P L I C A T I O N S AND PERSPECTIVES
w
w
w
w
~i74.
w
A logical theory of constructions: informal motivations
A logical theory of constructions: basic syntax
Axioms for the computation relations
Extending the logical theory of constructions with higher
reflection
Proof-theoretic reduction
Perspectives: related work in Artificial Intelligence and
Theoretical Linguistics
Sense and denotation as algorithm and value: subsuming theories
of reflective truth under abstract recursion theory
Confronting a theoretical piece of work with applications is always
useful for a critical assessment. For this reason, we address the question of
relating the systems of reflective truth that we have been investigating so
far, with applications in Theoretical Computer Science (TCS), Artificial
Intelligence (AI), Linguistics.
We are concerned only with potential connections, and not with direct,
well-established applicatzons, already available in the literature. We shall
consider three examples: a) a logical theory of constructions, arising from
TCS, and its modeling in the systems of chapters X-XI; b) some logics,
motivated by knowledge representation and the semantics of natural
languages; c) Moschovakis's intensional approach to the foundation of the
theory of algorithms. We underline that our choice is largely a matter of
taste and strongly bound to the limited competence of the writer. Thereby,
the aim of the present chapter is rather that of putting the content of the
book in a wider perspective and suggesting new problems; there will be no
attempt of systematization, nor we try to supply complete details. The only
relative exception is the first example, dealing with the logical theory of
constructions, LTCw; but this is due to the fact that LTCw fits nicely with
the material of chapters VIII-XI. As to the examples of part b), we hope
that some applicative-minded reader will find the results of chapters XIIXIII, as well as those of chapter VI, of some interest. The final example,
which discusses Moschovakis's lower predicate calculus with reflection, is to
us highly suggestive: it should lead to reflections, embracing both the
foundations of recursion theory and formal semantics.
402
Epilogue: Applications and Perspectives
[Ch.14
w70A. A logical theory of constructions: informal motivations
In TCS new logical formalisms are currently investigated:
(i) as tools for representing, stating and establishing properties of
programs (e.g. equivalence, termination and correctness);
(ii) as tools for program extraction;
(iii) as tools for reasoning about the specification of programs and their
typing (foundations of type theories);
(iv) as abstract theories of computation over abstract data types.
This list is not exhaustive and the single aims (i)-(iv) are usually integrated,
being a distinctive feature of the logic methodology its unifying power. In
this respect, we may mention Martin-LSf's type theories (Martin-LSf 1984,
B.NordstrSm et al. 1990), the ELF-approach (Harper-Honsell-Plotkin 1987),
the theories of constructions (Coquand 1985), NUPRL (Constable et al.
1986), the logical theories of constructions of Aczel et al. (1991), Feferman's
theories and its outcomes (Feferman 1979, Hayashi-Nakano 1988, Feferman
1990, 1991a, 1992, Talcott 1992), the proofs-as-programs approach, as
developed by Schwichtenberg (1991).
We concentrate upon a single example, which appears close to the spirit
of this work: the logical theory of constructions LTC, as it is outlined in
Aczel-Carlisle-Mendler (1991). On the conceptual side, LTC-theories are
motivated by "the idea that the notions of proposition and truth are, after
all, the fundamental ones for logic and that the logical notions are the
fundamental ones for a deductive system for mathematics. According to this
idea, although the notion of type is also essential for mathematics and
computer science, it is less fundamental conceptually" (see p. 5, cit.).
Technically, we can summarize the basic features of LTC in the
following points:
1) LTC includes the values of a functional programming language, as
well as the propositions of a reflective logic; in particular, in LTC there is a
truth predicate, which expresses the fact that a proposition, as an object of
our universe, is true; since the underlying logic is constructive, LTC-higher
systems are endowed with predicates expressing the fact that certain objects
are propositions (of a given level);
2) the functional language is untyped; but the semantics is operational,
explicitly controlled by a "lazy evaluation" relation;
3) the basic equality relation should be decidable; however, in the
strongest theory of Aczel et al., there is a mixed approach: recursive aspects
XIV.70]
A Logical Theory of Constructions
403
are handled as equalities, while discrimination and selection aspects are
maintained at an operational level; we do not know whether the resulting
conversion relation is decidable.
An interesting feature of LTC is that it points to possible refinements of
the underlying combinatory logic of OP, which make sense of important
distinctions for applications, apparently inaccessible within M F - and its
extensions. In fact, a limit of our systems concerns equality: we only deal
with a single basic equality = , which is interpreted as equality in
combinatory algebras; hence - is generally undecidable.
Moreover, one would like to have a notion of "value" and hence a predicate
of definedness, in order to explicitly control the main properties of
programs. As we shall see, the system LTC 0 of the next section offers a
viable alternative, by introducing a different semantics underlying the
theory of programs. A final point of interest is that LTC-theories establish a
sort of natural bridge between Martin-Lbf's type theories and the
predicative systems of reflective truth with variable levels of part D.
w70B. A logical theory of constructions: basic syntax
In this section we are going to introduce expressions with arities and their
basic definitional equality - , together with the notion of canonical
realization (term models) for the resulting formalism.
70.1. (i) Arities: they are inductively generated by the following clauses:
OB (individuals), BOOL (formulas) are basic arities; if c~, /3 are arities, so
By currying we also write (ch...c~n)---,c~ for (ch~(c~ 2 . . . ( % ( - o a ) . . . ) . Every
symbol is assigned an arity. If we understand BOOL as the arity of
formulas, (OB~BOOL), (OB, OB)~BOOL will obviously represent the
arities of unary and binary predicates (in the given order); on the other
hand (OB, OB)~OB is the arity assigned to binary function symbols. It is
clear that the stock of basic arities can be conveniently expanded, insofar as
we need new basic sorts of entities.
(ii) The formal language s
is given by specifying a list of primitive
symbols, together with their corresponding arities.
Individual symbols (arity OB): a denumerable list of individual variables
(x,y,z,u syntactical variables); the constants 0 and J_ (the object
representing the absurd proposition 1 );
Propositional symbols (arity POOL): I (absurd proposition);
Epilogue: Applications and Perspectives
404
[Ch.14
Logical symbols: ---, (implication of arity (BOOL, BOOL)-,BOOL); Vo
(universal individual quantification of arity (OB---,BOOL)---,BOOL); V1
(universal function quantifier of arity ((OB~OB)~BOOL)---,BOOL);
Predicate symbols:
arity (OB, OB)~BOOL: = (equality), LEV (lazy evaluation), N E Y (full
evaluation to numbers);
arity (OB---,BOOL): T (truth); Pi (proposition of level i), for any i > 0.
Function symbols:
arity (OB---,OB): a denumerable list of unary function variables f (f, g, h
syntactical variables); S (successor); Inl, Inr (projections); V0; Vl (internal
quantifications); P i , for each i > 0; T;
arity (OB, OB)---,OB: Pair; -=, ; LEV, NEV, - ;
arity (OB~OB)---,OB: A (abstraction);
arity (OB, (OB, OB)---,OB)~OB: Spread;
arity (OB, OB---,OB, OB--,OB)~OB: Decide;
arity (OB, OB, OB--,OB)---,OB: Decidenat;
arity (OB, OB, OB---,OB)--,OB: Ind (primitive recursion operator);
arity (OB, (OB---,OB)~OB)---,OB : Pa (permuted application).
70.2. Expressions of "~TC: they are inductively generated from the set of
basic symbols by means of the inductive clauses for abstraction and
application:
(i) if E is an expression of arity a and x is a variable of arity fl (hence
t3 = OB or ~ = OB~OB), then (x)E is an expression of arity (fl---,a);
(ii) if E is an expression of arity a---,~ and E' is an expression of arity
a, then E(E') is an expression of arity/3.
70.3. Notations. Expressions of arity BOOL are identified with formulas
and A, B, C play the role of metavariables for them; VxA := V0((x)A ) and
Vlf. A := V((f)A). Expressions of arity OB are the usual individual terms;
we let Ax.t stand for A((x)t). Multiple abstraction is reduced to iterated
abstraction (currying)in the usual fashion; for instance ( x y ) f ( x , y ) i s an
abbreviation for (x)((y)f(x, y)).
70.4. Dotted symbols: they are function symbols that allow to associate to
each expression of arity POOL an expression of arity OB. In particular, we
have:
70.4.1. FACT. To each formula A of s
we can effectively associate a
term Jl such that )1 has exactly the same free variables of A.
The intuitive meaning of the basic function symbols can be clarified by
anticipating that the following defining equations are valid in the standard
denotational semantics:
A Logical Theory of Constructions
XIV.70]
405
70.5
(Re):
Pa()~(f),h)- h(f);
(Decidenat):
Decidenat(O, x, f) - x and Decidenat(S(y), x, f) - f ( y ) ;
(Spread):
Spread(Pair(xl, x2),h ) - h(Xl, X2);
(Decide):
Decide(inl(x), f, g) - f(x); Decide(inr(x), f, g) - g(x);
(Ind):
Ind(x, y, h) - necidenat(x, y, (x)h(x, Ind(x, y, h))).
It is understood that the terms involved have the appropriate arity. If we
define
Ap(x, y) := Pa(x, (f)f(y)),
then (Re)implies (/3)-conversion. Ap(~(f), y ) = f(y).
In the present proposal denotational equality is split into finer relations,
which also take care of the operational level. More precisely, while the
Ind-equation is integrated in a suitable definitional equality on expressions,
the remaining equations are transformed into inductive clauses, which define
an appropriate evaluation relation. The first step takes inspiration from
Martin-Lhf's theory of expressions.
70.6. D E F I N I T I O N
(i) If ~ is an arity and E, E ' are expression of arity ~, we inductively
define the (ternary) relation E - E':~, to be read as E and E' are equal
expressions of arity ~. We write E: cr as an abbreviation of E - E: v~; E: cr
means that E is of arity ~.
E - E': cr is the smallest relation, which meets the following conditions:
1. Reflexivity: E -
E: ~;
2. S y m m e t r y : E -
E " ~ implies E ' -
3. Transitivity: E - E': a and E ' -
E: or;
E": cr imply E - E": a;
4. c~-conversion: if x and y are variables of arity cr and y does not
occur in E, then (x)E - ( y ) E [ x :-- y ] : a;
5. /?-conversion: if E': a, x: a, E: 5, then ((x)E)(E') =_E[x := E'] : ~;
6. ~-conversion: if E:cr-.5, x:cr, then ( x ) E ( x ) - E:~-~5;
7. ~-conversion: if x: a and E - E': 5, then (x)E - (x)E': (r--,5;
8. application: if E -- E': a - ~ i and H -- H': or, then E(H) - E ' ( H ' ) : li;
9. definiendum- definiens: E -
E': OB, provided E, E ' are the
expressions involved in the (Ind)-equation of 70.5 above.
406
[Ch.14
Epilogue: Applications and Perspectives
(ii) Let - ~ := { ( E , E ' ) " E -
E':cr}; [E]~ "- { E " E -
~E'};
M~ "- {[E]~: E closed expression}.
If c r - - ~+7, [E]~ E M~, and [F]~ E M~, we put [F]a([E]~)- [F(E)]~.
According to the last definition, each element of M ~ + ~ represents a unique
function from M~ into M~. Henceforth we use the families M ~ , - ~, where
a is an arity, in order to define a Tarskian semantics for formulas of LTC o.
70.7. DEFINITION. (i) A canonical
. A "- (M,s,Mo.,~), such that"
1. ~ . - O B ,
realization
for s
is a triple
o-.-OB--.OB;
2. r is an interpretation function satisfying the conditions:
2.1. for each expression E of arity a, r
[E]~;
2.2. ~( - ) - the relation - ~;
2.3 for each i E ~o, r
2.4. r
r
and r
are subsets of M0;
are subsets of M~ x M~.
(ii) The relation v~l=A (A sentence of LTC0) is inductively defined
according to the standard Tarskian clauses, once we stipulate that:
1. variables of arity O B range over M6, while variables of arity
O B - ~ O B range over M~;
2..At, l: (t - s) iff t - s "~; .Atl= T t iff [t]~ E r
~ l = P i t iff [t]~ E O(Pi) (i E or, ~ -- OB, t, s closed terms, i.e. closed
expressions of arity OB).
We conclude with a remark: if we omit the defining equations for Ind, the
expressions with the family { - ~ ' a arity} yield a (version of) typed
A-calculus. As usual, the definitional equality relations can be generated by
the corresponding natural reduction relations, which are introduced by
regarding the basic --clauses as contractions. Moreover, every expression
reduces to a unique normal form (and actually every reduction sequence is
finite; strong normalization). As a corollary, one has:
70.8. PROPOSITION. The relations
are decidable.
-~
without the Ind-defining clause
It is not known to us whether the unrestricted relation with Ind-clause
E-E':OB
is decidable. The problem is left open by Aczel, Carlisle,
Mendler (1991); according to them, let =~+ be the reduction, generated by
XIV.71]
407
Axioms for the Computation Relations
(a)-, (fl)-, (~/)-contractions, extended with the (Ind)-contraction:
Ind(t,s,h) :=~+ Decidenat(t,s, (x)Ind(x,s,h)).
Then :=:~+ is still consistent, in the sense that a corresponding ChurchRosser property holds (if E:=~+E ' and E=:~+E '', then E'=:~+H and E"=:~+H,
for some expression H). As a consequence"
70.9.
The following basic special equality axioms become true in the
corresponding term model:
F ( f l , ' " , f n) -- F ( g l , ' " , gn): OB =~f i(ul,..., urn) - gi(ul,..., urn): OB;
-~F(ci. . .cn) - G(hi. . .hn): OB;
here F, G are primitive function symbols ~ Ind, and fi,
appropriate arities.
gi, uj have
w71. Axioms for the computation relations
The basic syntax contains a functional programming notation; so we
have to explain how to compute programs and how to identify them. Of
course, one could stick to a denotational semantics: the resulting
interpretations would be essentially the (many sorted versions of) models of
A-calculus, which were introduced in Aczel (1980) under the name of
lambda structures.
By contrast, following Aczel-Carlisle-Mendler(1991), we outline a
semantics which lies between denotational and operational semantics. First
of all, we specify the space of values. This is done by exploiting the idea of
lazy evaluation and Martin-LSf's distinction between canonical and noncanonical expressions. Roughly speaking, a canonical expression is an
expression, which directly manifests the data type it belongs to and can be
immediately understood in terms of the givens we are dealing with
(numbers, functions, lists, etc...); as such, it is a static object. On the
contrary, non-canonical expressions involve control features, that have to be
eliminated, in order to understand the direct meaning in terms of the
givens. Of course, this is vague, but it will be made precise by specifying
canonical forms and by inductively defining the appropriate evaluation
relations.
71.1. DEFINITION. (i). Canonical symbols" 0, S, A, Pair, Inl, Inr, -:~, _J_,
~]0, ~/1' --'
L E ? , N E V , Pi (i > 0), T;
(ii) a term t of arity OB is canonical (or is in canonical form) iff either
t is 0, or else its outermost function symbol is canonical.
Epilogue: Applications and Perspectives
408
[Ch.14
The computation process is formalized by the predicate L E V of lazy
evaluation: L E V ( t , s ) holds if t evaluates to s and s is in canonical form.
L E V is inductively defined by appropriate inductive clauses that involve
non-canonical symbols. But L E V does not suffice in general: for instance,
S(t) is canonical, but we cannot directly read off from S(t) that S(t) truly
represents a natural number, unless we already know that t represents a
natural number, i.e. either t is 0 or has the form S(r). The second case may
require further evaluation and so on. To sum up, we also need a primitive
predicate N E V ( t , s ) which holds exactly when t fully evaluates to a numeral
and hence we properly have that t represents a natural number. Again, this
leads to an inductive specification of N E V , which involves primitive
recursion and allows to introduce a natural number predicate. Since the
reduction relations associated to L E V and N E V are Church-Rosser, we
shall be in the position to define the appropriate equivalence relations on
programs and numbers.
71.2. D E F I N I T I O N (Computation triples). We say that (a,b,k) is a
computation triple (in short ( a , b , k ) E C O M P T ) iff one of the following
(mutually exclusive) cases holds:
(i)
a -- )~(f), b -- h ( f ) and k - ( x ) P a ( x , h ) ;
(ii)
a - Pair(xl, x2) , b - h(Xl,X2) and k - (x)Spread(x,h);
(iii)
a-
Inl(u), b - f ( u ) and k - ( x ) n e c i d e ( x , f, g);
(iv)
a-
Inr(u), b - g(u) and k - ( x ) D e c i d e ( x , f, g);
(v)
a - 0 , b - u and k - (x)necidenat(x, u, g);
(vi) a - S(z) , b - g(z) and k - ( x ) n e c i d e n a t ( x , u, g).
{For a motivation of 71.2 simply recall the equations of 70.5}.
71.3. The ground system LTC 0 (without reflection)
LTC 0 is the theory in the given language LTC above, which contains the
following axioms:
71.3.1. two sorted intuitionistic logic with standard equality axioms (for
objects); recall that the basic sorts correspond to variables of arity O B (x,
y, z...) and to arity of O B - ~ O B (f, g, h,...), respectively.
In particular, we have the axiom schema:
V f A ( f ) - - - , A [ f "- g] (g term of arity O B - ~ O B , free for f in A).
We also postulate two special principles for - , which are related to the
fact that the intended model is a term model (see 70.9):
Axioms for the Computation Relations
XIV.71]
409
SEI:
F ( f l , " ", f n) - F(gl," " "' gn) ~ f i(ul, "'" , urn) -- g i ( u l , ' ' ' ' urn);
SE2:
-~F(Cl, . . ., ca) - G(hl, . . ., ha);
here F , G are primitive canonical function symbols; fi, gi, uj have
appropriate arities (remember that - only applies to expressions of arity
OB).
71.3.2. Closure
(OB~OB)~OB:
(~)
under
definitions
for
arities
OB~OB
and
Vw(((v)t)(w) - ((y)t[v . - y])(w)) (provided y does not occur in t);
Vf(((g)t)(f)-
(fl)
explicit
((h)t[g "- h])(f)) (provided h does not occur in t);
Vu(((x)t)(u) - t[x : - u]) (provided u is free for x in t);
V f ( ( ( g ) t ) ( f ) - t[g "- f]) ( provided f is free for g in t).
71.3.3. LEV-axioms:
LEI"
L E V ( c , c), provided c is a canonical term;
L E V ( x , y ) ~ L E V ( y , y); (if x is evaluated to y, y is canonical);
LE2"
V x V y ( L E Y ( x , y) A L E V ( x , z) -~ y - z);
LE3"
LEU(x,a)--.(LEW(k(x),z)~
L E V ( b , z ) ) , for (a,b,k) C COMPT.
71.3.4. N E V - a x i o m s :
NE1-
L E V ( x , O) --~ N E V ( x , 0);
NE2"
LEV(x,S(y)) A iEV(y,z)-~
NEV(x,S(z)).
N E V - I n d u c t i o n : if A(x, y) is a formula,
m
V x ( L E V ( x , O ) ~ A(x, 0)) A V x V y V z ( L E V ( x , S(y)) A A(y, z ) ~ A(x, S(z))).--~
---, Y u V v ( N E V ( u , v ) ~ A(u, v)).
( N E V - i n d u c t i o n states that N E V is the least predicate closed under the
inductive clauses formalized by NE1, NE2).
71.3.5. Peano axioms:
wvy(
s(
) -
0 ^
-
9 -
y));
71.3.6. Primitive recursion:
I n d ( x , y, f ) - Decidenat(x, y, (x)h(x, Ind(x, y, h))).
71.4. DEFINITION
i ( x ) "- N E V ( x , x )
( x is a numerical value);
410
Epilogue: Applications and Perspectives
[Ch.14
Nat(x) "-- 3 y N E V ( x , y) (x denotes a natural number);
E q g a t ( x , y) "- 3 z ( g E Y ( x , z) A N E Y ( y , z));
xl "- 3y L E V ( x , y) ( - x has a value or x is defined).
71.4.1. REMARK. LTC 0 can easily interpret a logic of partial existence s la
Scott and distinguish between quantifying over all the entities of the
universe (just the usual Y and 3) and quantification restricted to values"
Yx+(...) "- Yx(xl---+...) and 3 x + ( . . . ) " - 3x(xl A...).
The intended models of LTC 0 are obtained as special canonical realizations.
71.5. DEFINITION. Let 5 - OB:
(i) LEV is the least relation C_ M ~ x M ~
following clauses:
which is closed under the
1. if t is canonical, then ([t]6 , [t]6) E kEY;
2. if (t, s, r) is a computation triple (71.2), ([P]6, [t]6) E ILEY and
([s]5, [q]5) E LEV, then ([r(p)]5, [q]5) E LEY.
(ii) NEV is the least relation C_ M 6 x M 6
following clauses:
which is closed under the
([t]6, [016) E LEV implies ([t]6 , [016) E NEV;
([t]6, IS(r)]6) C LEV and ([r]6, [p]6) E NEV imply ([t]6, S[p]6) E NEV.
(iii) The interpretation function ~o satisfies:
@o(T) - e~o(Pi) - 0, for each i E w;
~ o ( L E V ) - LEV and r
- NEV.
71.6. THEOREM. If .Ago " - ( M 6 , M6__,6,~o) , then alg0]=LTC 0 (i.e..Ag o is a
canonical model of LTCo-axioms ).
As to the proof, the crux is to extend the Church-Rosser theorem (w to
the definitional equality relation, in order to verify the special equality
axioms SE1-SE2 (see Aczel-Carlisle-Mendler 1991). The equality axioms,
(c~), (fl), the Peano axioms and primitive recursion are immediately
verified, as ~0 is a canonical interpretation. By choice of LEV and NEV,
L E V - and N E V - a x i o m s are valid in ~0"
What can be said about the strength of LTCo? An answer is given by a
simple observation:
XIV.72]
Logical Constructions with Higher Reflection
411
71.7. PROPOSITION
(i) PA, the first-order system of Peano arithmetic, is interpretable
into LTC o.
(ii) LTC 0 is interpretable in PA.
PROOF (hint). (i): the domain of the interpretation is the defined predicate
Nat, which is closed under successor and contains zero; EqNat interprets
equality. The Ind-axioms imply that there are functions + and 9 under
which Nat is closed; NEV-induction implies Nat-induction.
(ii) One has to formalize the term model construction of w with the
corresponding Church-Rosser theorem (see appendix to chapter I). This is
possible since the inductive definitions of definitional equality - , and the
relations LFV and NFV, being given by existential positive clauses, can be
explicitly defined and arithmetized in PA. E!
71.7.1. REMARK. Assume that LTC o F-vx(gat(x)--, Nat(t(x))): then 71.7
implies that the number-theoretic function defined by the term (x)t(x) is
provably recursive in PA.
w72. Extending the logical theory of constructions with higher reflection
Up to now, the predicate symbols Pi and T have been left undetermined.
We wish to add axioms interpreting Pit as "t is a proposition of level i"
and Tt as "t is a true proposition".
72.1. DEFINITION. LTCw is the extension of LTC 0 with the following
propositional and truth axioms (for each i with 1 ~ i < w):
PTli
If c - [ A ] and A - ( r - s), LEV(r,s), NEV(r,s), L ,
LEV(t,c)--*Pi(t);
LEV(t,c)---~(A~Tt);
PT2i
(LEV(r, t-~s) A Pi(t) A (T(t)--,Pi(s))) ~ Pi(r);
((LEY(r, t-,s) A Pi(t) A (T(t)--+Pi(s))) ~ (T(r) ~-, (T(t) ~ T(s)));
PT.3i
(LEY(r,~/(f)) A V x P i ( f ( x ) ) ) ~ Pi(r);
(LEY(r, ~/(f )) A VxPi( f (x)) ) --, (VxT( f (x)) ~ T(V(f))).
PT.4i, j if 1 _< j < i,
LEV(r, Pj(t))--,(Pi(r ) A ( T ( r ) ~ nj(t)));
Pj(t)~Pi(t).
Epilogue: Applications and Perspectives
412
[Ch.14
72.2. DEFINITION. Let 1 < i < k.
(i)
Internal truth of level i:
Ti(t)'-T(t)APi(t
(ii)
).
LTC k is the subsystem of LTC~, which only contains the predicate
symbols P I , . . . , P k .
At first sight, one might conjecture that each fragment LTC k is directly
interpretable in the system STLR k with reflective truth predicates up to k
(see w
one would simply be tempted to identify the truth predicate T
with Tk, the truth predicate of level k, and to define the notion of
proposition of level i (for 1 _< i __ k) with the classical Pit "- Tit Y Fit. The
idea, though roughly sound, is not viable, since P i and T must be
well-behaved with respect to L E V and N E V ; but these predicates cannot be
simply reduced to usual conversion equality, due to the special equality
axioms. Of course, we can define new systems corresponding to TLR, ITnr
RS n of Chapters VIII-XI, which are based on LTC 0 instead of OP. Then the
proof-theoretic reduction of Chapters X-XI can be adapted to the new
systems without any difficulty, in order to show:
72.3. THEOREM. For each k, LTC k is proof-theoretically reducible to T L R
(i.e. the formal consistency of LTC k is provable in OF + TI(a), for some
a < F 0 and hence in TLR; see Ch. XI)
The reader not interested in further dreary details, can directly skip to w74.
In the rest of w72 and in w73 we illustrate a different route to the theorem,
which consists of building up a model of LTCk, directly in the available
systems. Since the verification of 72.3 is rather lengthy, we split it in a few
steps; in this section we restrict our attention to the interpretation of LTC 0.
First Step. We simulate the term model of LTC 0 within an untyped model
of OP by direct use of the combinatory structure available. Abstractions
(x)t, (f)t are represented by h-abstraction; the basic identity of LTC k is
sent into combinatory equality" thus the interpretation is certainly
unfaithful. Nevertheless, the interpretations of LEV, N E V and the basic
function symbols are chosen in such a way that the special equality axioms,
together with LEV, NEV-axioms, become true.
72.4. DEFINITION
(i) We associate to each individual constant and each primitive
function symbol G of LTC k a corresponding combinator G* (of OP).
)~* "--)~f.(-8,f); Pa* "-~x~h.(9,(x, hl); (.j_ )* . - ( 2 0 , 1 / ;
Inl* "-- )~x.(]-O,x); Inr* "-- Ax.(-H, x); Pair* "- AxAy.(-~, (x, y));
XIV.72]
Logical Constructions with Higher Reflection
413
Spread* "- )~x)~h.(13, (x, h)); Decide* "- )~x)~f )~g.(14, (x, f , g));
Decidenat* : - )~x)~yAf .(1--5, (x, y, f));
(LEfZ)* "- AxAy.(1---6,(x, y));
( N E f / ) * "-- )~x)~y.<l---7,(x, y>>; (Va)* "-- )~x.(1---g,x); S* "- )~x.(1---9,x>;
O* "--(20,0);
- * := ID;
(Vo)* "- ALL;
~ * "- )~x$y.IMPxy;
(~Pi)* "- )~x.[P*(x)]; (T)* "- )~x.[T~(x)],
where 1 < i < k and P*, T i are defined in 73.1 below;
Ind* "- FP(~g~x)~yAh.Decidenat*(x, y, )~u.g(u, y, h))).
N.B. It is understood that I M P ' - ~ x J ~ y . N E G ( A N D x ( N E G y ) ) ;
NEG,
A N D , I D , A L L are the combinators of Ch. II, 7.1. F P is the fixed point
combinator of Ch. I,w 2. Notice that * actually depends on k.
(ii) We inductively extend * to arbitrary (also n o n - p r i m i t i v e ) f u n c t i o n
symbols and terms of L T C k of arity OB:
1. if c is a constant (c)* - c*, as defined in 72.4 (i) ; (xi)* - x2i+l if x is a
variable of arity OB; (fi)* - x2i if f is a variable of arity OB--~OB;
2. F ( t l , . . ., tn) * " - F* t l * . . tn* (where F* is defined in 72.4 (i), if F is a
primitive function symbol of LTCk; F* is defined according to the
preceding clause, if F is a function variable f).
3.
(x)t* "- )~x.(t*) and (f)t* - )~f.(t*).
N.B. The use of variables of odd and even index, in order to interpret
variables of arity O B and (OB--.OB) respectively, is only required to avoid
undue identification of variables (e.g. we want that the translation (f(x))*
has two distinct variables corresponding to f and x). Henceforth we omit
explicit mention of indices and we still keep using f, g, h as metavariables
for variables in function position.
By inspection of the definitions above, we obtain the expected
independence properties, which also imply the counterparts of the special
axioms SE1-SE2 and the standard Peano Axioms:
72.5. LEMMA. I f C and E are distinct function symbols of LTCk, then
OP ~- -~C* - E*. Moreover OP proves:
SEI*"
F(fi,"',
fn)* - F ( g l , ' " , gn) --*fi(ul, "'" , urn)* - gi(ul, "'" , urn)*;
SE2*"
-~F(ci...Cn)* - G(hl...hn)*;
here F, G are distinct primitive function symbols ys Ind, and f i, gi, uj
have the appropriate arities (remember that - only applies to expressions
of arity OB);
Epilogue: Applications and Perspectives
414
suc*:
-
5")
A
wvy(s*
-
s*y
-
[Ch.14
y).
Second step. We introduce predicates L E V ~ and N E V ~ , which work as
"interpreters" of LTCk-terms. The notion of canonical object is explicitly
definable by means of the following formula of STLR k-
72.6. DEFINITION (using 72.4; k >_ 1)
C a n k ( x ) :-- 3 y 3 z ( ( x -- .~*y) V (x -- S ' y ) V (x = -0") V (x = I n l * y ) V
V (x - I n r * y ) Y (x - ( i )*) V (x - N E G y ) V
(x - (V0)*Y)V
V (x -- (V1)*Y)V (x - ( - ) * y z ) V (x - ( L E V ) * y z ) V
Y (x -- ( N E V ) * y z ) V
(x - A N D y z ) Y
V (x ----[Tl(Y)] ) V . . . V (x ---- [Tk(Y)] )
(x - P a i r * y z ) Y
V
V (X : [Fl(Y)] ) V . . . V (x -'- [Fk(Y)])).
72.6.1. FACT: we can find a formula C P T * ( y , w, z), which translates the
condition "(y, w, z) is a computation triple" (in the sense of 71.2 above) into
the language of STLRk:
C P T * ( y , w, z) := 3 f 3 h ( y = ~ * f A h f = w A z = ~x.Pa*xh) V
V 3 u3v3h(y = Pair*uv A w -- huv A z = A x . S p r e a d * x h ) V
Y 3u3v3n3g(z=)~xDecidenat*xvg A ((y=O* A w=v) Y ( y = S * n A w = g n ) ) ) V
v 3 3f3g(z=
.D cid * fg n
n
v
n
72.7. LEMMA. STLR k F C P T * ( y , w, z) A C P T * ( y , w', z) ~ w - w'.
PROOF. We essentially apply lemma 72.5. Assume the antecedent of 72.7
and also z = ~x.Decide*x f g, y = Inl*u, w = f u, for some u, f, g.
Then C P T * ( y , w', z) implies by SEI*: z - ~x.Decide *x f ' g', y - Inl(u'),
w'-f'u',
for some f', g', u'. Therefore D e c i d e * x f g = D e c i d e * x f ' g ' and
Inl*u-Inl*u',
which yield by SE2* f - - f ' ,
g=g',
u=u',
whence
f u = f'u', i.e. w = w'. The other cases are similar. 0
Let B k ( x , y , v) be the formula:
3 z 3 a 3 c 3 w ( C P T * ( c , w, z) A x = z(a) A (a, C)~ l v A (w, Y)~71v)
and define L Y k ( X , y, v) := ( C a n k ( x ) A x = y) V Bk(X , y, v).
By inspection, we see that ~ v . { ( x , y ) : L Y k ( X , y , v)} is an existential operator
(see Ch. II, 10.9) and actually that L V k ( x , y, v) is elementary in v. By 10.9.1
we find a term A u . L E V ~ ( u ) such that:
Logical Constructions with Higher Reflection
XIV.72]
415
LEV~(O)- 0 and L E V ~ ( m + I ) - {(x,y)" LVk(X,y, LEV~(m))};
finally we put LEV~ "- {(x,y)" 3m. (x,y}TI1LEY~(m)}.
Now we claim"
72.8. LEMMA. Let k > 1.
D
(i) STLR k F Cll(LEV~);
(ii) STLR k F (x,y)TIILEV~ ~ LVk(X,y, LEV~);
(iii) STLR k F (Cank(x)---~(x,x)~llLEY~)A ((x,y)TI1LEY ~---*Cank(y));
(iv)
(v)
STLR k F VxVy((x,y)~llLEY~ A (x,z)qlLEY~---*y - z);
STLR k F CPT*(y, w, g) A (x, y)~IILEV~---+
((gx, z)~I1LEV~ ~ (w, z>r/1LEY~).
PROOF. As to (i)-(ii), STLR k has A0-N-induction, i.e. number-theoretic
induction for formulas built up from atoms of the form Tit, Fit (1 < i < k),
t - s , Nt, by means of standard logical operations. Now Cll(LEV~(y)) is
A o and we can verify with elementary comprehension for classes of level 1
and A0-N-induction that for every m, LEV~(m) is a class of level 1. Then
we apply 10.8-10.9.1.
(iii): the first conjunct is a corollary of (ii), while the second is obtained by
checking VxVy(Ix, y)711LEV~(m)~Cank(y)) by Ao-N-induction on m.
(iv): Ao-N-induction and lemma 72.7.
(v): apply (ii), (iii). O
72.9. DEFINITION. NVk(Z , y, v) is the formula:
(y=0* A (x,O*)~71LEYk) V 3z3w(y=S*w A (x,S z>~llLEVkA (z,w)YlV).
Clearly )~v{(x,y)" Nvk(x, y, v)} is an existential operator; hence elementary
comprehension and 72.8 (i) imply, for k > 1"
STLR k F Cll(V)----~Cll({(x,y > 9NVk(X,y,v)}.
If we recursively define
NEV~(O)- 0 and NEV*k(m+I ) - {(x,y)" Nvk(x,y, NEV~(m)) },
we
can
again
apply
A0-induction
on
N,
in
order
to
check
Cll(NEV~(m)) , for every m. If we argue as in 72.8 above, we conclude:
that
Epilogue: Applications and Perspectives
416
[Ch.14
72.10. LEMMA. Let k > 1.
(i)
STLR k F- Cll(NEV~) ,
where N E V ~ "- {(x,y)" 3m((x,y)r/aNEV~(m))};
(ii)
STLR k F- (x, y)rllNEY~ ~ NVk(X , y, NEV~);
(iii)
STLR k ~ (x,-O*)rllLEY~ --, (x,-O*)rllgEY~;
(iv)
STLR k I- (x,S*Y)rllLEV~ A (y,z)rllgEY~ ~ ( x , S * z ) r l l g E Y ~.
72.10.1. CONVENTION: henceforth we adopt the abbreviations
LEV~(x, y) "- (x, y)~ILEV~;
g E Y ~ ( x , y) "- (x, y)rllgEY~.
72.11. LEMMA (gEY~-induction). If k > 1 and A(x,y) is a Ao-formula of
STLR k (i.e. built up by means of A, ~, Y from atoms of the form Nt,
t - s, Tit , Fit , where 1 <_i < k), STRL k proves:
YxVyYz((LEY~(x, O*)---~A(x, 0")) A (LEYk(X, S'y) A A(y, z)---,A(x, S'z))) ---,
~YuYv(NEV~(u, v)~A(u, v)).
P R O O F . By hypothesis on A, we can apply induction on m:
VuVv((u, v)rllNEV~(m ) ~ A(u, v))
(,)
but ( , ) is immediate under the assumption of the antecedent of 72.11. V1
w73. Proof-theoretic reduction
We now proceed to the
Third Step
As a preliminary move, we introduce a term V(i) for each 1 < i <k, in
STLRk, such that, if we set"
T~(x) "-(x,O)~liV(i) and F * ( x ) "- (x, 1)~hV(i),
73.1.1
P~t "- (T~t V F~t),
73.1.2
r162
r162
then T k and P i are the required interpretations of the truth and
proposition predicates of LTC k within a slight extension of system STLR k.
This extension turns out to be proof-theoretically reducible to the ramified
system RS k of Ch. XI.
We first apply the fixed point theorem for predicates relativized to
level i (use 10.1 and 37.2)"
Proof-theoretic Reduction
XIV.73]
417
73.2. LEMMA. We can find a term V(i), and a formula A(a, u, v) such that
the following conditions on Y(i) , Ti,* F i* (as defined in 73.1.1-73.1.2 above)
hold, provably in STLR k (for k >__1)"
1.
(a, u)~iV(i ) ~ A(a, u, V(i));
2.
if t is a term of the form ( - *xy), ( ( L E ? ) * x y ) , ( i E ? ) * x y ) ,
[T~x], [F~x], for 1 <_ j < i) and A T is the formula x - y
(LEU~(x,y), N E U ~ ( x , y ) , T j x , F i x respectively, for 1 <_ j < i),
then L E V i ( a , t) ~ (T*a ~ A T) A (F*a ~ -~AT);
3.
if m - O, 1,
LEV~(a, VmX) ---+(T*a ~-~VuT*(xu)) A (F*a ~ 3uF*(xu));
4.
L E V i ( a , A g D x y ) ---+(T* a ~-~T* x A T~y) A (F~a ~ F* x V F'y);
5.
LEY~(a, g E G x ) ---+(T~a ~-. F~x) A (F~a +--,T ' x ) ;
6.
if 1 < j < i, then (T~a---+ T*a ) A (F~a ~ F'a).
7.
Let STLR~ be STLR k plus C O N S * ( 1 ) , . . . , C O N S * ( k ) ,
where
C O N S * ( i ) : - VaVuVv((a, u)r]iY(i ) A (a, v)~iV(i ) - . u - v)
( i - 1,... ,k). Then STLR i ~ Vx -~(T~x A F~x).
As to the proof, we simply define truth and falsehood of level i, in such
a way that they are invariant under the computation predicates L E V ~ and
N E V ~ and meet the conditions (2)-(6); we then repeatedly apply 72.8-72.10
and C O N S * ( i ) .
73.3. THEOREM. For each k >_ 1, we can define a translation A~-~(A)k of
LTCk-formulas into STLRk-formulas such that:
LTC k F A implies STLR~ F (A)k.
PROOF. If t is a term of LTCw of arity OB, let t* be the term of STLRk,
obtained, according to definition 72.4 (ii). Then we inductively set"
1.
( t - S)k "--(t* -- s*); ( L E Y ( t , s ) ) k "- LEY~(t*,s*);
( N E Y ( t , s ) ) k "- NEY~(t*,s*);
2.
(Pit)k "-- P*(t*) ( 1 <_ i <_ k) and (Tt)k "-- T~(t*); ( J_ )k "-- (-0- 1);
3.
(VoxA)k "- Vx(A)k; ( V i f . A ) k "- Vy(A)k ; ( B - ~ C)k - ( B ) k - . ( C ) k .
If A is a successor axiom, a special equality axiom SE.1-SE.2 of LTC 0 or
the defining equation for Ind, (A)k is provable by lemma 72.5 and the fixed
Epilogue: Applications and Perspectives
418
[Ch.14
point theorem for operations. If A is any LEV- or NEV-axiom,
STLR k F (A)k is a straightforward consequence of 72.8, 72.10-72.11. If A is
an axiom among PT1-PT4, we apply lemma 73.2 above. As a sample, let
us check (PT.2)k , which has the form:
(LEV~(a, x~y) A P*(x) A (T~(x)~P*(y)))---, P*(a);
(1)
(LEV~(a,x~y) A P*(x) A (T~(x)---,P*(y)))---,
---,(T~a~--~(T~x----~TkY)).
(2)
First of all, observe that lemma 73.2 implies:
LEV~(a,x--~y) ~ (T*a ~ (F~x Y T'y));
(3)
LEV~(a, x~y) ~ (F* a ~ (T~x A F'y)).
(4)
(1)" assume the antecedent of (1) and let ~T~a: then ~F*x A~T~y by (3),
whence T~x A F'y, i.e. F~a by (4).
(2)" again assume the antecedent of (2) with T~a, T~x and by contradiction
--,T~y. By persistence ~T*y, hence F*y (with the assumption P'y). Were
F'x, also F~ x" against consistency (lemma 73.2.7). Hence T'x, i.e. F*a by
(4), and finally F~a (persistence), which contradicts T~a. Conversely,
assume the antecedent of (2) and T~x ~ T*kY together with =F~x. Then by
persistence ~F*x, whence T~x with the assumption P*x and finally T~x by
persistence, i.e. by assumption T~y. The conclusion follows by (3). [3
Final step: concluding the proof of theorem 72.3. We rely upon the proof
theory of chapters X-XI. First of all, let ITS~
be the system IT~ ~ of w50
extended by initial sequents, which correspond to the axioms CONS*(i),
written in the language of I T ~ (1 _< i _< k). They have the form
S-CONS*(i):
~(t, s)~iV(i), ~(t, r)rliV(i), s - r.
By inspection we see:
73.4.1. the embedding theorem 51.6 trivially holds for the pair STLR~ and
I T S ( . ) as well: hence the first system is interpretable in the second with
finite cut rank and length < w2.
73.4.2. The partial cut elimination theorem 55.1 carries over unchanged to
I T S ( . ) , due to the form of S-CONS*(i).
73.4.3. S-CONS(k)is sound for the asymmetric interpretation of I T S ( . )
into RSk, the ramified system of w52; more precisely, we obtain:
~i~+1
~
RSk~ w,
"I -~(t, s)y k~ Y(i), -~(t, r)~lk Y(i), s - r.
(The verification runs by induction on a; if k = 1, as a preliminary step,
one checks the RSk-derivability of sequents of the form:
Perspectives
XIV.74]
419
-~(t, s)r]~ LEV~, ~(t, r ) ~ LEV~, s - r.
In addition, if k - i+1, one needs the derivability of the sequents:
-,(t,
(t,
v(i).
73.4.4. OP + TIop ( < F0) proves the consistency of LTC k.
In fact, any derivation of _k in LTC k is sent into an RS0-derivation of
0 " - 1 " . But this is impossible by w167
provably in OP + TI(a), for a
sufficiently high < P 0. D
The same method might be used to classify the number-theoretic
functions, which are provably total in LTC~, and to obtain additional
recursion-theoretic information. Theorem 72.3 can be inverted as well: as
sketched by Aczel-Carlisle-Mendler(1991), LTC,; interprets Martin-Lhf's
theory TT~ with arbitrary finitely many universes and without wellordering types. It is known that TT,; carries out the well-ordering proof for
F o (see Jervell 1978).
w
Perspectives:
Linguistics
related
work in Artificial Intelligence and Theoretical
We conclude with a short survey of contributions, which point out
applications of type-free systems to areas related with Theoretical Computer
Science, Artificial Intelligence and Theoretical Linguistics. Once more, we
do not offer well-established results, but only a few hints that should
stimulate towards further investigation.
A. Illative Theories of Relations. Plotkin (1990) describes a system, in which
one can formalize ideas from Situation Theory (cf. Barwise-Etchemendy
1987), namely the notions of relation, assignment, predication, slate-ofaffairs (soa, in short), fact. Plotkin's system-which we name PS for
simplicity-is actually a type-free theory, formalized in a language of
expressions with arities (see w
Expressions are meant to (possibly)
describe soas and true soas ( - facts). PS is given in the style of an illative
calculus (recall the system with levels of implications of w
which is
devised to produce two sorts of judgments: 1) E is a fact; 2) E is a soa. Of
course, the rules for PS depend on the basic constructors, which include,
among other logical operations, restriction. The specific feature of PS
consists of extending the basic theory of soas and facts with the notion of
predication, relation, assignment, relation abstraction. It turns out that the
theory derives a fixed point theorem and the undefinability of the notions of
fact and soa, very much in analogy with the well-known results from the
theory of propositions and truth s la Aczel.
From the metamathematical point of view, Plotkin shows that PS can be
420
Epilogue: Applications and Perspectives
[Ch.14
embedded into a conservative first-order standard theory of facts and soas.
This is achieved by expanding the illative language with a new predicate
Fact(x) and then by axiomatizing the properties of Fact(x), as established
by the PS-rules. On the other hand, he also sketches a three-valued logic
approach PS3, which is much in the spirit of the present work; not
surprisingly," the notion of "facticity" becomes internally definable in PS3,
but the predicate of being a fact has to be partial. The interested reader
might try to construct a model for PS or PS 3 within MF-. Of course, one
might wonder whether similar frameworks are useful as foundations of
situation semantics, in alternative to non-wellfounded sets. As far as we can
see, Plotkin's work is a move in this direction.
B. Artificial Intelligence, semantics, linguistics. It is clear from the heading
that the problems one has to cope with, are intertwined with highly
controversial themes: the logic of propositional attitudes, such as "to
know", "to believe", and, more generally, the foundations of property
theory and intensional logic. Consequently, this research trend is extremely
sensitive to the conceptual analysis of the basic notions involved, and to
philosophical discussions. It should be mentioned that there is a clear-cut
distinction between the modal approach, which sticks to the idea that
attitudes are to be rendered by modal operators, and the idea that
propositional attitudes can be expressed by means of predicates of objects
with some kind of logical structure (sentences, propositions in the sense of
Barwise and Etchemendy 1987, i.e. suitable non-wellfounded sets,
representations in the sense of Discourse Representation Theory). Be as it
may, we only refer to contributions that are immediately related to the
technical aspects discussed in this book.
Formally, the underlying problem is to design languages which are
capable of representing, to a certain extent, truth, knowledge, belief, as well
as reasoning about them by "reasoning agents". But the inherent
"reflective" aspect apparently calls for systems, that can consistently live
with circularity and self-reference. This need has naturally led some AIresearchers to consider, as a reasonable tool for the logics of knowledge
representation, formalisms which are primarily languages with selfreferential truth, supported either by a a partial semantics d la Kripke or by
semi-inductive semantics d la Herzberger. There is a hope that a suitable
combination of inductive and semi-inductive methods, and even modal
logic, can lead to a natural integration of self-referential truth, belief and
knowledge predicates, possibly constrained with indexical coordinates,
depending upon the different knowing agents. Typical attempts in the
axiomatic vein can be found in Perlis (1985), (1988), Turner (1990), Davies
(1991).
XIV. 74]
Perspectives
421
Perlis experiments with several systems, which at least include
first-order logic with partial truth axioms s la Kripke-Feferman and a
substitution predicate ensuring self-reference. These systems can be readily
considered as subsystems, say, of MF-(Ch. II). In the context of a theory of
reasoning agents, Perlis proceeds to enrich the ground system with
predicates for knowledge and belief (Know(x), Bel(x)). He then singles out
some options, trying to discover reasonable sets of axioms for Know(x),
Bel(x), also guided by analogies with extant work on provability predicates.
According to Perlis, an advantage of the first-order approach is that no
problem seems to arise with undesired substitution in non-extensional
contexts (e.g. t = s and Know(t) do not imply in general Know(s)).
Turner (1990) develops a systematic proposal for integrating problems,
stemming from knowledge representation and semantic theory, with
applications to linguistics. In contrast to Perlis, Turner somewhat inclines
towards a notion of truth, supported by semi-inductive definitions (see
Ch. XII) and refined with modal operators. Like the present work, selfreference is basically granted by the underlying lambda calculus. The
influence of Aczel's views is important here: propositions are essentially
objects, to be characterized axiomatically (either directly or via truth); in
any case, they should not simply be identified with sentences of any given
formal language; properties are explained away in favour of propositional
functions, defined in the underlined lambda structures. Further
developments are sketched in Davies (1991).
On the side of linguistic applications and the foundations of intensional
logic, we mention an attempt by Chierchia and Turner (1988). The authors
present theories of properties, relations and propositions, that should be so
powerful and expressive to support the semantics of natural languages and
to explain specific linguistic phenomena (e.g. nominalization). Technically
speaking, the theories in question can be considered specializations of the
formalisms in w
they are formulated in a language of expressions with
arities. In particular, there is a universal arity, together with special basic
arities for information units ( = propositional objects),
urelements,
nominalized functions ( i.e. denotations of )~((x)t)), possible worlds and lime
instants. Complex arities are generated by arrow application, while
expressions are inductively defined by means of applications of the standard
logical constructors (including quantification on variables of different sorts),
modal and tense operators, truth. The underlying logic is classical logic,
enriched by Turner's axioms and rules for truth, plus possibly special
axioms and rules for modal and tense operators. The intended
interpretations are suitable expansions of lambda calculus models in the
sense of Aczel, that satisfy the closure conditions required by the axioms.
The authors sketch a formalization of a fragment of English in such a
422
Epilogue: Applications and Perspectives
[Ch.14
framework, thus extending Montague's type-theoretic approach. They also
argue that certain technical features of their model directly correspond to
specific semantic problems. In particular, the distinction between a
propositional function f and its individual correlate )~(f) is motivated:
1) by the need of a reasonable semantic for infinitivals and gerunds;
2) by the search for an explanation of syntactical phenomena (e.g. why in
English verbs sometimes are flected and sometimes not).
w
Sense and denotation as algorithm and value: subs-ming theories of
reflective truth under abstract recursion theory
It is in harmony with the role of recursion-theoretic intuitions in this book
that we conclude by mentioning a recent paper, which leads us again into
the realm of generalized recursion. In a series of papers Moschovakis (see
Moschovakis 1984, 1989a, 1989b) has developed a foundation for the theory
of computation, which is based on the notion of recursive algorithm and
which is sufficiently strong to subsume recursion on finite structures, as well
as generalized recursions on abstract infinite structures.
The starting point is that recursive definitions, as specified by systems
of partial monotone functionals (with respect to the appropriate partial
orderings generated by the relation "to be more defined than"), are taken as
primitive; a crucial and original aspect of Moschovakis's approach is that it
yields, together with the usual denotational semantics, a mathematical
definition of intension for recursive algorithms by means of certain settheoretic objects, the so-called recursors. Recursors are directly related to
the denotations of the (uniquely determined) terms, resulting from an
abstract "compilation process" of the given recursive algorithm. Of course,
this idea requires a formal language of recursion (FLR), in which algorithms
are codified and "compiled" by means of a precise reduction calculus and
for which a unique termination property is established. Syntactically,
recursors are immediately read off from the normal forms of the FLRformalizations of the given algorithms (see Moschovakis 1989a).
A stimulating feature of the theory of recursive algorithms is that the
recursion-theoretic definition of intension can be readily applied to logical
languages and to the modeling of the Fregean notion of sense. The idea is
that the sense of a sentence is essentially the method we follow, in order to
establish the truth value of the sentence itself, and that such a method
reduces to compute a generalized recursive algorithm; but this algorithm
has an associated referential intension, which can be precisely defined in
terms of Moschovakis theory. Thus, if we identify the Fregean notion of
sense with the referential intension, we have a mathematically precise
notion of intensional identity at hand.
XIV.75]
Sense and Denotation
423
The connection with the present work is made apparent by
Moschovakis' extension of lower predicate calculus with reflection, i.e. the
language LPCR. In fact, LPCR easily subsumes Kripke's languages for selfreferential truth and it can be shown that LPCR has enough expressive
power to characterize inductive and hyperelementary sets on a given
structure ~ (see Ch. III, w
Let us briefly and informally sketch
LPCR. First of all, assume we are given a first-order language LPC of a
given relational signature, which includes - and the standard logical firstorder operators (for definiteness, think of the OP-language, formalized
without function symbols). The language of LPCR is obtained form LPC
by first adding a new denumerable list of predicate variables (in each finite
arity, including 0; P, Q, are used as metavariables). P, Q, R are to be
interpreted as partially defined relations on the given interpretation domain;
if P is 0-ary, P plays the role of a propositional variable, whose truth value
is possibly undefined. Then LPC is further extended with a new formula
constructor W H E R E for introducing direct self-reference; formally the
inductive definition of LPCR-formulas also includes the following clause:
if A0, A1,... ,A n are LPCR-formulas, P 1 , ' " , P n are predicate variables,
each Pi has arity ki, u(i) is a string of k i individual variables (1 < i < n),
then the string
A "- A 0 W H E R E {Pl(U(i)) _~ A1,...,Pn(u(n))~
An}
is a formula of LPCR; the bound occurrences of A are the bound
occurrences in the head A o and in the parts A1,...,An, all the occurrences
of P1,...,Pn and the occurrences of the individual variables in each list u(i)
(for 1 < i < k). Note that the list u(i) may be empty, if Pi is 0-ary, i.e. a
propositional variable; this is important to get self-reference at a
propositional level.
On the semantic side, since partial predicates are around, one adopts
Kleene's strong three-valued logic for -~, A, V, in order to evaluate complex
sentences; therefore the corresponding truth functions (for sentential
connectives) and functionals (for individual quantification) are monotone.
The interpretation of the WHERE-constructor is readily explained by fixed
point semantics. To each formula Ai, we can canonically associate a partial
functional F[Ai] which is built up from the monotone partial functions and
functionals corresponding: 1) to logical operators; 2) to the partial
predicates P1,...,Pn and eventually to the basic predicates of LPC. F[Ai]
will possibly depend on the free individual variables of A i and the partial
characteristic functions Pl,'",Pn of P1,...,Pn. Since each FlAil is
monotone, we can apply the Knaster-Tarski theorem and show that there
are simultaneous minimal solutions I(A1),...,I(An) to the system of
equations:
424
Epilogue: Applications and Perspectives
[Ch.14
FI[A1](Pl,..., Pn, a(1)) ~_ Pl(a(1));
o
.
.
.
.
.
.
.
.
.
,
.
.
o
~
Fn[An](Pl,..., Pn, a(n)) ~_ Pn(a(n)).
The truth value of A := AoWHERE{PI(U(1))~_ AI, ...,Pn(u(n)) ~- An} is
then essentially given by computing F[A](I(AI),...,I(An),a(O)).
In the frame of LPCR, one verifies that the usual self-referential
constructions of the Liar, the SameSayer, etc., can be directly carried out;
moreover, Kripke's partial self-referential truth can be adequately defined
by simply writing down the appropriate WHERE-formulas by means of the
operator of Ch. II, w7: LPCR is able to model reflective truth. In addition,
let us mention a central result of Moschovakis (1990): the notion of
intensional identity, as defined for LPCR-formulas, is decidable (though at
least hard as the isomorphism problem for finite graphs). Thus, given two
arbitrary expressions of LPCR, we can effectively compute their referential
intensions and decide if they coincide. On the other hand, the problem of
axiomatizing LPCR and the corresponding logic of intensions is left open.
As to the relation with the present framework, the WHERE-construct
can be simulated, say in MF-, by choosing a suitable simultaneous fixed
point operator, which is obtained by well-known
techniques (e.g. see
Barendregt 1984, p.142). However, we warn that in M F - we only get an
unfaithful modeling of WHERE. In fact, like the more general formalism
FLR, LPCR is based on a primitive idea of recursive computation, not
involving a sequential order in the computation of the fixed points
(Moschovakis's theory is motivated by applications to the analysis of
concurrency).
We wish to conclude with a note of self-criticism. If we look backwards,
in the light of the previous considerations, we have to admit that the
systems based on combinatory logic are at the same lime too simple and
strong for capturing important distinctions one would like to have at hand
for computational applications. This can be partly remedied by adding
more structure at the ground level, as suggested by LTC-systems, and there
seems to be room for experimentation and research here. On the other hand,
at a higher level, the consideration of LPCR suggests that there are
important aspects, which are left untouched by the present work. One
might try to integrate the systems of reflective truth with subtler tools
coming from Moschovakis (1990), that could be relevant toward a deeper
foundation of a theory of abstraction and properties, and the mathematical
treatment of intensional constructions. We hope that these notes will
encourage someone to new investigations.
BIBLIOGRAPHY
Abbreviations for journals and book series, to be used below:
JSL
JPL
ZML
Journal of Symbolic Logic;
Journal of Philosophical Logic;
Zeitschrift ffir Mathematische Logik und die Grundlagen der
Mathematik;
MLQ
Mathematical Logic Quarterly (formerly ZML);
AML
Archive for Mathematical Logic (formerly Archiv f/Jr
Mathematische Logik und die Grundlagenforschung);
APAL Annals of Pure and Applied Logic (formerly Annals of
Mathematical Logic);
SLNM Lecture Notes in Mathematics, (Springer, Berlin Heidelberg);
SLNCS Lecture Notes in C o m p u t e r Science (Springer, Berlin Heidelberg);
TCS
Theoretical C o m p u t e r Science;
N D J F L Notre Dame Journal of Formal Logic.
ACKERMANN, W.
(1941) System der typenfreier Logik I, Forschungen zur Logik und zur exakten
Wissenschaflen, Heft 7.
(1950) Widerspruchsfreier Aufbau der Logik. Typenfreies System ohne Tertium
non datur, JSL, 15, 33-37.
(1952-53) Widerspruchsfreier Aufbau einer typenfreier Logik I, II, Mathematische
Zeitschrif2, 55, 364-384, 57, 155-166.
ACZEL, P.
(1977) The strength of Martin-LSf's intuitionistic type theory with one universe,
in: S. Mietissen and J. V~n~nen (eds.), Proceedings of the Symposiums in
Mathematical Logic in Oulu 1974 and Helsinki 1975, (Report no.2, Dept.of
Philosophy, University of Helsinki), 1-32.
(1977a) An introduction to inductive definitions, in: J. Barwise (ed.), Handbook o]
Mathematical Logik (North Holland, Amsterdam), 867-895.
(1980) Frege structures and the notions of proposition, truth and set, in:
J. Barwise, H.J. Keisler, K. Kunen(eds.), The gleene Symposium (North
Holland, Amsterdam), 31-59.
ACZEL, P., P. CARLISLE and N. MENDLER
(1991) Two frameworks of theories and their implementation in Isabelle, in:
426
Bibliography
G.M.Plotkin et al. (eds.), Logical Frameworks (Cambridge University Press,
Cambridge), 5-39.
A CZEL, P. and S. FEFERMAN
(1980)
Consistency of the unrestricted abstraction principle using an intensional
equivalence operator, in: J.P. Seldin and R. Hindley, (eds.), To H.B. Curry:
Essays on Combinatory Logic, Lambda Calculus and Formalism (Academic
Press, New York), 67-98.
ACZEL, P. and W. RICHTER
(1974)
Inductive definitions and reflecting properties of admissible ordinals, in:
J.E.Fenstad and P.G.Hinman(eds.), Generalized Recursion Theory (North
Holland, Amsterdam), 301-381.
APT, K. and W. MAREK
(1974)
Second order arithmetic and related systems, APAL, 6, 177-229.
ASPERTI, A. and G. LONGO
(1990)
Categories, Types and Structures (Foundations of Computing Series,
M.I.T. Press, Cambridge Mass.).
BARENDREGT, H.
(1973)
Combinatory logic and the axiom of choice, Indagationes Mathematicae, 35,
203-221.
(1984)
The Lambda Calculus: its Syntax and Semantics (North Holland,
Amsterdam).
BARWISE, J.
(1975)
Admissible Sets and Structures (Springer, Heidelberg) .
BARWISE, J., (ed.)
(1977)
Handbook o] Mathematical Logic (North Holland, Amsterdam).
BARWISE, J. and J. ETCHEMENDY
(1987)
The Liar. An Essay on Truth and Circularity (Oxford University Press,
Oxford).
BARWISE, J., R. GANDY and Y. N. MOSCHOVAKIS
(1971)
The next admissible set, JSL, 36, 108-120.
BARWISE, J. and J. SCHLIPF
(1975)
On recursively saturated models of arithmetic, SLNM 498, 42-55.
Bibliography
427
PEALER, G. and U. MONNICH
(1989)
Property theories, in : D.Gabbay and F.Guenthner (eds.), Handbook o]
Philosophical Logic IV (Reidel, Dordrecht), 133-251.
BEESON, M.
(1985)
Foundations o] Constructive Mathematics (Springer, Berlin).
BEHMANN, H.
(1931)
Zu den Widerspriichen der Logik und der Mengenlehre, Jahresberichte der
Deutschen Mathematischen Vereinigung, 40 , 37-48.
(1937)
The paradoxes of logic, Mind, 46,218-221
(1959)
Der Priidikatenkalkiil mit limitierten Variablen : Grundlegung einer
natiirlichen exakten Logik, JSL, 24, 112-140.
BELNAP, N.
(1977)
A useful four-valued logic, in: M. Dunn and R. Epstein (eds.), Modern Uses
o] Multiple-Valued Logic (Reidel, Dordrecht), 9-37
BIRKHOFF, G.
(1967)
Lattice Theory (American Mathematical Society, Colloquium
Publications, vol.XXV, 3rd edition Providence, R.I).
BISHOP, E.
(1967)
Foundations o] Constructive Analysis (Mc Graw Hill, New York).
BOYD, R., G. HENSEL and H. PUTNAM
(1969)
A recursion-theoretic characterization of the ramified analytical hierarchy,
Trans. Amer. Math. Soc., 141, 37-62.
BUCHHOLZ, W.
(1991)
Notation systems for infinitary derivations, AML, 30, 277-296.
(1992)
A simplified version of local predicativity, in: P. Aczel, H. Simmons,
S. Wainer, (eds.), Proo] Theory (Cambridge University Press, Cambridge),
115-147.
BUCHHOLZ, W., A. CICHON and A. WEIERMANN
(1994)
A uniform approach to fundamental sequences anf hierarchies, MLQ, 40,
273-286.
BUCHHOLZ, W., S. FEFERMAN, W. POHLERS and W. SIEG
(1981)
Iterated Inductive De]initions and Subsystems o] Analysis: Recent Proo]
Theoretical Studies, SLNM 897.
Bibliography
428
BUCHHOLZ, W. and S. WAINER
(1987)
Provably computable functions and the fast growing hierarchy, in:
S. G. Simpson (ed.), Mathematical Logic and Combinatorics (AMS,
Contemporary Mathematics, vol.65), 179-198.
BULL, R. and[ K. SEGERBERG
(1983)
Basic Modal Logic, in: D. Gabbay and F. Guenthner (eds.), Handbook of
Philosophical Logic I (Reidel, Dordrecht), 133-188.
BURGE, T.
(1979)
Semantical paradox, Journal o] Philosophy, 76, 169-198.
BURGESS, J.
(1986)
The truth is never simple, JSL, 51,663-681.
CANTINI, A.
(1980)
A note on three-valued logic and Tarski theorem on truth definitions,
Studia Logica, 39, 405-415.
(1982)
Non-extensional theories of predicative classes over PA, Rendiconti del
Seminario Matematico dell' Universit~ e del Politecnico di Torino, 40, 47
(1983)
(1983a)
(1985)
(1985a)
(1987)
-79.
A note on the theory of admissible sets with E-induction restricted to
formulas with one quantifier, Bollett. Un. Mat. Italiana, 6 (2-b), 721-737.
Proprieta' e Operazioni (Bibliopolis, Naples).
On weak theories of sets and classes which are based on strict 1-I~-reflection,
ZMLG, 31,321-332.
A note on a predicatively reducible theory of elementary iterated induction,
Bollett. Un. Mat. Italiana, 4-b, 413-430.
Su una teoria generale delle proprietY, basata su schemi di comprensione
iterati e privi di tipi, in: R. FERRO, A. ZANARDO (eds.), Atti degli
Incontri di Logica Matematica, vol.3 (Cluep, Padova ), 53-57.
(1988)
Two impredicative theories of sets and properties, ZMLG, 34, 403-420.
(1989)
Notes on formal theories of truth, ZMLG, 35, 1989, 97-130.
(1990)
A theory of formal truth arithmetically equivalent to ID 1 , JSL, 55 , 244
(1991)
-259.
A logic of abstraction related to finite constructive number classes, AML,
(1992)
31,69-83
Levels of implications and type-free theories of partial classifications with
approximation operator, ZMLG, 38, 107-141.
(1993)
Extending first order theory of combinators with self-referential truth, JSL,
(1995)
58, 477-513.
Levels of Truth, NDJFL, to appear
Bibliography
429
CHANG, C.C. and H. J. KEISLER
(1990)
Model Theory (North Holland, Amsterdam; third edition)
CHIERCHIA, G. and R. TURNER
(1988)
Semantic and property theory, Linguistics and Philosophy 11,261-302.
CHURCH, A.
(1941)
The Calculi of )~-Conversion (Ann. Math. Studies, no.6, Princeton
University Press, Princeton, N J).
CHURCH, A. and J. B. ROSSER
(1936)
Some properties of conversion, Trans. Amer. Math. Soc., 39, 472-482.
CONSTABLE, R., S. F. ALLEN, H. M. BROMLEY, W. R. CLEVELAND, J. F.
CREMER, R.W. HARPER, D. J. HOWE, T. KNOBLOCK, N.P.
MENDLER, P. PANANGADEN, T. SASAKI and SMITH, F.S.
(1986)
Implementing Mathematics with the Nuprl Proof Development System
iPrentice-Hall, Englewood Cliffs N J)
COQUAND, T.
(1985)
Une Th~orie des Constructions (Th~se de troisi~me Cycle, Universit6 Paris
VII).
CROSSLEY,
(1969)
J.N.
Constructive Order Types (North Holland, Amsterdam).
CURRY, H.B.
(1942)
The inconsistency of certain formal logics, JSL, 7, 115-117.
CURRY,
H. and R. FEYS
(1958)
Combinatory Logic I (North Holland, Amsterdam).
CURRY, H., R. HINDLEY and J. SELDIN
(1972)
Combinatory Logic II (North Holland, Amsterdam).
DAVIES,
N.
(1991)
A first-order theory of truth, knowledge and belief, in: J.van Eijck, (ed.),
Logic in AL (Springer, Berlin), 170-179.
DERSHOWITZ,
(1990)
N. and J. P. JOUANNAUD
Rewrite systems, in: J. van Leeuwen (ed.), Handbook of Theoretical
Computer Science, chapter VI (North Holland, Amsterdam).
430
Bibliography
DERSHOWITZ, N. and M. OKADA
(1988)
Proof theoretic tecniques for term rewriting theory, 3rd Annual Symposium
on Logic in Computer Science (Edinburgh, Scotland, July 1988, IEEE
Computer Science Press, Silver Spring, MD), 104-111.
DEVLIN, K.
(1977)
Constructibility, in: J. Barwise (ed.), Handbook o] Mathematical Logic
(North Holland, Amsterdam), 453-502.
ENGELER, E.
(1981)
Algebras and combinators, Algebra Universalis, 13, 389-392.
(1988)
Cumulative logic programs and modelling, in: F. R. Drake, J. K. Truss
(eds.), Logic Colloquium '86 (North Holland, Amsterdam), 83-93.
ERSHOV, Y. L.
(1977)
Model C of partial continuous functionals, in: R. O. Gandy, M. Hyland
(eds), Logic Colloquium '76 (North Holland, Amsterdam), 455-467.
(1985)
]~-predicates of finite type over an admissible set, Algebra i Logika, 24, 499
-536 (English translation: Algebra and Logic, 1986, 326-351)
FEFERMAN, S.
(1964)
Systems of predicative analysis I, JSL, 29, 1-30.
(1968)
Systems of predicative analysis II, JSL, 33, 193-200.
(1974)
Non-extensional type free theories of partial operations and classifications I,
in: J. Diller and G. H. Miiller (eds.), Proof Theory Symposium, SLNM 500,
73-118.
(1975)
A language and axioms for explicit mathematics, in : SLNM, 450, 87-139.
(1977)
Theories of finite type related to mathematical practice, in: J. Baxwise
(ed.), Handbook of Mathematical Logic (North Holland, Amsterdam), 913
-971.
(1977a) Categorical foundations and foundations fo category theory, in : R. Butts,
J. Hintikka (eds), Logic, Foundations o] Mathematics and Computability
(Reidel, Dordrecht), 149-169.
(1979)
Constructive theories of functions and classes, in: D. Van Dalen, M. Boffa
(eds.), Logic Colloquium '78 (North Holland, Amsterdam), 159-224.
(1982)
Iterated inductive fixed point theories: application to Hancock's
conjecture, in: G.Metakides (ed.), Patras Logic Symposion (North Holland,
Amsterdam), 171-195.
(1984)
Towards useful type-free theories I, JSL, 49, 75-111.
(1985)
Intensionality in mathematics, JPL, 14, 41-55.
(1985a) A theory of variable types, Revista Colombiana de Matematicas, vol.XIX,
95-105.
(1990)
Polymorphic typed lambda-calculi in a type-free axiomatic framework, in:
Bibliography
431
Logic and Computation (Contemporary Mathematics 104, AMS, Providence
101-136.
Reflecting on incompleteness, JSL, 56, 1-49.
Logics for termination and correctness of functional programs II. Logics of
strength PRA, in: P. Aczel, H. Simmons, S. Wainer eds. Proof Theory
(Cambridge University Press, Cambridge), 195-225.
Logics for termination and correctness of functional programs, in:
Y. N. Moschovakis (ed.), Logic from Computer Science (Springer, Berlin
RI),
(1991)
(1991a)
(1992)
-Heidelberg).
FEFERMAN, S. and W. SIEG
(1981)
Iterated inductive definitions and subsystems of analysis , in:
W. Buchholz, S. Feferman, W. Pohlers and W. Sieg, SLNM 897, 16-77.
FITCH, F. B.
(1936)
A system of formal logic without an analogue to Curry W-operator, JSL, 1,
(1942)
(1946)
(1948)
(1950)
(1951)
(1963)
(1964)
(1967)
(1980)
92-100.
A basic logic, JSL, 7, 105,-114.
Self-reference in philosophy, Mind, 55, 64-73.
An extension of basic logic, JSL, 13, 95-106.
A demonstrably consistent mathematics I, JSL, 15, 17-24.
A demonstrably consistent mathematics II, JSL, 16, 121-124.
The system CA of combinatory logic , JSL, 28, 87-97.
Universal metalanguages in philosophy, Review of Metaphysics, 17, 396-402
A complete and consistent modal set theory, JSL, 32, 93-103.
A consistent combinatory logic with an inverse to equality, JSL, 45, 529-543.
FITTING, M.
(1981)
Fundamentals of Generalized Recursion Theory (North Holland,
Amsterdam).
(1986)
Notes on the mathematical aspects of Kripke's theory of truth, NDJFL,
27, 75-88.
FLAGG, R.
(1987)
n-continuous lattices and comprehension principles for Frege structures,
APAL, 36, 1-16.
FLAGG, R. and MYHILL, J.
(1987)
Implications and analysis in classical Frege structures, APAL, 34, 33-85.
(1987a) An extension of Frege structures, Lecture Notes in Pure and Applied
Mathematics, 106, (M.Dekker, New York ), 197-217
(1989)
A type-free system extending (ZFC), APAL, 43, 79-97.
432
Bibliography
FRAASSEN, B. van
(1968)
Presupposition, implication and self-reference, Journal o] Philosophy, 65,
135-152
(1970)
Inference and self-reference, Synth~se, 21,425-438.
FREGE, G.
(1903)
Grundgesetze der Arithmetik I, II (Olms, Hidesheim, 1962).
(1975)
Funktion, Begri]], Bedeutung (G.Patzig, ed., Vandenhoek & Ruprecht,
GSttingen)
(19842) Nachgelassene Schriften (H. Hermes, F. Kambartel, F. Kaulbach, eds.,
F.Meiner, Hamburg).
FRIEDMAN, H.
(1969) Bar Induction and HI-cA, JSL, 34, 353-362.
(1975)
Some systems of second order arithmetic and their use, Proc. 1974
International Congress of Mathematicians, Vancouver, vol.1, 235-242.
(1976)
Uniformly defined descending sequences of degrees, JSL, 41,363-367.
(1976a) Subsystems of second order arithmetic with restricted induction I, II
(abstracts), JSL, 41, 557-559.
(1980)
A strong conservative extension of Peano arithmetic, in : J. Barwise,
H. J. Keisler, K. Kunen (eds.), The Kleene Symposium (North Holland,
Amsterdam), 113-122.
(1981)
On the necessary use of abstract set theory, Advances in Mathematics, 41,
209-280.
FRIEDMAN, H., K. MC ALOON and S. G. SIMPSON
(1982)
A finite combinatorial principle equivalent to the 1-consistency of
predicative analysis, in: G. Metakides (ed.), Patras Logic Symposion
(North Holland, Amsterdam), 197-230.
FRIEDMAN, H. and M. SHEARD
(1987)
An axiomatic approach to self-referential truth, APAL, 33, 1-21.
FRIEDMAN, H., S. G. SIMPSON and R. SMITH
(1983)
Countable algebra and set existence axioms, APAL, 25, 141-181.
GALLIER, J.
(1991)
What's so special about Kruskal's theorem and the ordinal F 0 ? A survey
of some results in proof theory, APAL, 53, 199-260.
GANDY, R.
(1974)
Inductive definitons, in: J.E.Fenstad, P.G.Hinman (eds.), Generalized
Recursion Theory (North Holland, Amsterdam), 265-299.
Bibliography
433
GILMORE, P. C.
(1974)
The consistency of partial set theory without extensionality, in:
T. Jech(ed.), Axiomatic Set Theory II (Proceedings of Symposia in Pure
Mathematics, 13, A.M.S, Providence R.I), 147-153.
(1980)
Combining unrestricted abstraction with universal quantification, in:
J.Seldin, R.Hindley(eds), To H.B. Curry: Essays on Combinatory Logic,
Lambda Calculus and Formalism (Academic Press, New York), 99-123.
(1986)
Natural deduction based set theories : a new resolution of the old
paradoxes, JSL, 51,393-411.
(1990)
How many real numbers are there ?, (Amsterdam, preprint)
GIRARD, J. Y.
(1982)
A survey of 1-II-logic, in: L. Cohen, J.Los, H. Pfeiffer, K. P. Podewski (eds.),
Logic, Methodology and Philosophy of Science VI (North Holland,
Amsterdam), 89-107.
(1987)
Proof Theory and Logical Complexity (Bibliopolis, Napoli, 1987, vol.1).
GIRARD, J. Y. and J. VAUZEILLES
(1981)
Les premieres r~cursivement inaccessible et Mahlo et la th~orie des
dilatateurs, AML, 24, 167-191.
GODEL, K.
(1944)
Russell's mathematical logic, in: P.A.SCHILPP (ed.), The Philosophy of
Bertrand Russell (Northwestern University Press, Evanston, Ill.), 125-153.
GORDEEV, L.
(1989)
Proof theoretic analysis: weak systems of functions and sets, APAL, 38,
1-121.
GUPTA, A.
(1982)
Truth and paradox, JPL, 11, 1-60
HALBACH, V.
(1994)
A system of complete and consistent truth, NDJFL (to appear)
HAREL, D. and D. KOZEN
(1984)
A programming language for inductive sets and applications,
Inform. Control, 63, 118-139.
HARPER,R., F. HONSELL and G. PLOTKIN
(1987)
A framework for defining logics, Proceedings of the 2nd annual conference
on Logic in Computer Science (IEEE Computer Press, Washington DC)
434
Bibliography
HARRISON, J.
(1968)
Recursive pseudowellorderings, Trans. Amer. Math. Soc., 131, 526-543.
HAYASHI, S. and H. NAKANO
(1988)
PX: A Computational Logic (Foundations of Computing Series, MIT Press,
Cambridge Mass.).
HERZBERGER, H.
(1982)
Notes on naive semantics, JPL, 11, 1-60.
HILBERT, D. and P. BERNAYS
(1938)
Die Grundlagen der Mathematik I, II (Springer, Berlin-Heidelberg).
HINDLEY, R. and J. SELDIN
(1986)
Introduction to Combinators and A-Calculus (London Mathematical
Society Student Texts 1, Cambridge University Press, Cambridge).
HINMAN, P. G
(1978)
Recursion-Theoretic Hierarchies (Springer, Berlin Heidelberg).
JAGER, G.
(1980)
Beweistheorie von KPN, AML, 20, 53-64.
(1982)
Zur Beweistheorie der Kripke Platek Mengenlehre fiber den naturlichen
Zahlen, AML, 22, 121-141.
(1984)
A version of Kripke Platek set theory which is conservative over Peano
arithmetic, ZMLG, 30, 2-9.
(1984a) The strength of admissibility without foundation, JSL, 49, 867-879.
(1986)
Theories ]or Admissible Sets (Sibliopolis, Naples).
(1987)
Induction in the elementary theory of types and names, SLNCS, 329.
J)kGER, G. and W. POHLERS
(1982)
Eine beweistheoretische Untersuchung von A12-CATBI und verwandter
Systeme, Sitzungsberichten der bayerischen Akademie der Wissenscha]ten,
1-28.
JAGER, G. and T. A. STRAHM
(1994)
Totality in applicative theories, APAL, to appear
JERVELL, H.R.
(1978)
Constructive universes I, in: G. H. Miiller and D. Scott, (eds.), Higher Set
Theory, SLNM 69, 73-98.
Bibliography
435
JOCKUSCH, C.jr. and S. G. SIMPSON
(1975)
A degree theoretic definition of the ramified analytical hierarchy,
APAL,10, 1-32.
KELLEY, J. L.
(1955)
General Topology (Van Nostrand, New York).
KLEENE, S. C.
(1958)
Extension of an effectively generated class of functions by enumeration,
Colloq. Math., 7, 67-78.
(1959)
Quantification of number theoretic functions, Compositio Mathematica, 7,
67-78.
(1952)
Introduction to Metamathematics (Wolters-Nordhoff and North Holland,
Amsterdam).
KOYMANS, K.
(1984)
Models of the Lambda Calculus (Math. Center, Tracts, no.9, Amsterdam).
KREISEL, G.
(1967)
Informal rigour and completenes proofs, in: I.Lakatos (ed.), Problems in the
Philosophy of Mathematics (North Holland, Amsterdam), 138-171.
(1968)
A survey of proof theory I, JSL, 33, 321-88.
(1971)
Some reasons for generalizing recursion theory, in: R. O. Gandy,
C. M. Yates (eds.), Logic Colloquium '69 (North Holland, Amsterdam),
139-198.
KREISEL, G. and G. SACKS
(1965)
Metarecursive sets, JSL, 30, 318-338.
KRIPKE, S.
(1975)
Outline of a theory of truth, Journal of Philosophy, 72,690-716.
LAMBEK, J. and P. SCOTT
(1986)
Introduction to Higher Order Categorical Logic (Cambridge University
Press, Cambridge).
LONGO, G.
(1983)
Set theoretical models of X-calculus: theories, expansions, isomorphisms,
APAL, 24, 153-188
LORENZEN, P. and J.MYHILL
(1959), Constructive definitions of certain analytic sets of numbers, JSL, 24, 37-49
Bibliography
436
MARTIN, R. M. (editor)
(1984)
Recent Essays on Truth and the Liar Paradox (Oxford University Press,
Oxford).
MARTIN-LOF, P.
(1984)
Intuitionistic Type Theory (Bibliopolis, Naples).
MARZETTA, M.
(1993)
Universes in the theories of types and names, in: E.BSrger et alii (eds.),
Computer Science Logic '9~, SLNCS 702, 340-351.
MC-GEE, V.
(1985)
How truthlike can a predicate be? A negative result, JPL, 14, 399-410.
(1991)
Truth, Vagueness, and Paradox (Hackett, Indianapolis-Cambridge).
MEYER, A.
(1982)
What is a model of the lambda calculus, Inform. Control, 52, 87-122.
MINTS, G.
(1975)
Finite investigations of transfinite derivations, J.Soviet Math., 10 (1978),
548-596 (English translation from : Zap. Nauchn. Semin. LOMI, 49).
MINARI, P.
(1987)
Some aspects of Kripke's theory of truth, (unpublished notes, Florence)
MONTAGUE, R.
(1963)
Syntactic treatment of modality, with corollaries on reflection principles
and finite axiomatizability, Acta Philosophica Fennica, 16, 153-167.
MOSCHOVAKIS, Y. N.
(1974)
Elementary Induction on Abstract Structures (North Holland, Amsterdam).
(1984)
Abstract recursion as a foundation of the theory of the algorithms,
Computation and Proof Theory, SLNM 1104, 289-364.
(1989a) A formal language of recursion, JSL 54, 1216-1252.
(1989b) A mathematical modeling of pure, recursive algorithms, in : A. R. Meyer
and M.A.Taitslin, Logic at Botik '89, SLNCS 363, 208-229.
(1990)
Sense and reference as algorithm and value, Logic Colloquium '90, to
appear.
MUSIL, R.
(1913)
Der mathematische Mensch, in: R.Musil, Gesammelte Werke, Bd.8
(Rowohlt, Reinbek bei Hamburg, 1978), 1004-1008.
Bibliography
437
MYHILL, J.
(1950)
A system which can define its own truth, Fund.Math., 37, 190-92.
(1972)
Levels of implications, in: A. Ross, R. Barcan Marcus, R. M. Martin (eds.),
The Logical Enterprise (Yale University Press, New Haven).
(1984)
Paradoxes, Synth~se, 60 , 129-142.
NEPEIVODA, N.
(1973)
A new concept of predicative truth and definability, English translation
from: Matematischskie Zametki, 13, 735-745.
NORDSTROM, B., K. PETERSSON and J. SMITH
(1990)
Programming in Martin-L'5]'s Type Theory (Oxford University Press,
Oxford).
ODIFREDDI, P. G.
(1989)
Classical Recursion Theory (North Holland Amsterdam).
PARSONS, C.
(1972)
On n-quantifier induction, JSL, 36, 466-482.
PERLIS, D.
(1985)
Languages with serf-reference I. foundations, Artificial Intelligence 25,
301-322.
(1988)
Languages with selfreference II: Knowledge, belief and modality, Artificial
Intelligence, 34, 179-212.
PLOTKIN, G.
(1990)
An illative theory of relations, in : R. Cooper, K. Mukai, J.Perry (eds.),
Situation Theory and Its Applications (CSLI, Stanford 1990), 134-146.
POHLERS, W.
(1989)
Proof Theory. An Introduction, SLNM, 1407.
POINCARE, H.
(1913)
Derni~res Pens~es (Flammarion, Paris).
RATHJEN, M.
(1991)
(1992)
Proof-theoretic analysis of KPM, AML, 30, 377-403.
Fragments of Kripke-Platek set theory with infinity, in: P. Aczel,
H. Simmons, S. Wainer (eds.), Proof Theory (Cambridge University Press,
Cambridge), 251-273
438
Bibliography
RESSAYRE, J.P.
(1982)
Bounding generalized recursive functions of ordinals by effective
functors: a complement to the Girard theorem, in: J.Stern, Logic
Colloquium '82 (North Holland, Amsterdam), 251-280.
RICHTER, W.
(1971)
Recursively Mahlo ordinals and inductive definitions, in: R. O. Gandy,
C. M. Yates (eds.), Logic Colloquium '69 (North Holland, Amsterdam),
273-288.
ROGERS, H. jr.
(1967)
Theory of Recursive Functions and Effective Computability (Mc Graw-Hill,
New York).
SCHELLINX, H.
(1991)
Isomorphisms and nonisomox:phisms of graph models, JSL, 56, 227-249.
SCHONFINKEL, M.
(1924)
Uber die Bausteine der mathematischen Logik, Mathematische Annalen, 92,
305-316.
SCH[)TTE, K.
(1960)
Beweistheorie (Springer, Heidelberg).
(1977)
Proof Theory (Springer, Heidelberg).
SCHWICHTENBERG, H.
(1971)
Eine Klassifikation der e0-rekursiven Funktionen, ZMLG, 17, 61-74.
(1977)
Some applications of cut elimination, in: J. Barwise (ed.), Handbook of
Mathematical Logic (North Holland, Amsterdam), 867-895.
(1992)
Proofs as programs, in: P. Aczel, H. Simmons, S. Wainer (eds). Proof
Theory (Cambridge University Press), 79-113.
SCOTT, D.
(1975)
Combinators and classes, in: C. Bbhm (ed.),)~-Calculus and Computer
Science, SLNCS 37, 1-26.
(1975a) Some philosophical issues concerning theories of combinators, ibid. 346-366.
(1976)
Data types as lattices, SIAM Journal of Computing, 51, 522-587.
(1980)
Lambda calculus: some models, some philosophy, in: J. Barwise, H. J.
Keisler, K. Kunen, (eds.), The Kleene Symposium (North Holland,
Amsterdam), 223-265.
(1980a) Relating theories of A-calculus, in : J. P. Seldin and J. R. Hindley, (eds.),
To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and
Formalism (Academic Press, New York), 403-450.
Bibliography
(1982)
439
Domains for denotationM semantics, SLNCS 140, 577-613.
SELDIN, J. R.
(1976)
Recent advances in Curry's program, Rendiconti del Seminario Matematico
dell' Universith e del Politecnico eli Torino, 35,77-88.
(1980)
Curry's program, in : J. P. Seldin and J. R. Hindley (eds.), To
H.B. Curry:Essays on Combinatory Logic, Lambda Calculus and Formalism
(Academic Press, New York), 3-33.
SHOENFIELD, J.
(1967)
Mathematical Logic (Addison Wesley, Menlo Park).
SIMPSON, S. G.
(1982)
Set theoretic aspects of ATR0, in : D. van Dalen, D. Lascar, J. Smiley
(eds.), Logic Colloquium '80 (North Holland, Amsterdam), 255-271.
(1984)
Which set existence axioms are needed to prove the Cauchy- Peano
theorem for ordinary differential equations?, JSL, 49, 783-802.
(1987)
Subsystems of Z 2 and reverse mathematics, in: G.Takeuti, Proo] Theory
(North Holland, Amsterdam, 2nd edition), 432-446.
(1987)
Subsystems of Second Order Arithmetic, typewritten notes, chapters 1-5;
6-9.
SMITH, J.
(1984)
An interpretation of Martin-LSf's type theory in a type-free theory of
propositions, JSL, 49, 730-753.
SMULLYAN, R.
(1961)
Theory of Formal Systems (Annals of Mathematical Studies, Princeton
University Press, Princeton N J).
TAIT, W.
(1968)
Normal derivability in classical logic, in: J. Barwise, (ed.), The Syntax and
The Semantic of Infinitary Languages, SLNM 72 , 204-236.
TAKEUTI, G.
(1975)
Proo] Theory (North Holland, Amsterdam; 2nd edition 1985).
(1978)
Two Applications o] Logic to Mathematics (Princeton University Press,
Princeton, NJ and Publicat. Math.Soc.Japan) .
TALCOTT, C.
(1992)
A theory of program and data type specification, TCS 104, 129-159.
440
Bibliography
TARSKI, A.
(1956)
Logic, Semantics, Metamathematics: Papers from 1923 to 1938 (J.Woodger,
ed., Clarendon Press, Oxford).
(1965)
A simplified version of predicate logic with identity, AML 7, 61-79.
TROELSTRA, A. S.
(1973)
Metamathematical Investigation of Intuitionistic Arithmetic and Analysis,
SLNM 344.
TROELSTRA, A. S. and D. van DALEN
(1988)
Constructivism in Mathematics I, II (North Holland, Amsterdam).
TURNER, R.
(1987)
A theory of properties, JSL, 52,455-471.
(1990)
Truth and Modality for Knowledge Representation (Pitman, London).
(1990a) Logics of truth, NDJFL, 31,308-329.
VEBLEN, O.
(1908)
Continuous, increasing functions of finite and transfinite ordinals,
Trans. Amer. Math. Soc., 9, 280-293.
VISSER, A.
(1984)
Four-valued semantics and the liar, JPL, 13, 181-212.
(1989)
Semantics and the liar paradox, in: D. Gabbay and F. Guenthner (eds.),
Handbook of Philosophical Logic IV (Reidel, Dordrecht), 617-706.
WEYL, H.
(1918)
nas Kontinuum (Von VeNt, Leipzig).
WHITMAN, P.M.
(1943)
Splitting of lattices, American Journal of Mathematics, 65, 179-196.
WOODRUFF, P.
(1984)
Paradox, truth and logic. Part I: paradox and truth, JPL, 13,
213-233.
INDEX
The bibliography is not covered by the following index.
abstraction
explicit 60, 77
extended 222
internal 353
lambda (A-) 15, 30, 34, 36,
121, 219
local, 222
second order 28, 33, 65,
239, 248
arithmetical
comprehension 240, 243
formula 239
operator 66, 341
set 240
operation 2, 44
arity 403
principle 1, 56
Asperti 34
unrestricted 1
axiom(s)
Ackermann 217
bounded quantifier 25O
Aczel 5, 6, 8, 43, 53, 66, 163,
choice 128
217, 230, 254, 380, 402
admissible hull 125, 131
class N-induction 66, 352
combinatory 139
admissible ordinal 230
completeness 54
analysis
arithmetical 240
hyperarithmetical 240
predicative 7, 241
connection 220
consistency 49, 290 304, 314
enumeration 27, 32, 91,
151, 152, 163
recursive 241
application (operation) 14, 29,
extensional choice 129
dependent choice, 129
extensionality (for
operations), 16
30, 33, 121
approximation
a t h 181
axioms 99, 104
extensionality (for sets) 199
independence 164
operation (~') 98
N-induction 15
theorem 99, 109
level 220
limit 220
Apt 264
arithmetic
Peano 14, 27, 28, 39, 66, 70,
local consistency 219
local N-induction 219
90, 91, 101, 131, 151, 241,
local truth 219
243, 338, 355,356, 373, 380,
localization 220
383, 411
Loeb 389
primitive recursive 39, 40
natural number (also
Index
442
number-theoretic, Peano)
15, 39, 219 , 290, 304
Buchholz 278, 279, 312,335,
338, 373
negative soundness 220
Burge 218
ontological 139
Burgess 205, 206
operational 15, 219, 289, 314
pair 139
lambda (,~-) 13, 16, 34, 35
pairing 15
persistence 220
1r-axioms 104
positive soundness 220
potential completeness 220
projectibility 219
property N-induction 66, 357
restriction 49, 50, 54, 209
schema,
see
calculus
schema
T-axioms 49, 50
union 139
sequent 286, 288, 289, 290,
291,294, 303-305, 314
351,358, 361,369
canonical
enumeration 28
expression 407
model, realization 406, 410
term 407
term (for logical operators) 45
Cantini 101,144, 178, 199, 337, 355,
356, 373.
Cantor 42, 243
Baire 243
bar induction, 241
Cauchy 243
Barendregt 17, 18, 20, 21, 22,
characterization theorem 376
25, 28, 32, 34, 38, 39, 424
Barwise 131, 207, 230, 232,
233, 235, 256
Church 22, 25, 26
Church's thesis 20, 27, 28, 32
Church-Rosser theorem 13,
22, 23, 26, 38, 41, 151,
152, 162, 165, 169
class 55, 56, 58-64, 71, 73, 74, 75
explicit 79
basis
change of, 81
for a topology, 114, 117, 121
Beeson 27, 59, 75
Behmann 151, 162, 163
Belnap 55
of level i 218, 219, 222, 224,225
class compactness 104, 105, 111, 122
Bendixson 243
class finiteness 114
Bernays 238
closed
Birkhoff 11, 190, 192
Bishop 75
bounded formula, see formula
bounded level quantifier, 237,
249, 250, 287
bounded number quantifier
39, 64, 239
bounded set quantifier, 138,
198, 230, 238
boundedness theorem 147
Boyd 262
unbounded set 269
F-closed 47, 48, 50, 179-181
(I>0-closed 357, 358
club, s e e closed unbounded set
CODE 34O
coherent
element 204
poset 182
coinduction 177, 178, 207
combinator 14, 16, 20, 21, 27
K, S 14, 15
Index
paradoxical 17
443
Crossley 259
combinatorial operation 60, 77
Curry 2, 17, 21, 25, 43, 47, 53, 151
currying 403
combinatory algebra 2, 3, 5, 6,
13, 16
cut elimination theorem
for OP cr 327
combinatory logic 13, 14, 15,
16, 18, 21, 25, 34
complement 59
partial 324
complete
primitive recursive 347
for the calculus ~ 364
coherent poset (ccpo) 182
Schfitte's 319
lattice 183, 185, 186
Tait's 320
set 47, 208, 357
weak 295,301
completeness
cut rule 155, 291,315
~op-Completeness
9
363
w-completeness 376
Dalen, D. van 27
complexity
decomposition theorem 393
logical 10, 257, 264-68, 281
285, 293, 295,313, 327,
336, 370
abstract 153
n-complexity 303, 305,307,
324
definition by cases 14, 15
analytical 203
denotation 422
elementary 58, 60, 61, 62,
67,
223, 229, 242, 243, 253
second order 63, 178, 200, 239
type-free 44
computation relations 407
computation triples 408
concatenation 244, 272,341
connection axioms 220
conservation, s e e theorem
consistency axiom,
see
axiom
consistency lemma 322, 363,
consistency theorem 156
47,
29, 30, 31, 36, 103,
109, 112, 118-120, 144
continuous operator,
see
set
dependent choice,
see
schema
derivability relation for
I T ~ 3O5
OP cr 327
RSn316
]Eo o 158
STLR 291
STLR n 297
constructive completeness 115
27,
dense,
STLR c~ 294
179
Constable 402
continuity
of level n, 157
definability theory 85
comprehension
consistent set
Dedekind 152, 243
deducibility
see
operator
conversion 16
Dershowitz 283
Devlin 232
diagonalization 76, 77
domain operation 59
covering 147
creative set (property)
derivative 270
71-72
dual extensional membership
132, 133
444
Index
effective
inseparabilty 108
operator 118
elementary
atom 46
extensional 2 (operation) 110, 128
2-extensional 129
extensionality axiom,
see
axiom
extensionality
for properties, classes 73- 74
for sets 135, 199
comprehension 58
extensional 130
family
formula 57
J~-normal 387
predicate 60
RS- 116
elimination lemma 296
embedding
of admissible sets 135
theorem 190
encoding 68
of logical operators 43, 80
Engeler 13, 28, 33
enumeration
axiom 27, 126, 144, 151, 152,
163
theorem 32, 42
envelope 88
equality 14, 15
definitional 403-407
extensional 28, 60,
70, 108, 110,
intensional 429, 431
level 220
pointwise 113, 121,
set-theoretic 132, 133
Feferman 2, 5, 6, 7, 44, 50, 55,
57, 58, 59, 66, 75, 80, 151,
152, 162, 163, 165, 166, 199,
203, 217, 226, 238, 241, 243,
253, 257, 273, 278, 402,421
field (of a relation) 126
first recursion theorem
(analogue of) 104, 120
Fitch 2, 5, 43, 217, 379, 380,
386, 388, 396
Fitch's internal logic 388
Fitch's models 386
Fitch's theory 388
Fitting 70, 71, 110
fixed point
axiom 50, 66
complete 47, 179
consistent 47, 179
dual 181
P-, 48
Ershov 104, 114, 121
intrinsic 186,
exact representation 108, 122
largest 50, 181
expansion (operation) 60, 78
least 48, 88, 181
explicitly CL-continuous 118
maximal 186
explicitly open 116
~0-' 357
exponential (w-) 274
~1-' 375,
exponentiation 61
~2-' 387.
model 179, 186, 352,358, 379
extensional
choice, s e e choice
equality, s e e equality
fixed point theorem
for operations 16
membership 134
for monotone operations 112
model 34
for predicates or
extensional 1 (for properties) 70, 110
properties 63
Index
fixed point theory 50
Flagg 33, 39, 103, 151, 152,
156, 162,
445
28, 121
grounded element 204
Gupta 394
formula
analytical 199
arithmetical 239
Halbach 385
bounded 39, 40, 138, 230, 239
Harrison 258
Harper 402
elementary 57
Hayashi 402
elementary extensional 130
Hensel 262
u-free 199
Herbrand 365,367, 372
operative 64
Herzberger 380, 390, 394
positive elementary 88
hierarchy
quasi-elementary 57
stratified 199
T-negative 53,
ramified 261, 263, 267
Veblen 270
Hilbert 15, 238
T-positive 53, 220
Hindley 18, 21, 22, 34
Fraassen, van B. 352,357
Hinman 91, 230, 233, 237, 256
Frege 1, 2,422
Frege structures 2, 43, 53
Friedman 7, 217, 227, 238,
hyperarithmetical analysis 240
hyperelementary set 87
241, 243, 258, 261, 282-83
Honsell 402
hyperjump 237, 258
379, 380, 383, 384, 394
function
continuous on ordinals 269
C_-increasing 49, 181, 208
increasing on ordinals 269, 391
normal 269
uniformly A1- , ~1-' 231
fundamental sequence 275, 276
Gale 243
Gallier 282, 283
Gandy 91, 261, 262
Gilmore 73, 75, 198
Girard 7, 9, 253, 271
global consistency 222
GSdel 1, 60
GSdel numbering 42, 68, 82,
336, 338
Gordeev 73, 74
graph model 13, 28, 34
graph of a continuous operator
ideal completion 115
implication
levels of, 151, 152
R- 153
incoherent element 204
independence 45, 80, 81, 164,
292,320, 407, 413
induction, s e e axiom, schema
inductive
definition 63-67, 85, 87,
232, 257, 351,355
model 43, 58, 85, 86,
88, 91, 97
set 88
intensional equivalence 151,
162, 174
interpretation
asymmetric 7, 299, 311,
324, 325, 336
provability 369, 370, 372
intersection 59
Index
446
Levy 206
generalized 61
intrinsic element 204
liar (sentence) L 52, 204, 374, 424
intrinsic fixed point 186
local consistency 219
inversion lemma 234, 295,317,
local truth axioms 219
345,363
local truth lemma 370
involution 182
LOG- rule 353
iteration A-, 390
logic
Iterationsprinzip 217, 229
combinatory 13-16
deontic 386
J~,ger 27, 74, 144, 217, 226, 337
external 180, 399
Jockusch 262
join 58, 59, 79, 80, 223, 225,
Fitch's internal 388
Friedman-Sheard 384
226, 228, 229, 253
internal 180, 380, 394, 396, 400
modal 396
Kalmar 343
M-logic 376
Kelley 114
Kleene 3, 7, 20, 28, 80, 91, 98,
type free 6, 151, 174
104, 244, 245, 246, 248,
258, 262,343, 423
Knaster 113, 358, 424
KSnig 243
Koymans 34
Kreisel 4, 114, 238
Kripke 5, 6, 44, 139, 177, 178,
186, 203, 231,351,420,
421,423, 424
Kripke's classification 203
Kripke-Platek set theory 139
Kruskal 282
logical consequence 351,355,379,
384, 386, 388, 398
logical theory of constructions
401-403, 411
Longo 34, 35
Lorenzen 217
Lusin 243
Marek 262
Martin-LSf 217, 226, 402,403,
405,407, 419
Marzetta 74
Mc Gee 7, 380, 381,383
Meyer 17, 28
label 339
Minari 74, 112, 122, 194, 196
lambda calculus 16, 17, 18
minimalization 20
lattice 179
complete 183
Mints 312,335,338
model
involutive 182
/3-model 258, 262
non-modular 192
closed term 26
lazy evaluation 407, 408
level
Doo 35
Engeler's D-, 33
axioms 220, 290
Fitch's, 379, 386
induction, 293
fixed point, see inductive
lowering lemma 328
inductive, see inductive
of implication 151
open term 26
of t r u t h 215
w-model (w-standard) 47, 49,
Index
447
180, 194, 230, 237, 258, 358,
number-theoretic 20
388
of ordinal arithmetic
274-275,
Pw 28-32
recursion-theoretic 215, 230
recursive graph RE, 28, 32
recursively s a t u r a t e d 90
selection 127
~- 107
operator (formula) 64
semi-inductive 394
closure 29, 36
supervaluation 357
continuous 28
CL-continuous 117
monotonicity
of deducibility 153
ECL-continuous 118
of ordinal assignement 316, 362
effectively continuous 28
monotone,
see
operation, operator
Montague 383
Moschovakis 6, 8, 64, 80, 85, 88, 98,
131, 206, 207, 262,401,
RS-continuous 119
elementary 64
existential 65
monotone 29, 88, 120, 144, 207,
422 425
Musil 149
Myhill 1, 4, 6, 39, 44, 68, 71,
103, 109, 112, 147, 151,
152, 153, 156, 160, 162, 217
Myhill's theorem 71
Myhill-Shepherdson's theorem
(analogue of) 112
234, 371,387
non-monotone 394
ordering
connected (linear) 126, 227
directed u n b o u n d e d partial 220
onw
20
ordinal
admissible 230
closure 253, 392
Nakano 402
n a t u r a l ordinal sum 318
notation 272
nesting p r o p e r t y 277
n u m b e r 145
constructive 64
NordstrSm 402
of predicative analysis F 0 271
n o r m a l form
projectible 231
Cantor 271
recursively inaccessible 231
for c o m b i n a t o r y terms 22
recursively Mahlo 255
no solution l e m m a 367
Odifreddi 82
operation
basic (for predicates) 59
choice 128
combinatorial 60
extensional 2 110
K a l m a r elementary 343
lattice-theoretic 181
monotone 62, 109-110, 112,
113, 117
stabilization 392
pair 61, 77
axiom 15, 139
ordered 14
pairing
combinator 21
function 42
surjective 31
paradox
Curry's 53, 151
Gordeev's
74
448
Index
Russell's 56
acceptable 126, 259
locally decidable 259
paradoxical
combinator 17
element 204
unbounded 259
P u t n a m 262
p a r a m e t r i z a t i o n 56
quasi-well-ordering (qwo) 259
P a r k 37
Parsons 40, 41, 42
Peano 243,
see
quasi-elementary formula 57
arithmetic
period 392
periodicity 392
Perlis 420, 421
persistence
axioms 220, 290, 296, 304, 314
l e m m a 298, 321
ramified hierarchy 261-263
bounded 267
n-rank 313
Rathjen 256
recursion
arithmetical transfinite 240
Plotkin 13, 28, 402,419
A I - ' E l - ' 232 , 235-37
formal language of, 422
Pohlers 9, 217
on natural numbers 19
Poincar~ 7 5 , 3 4 9
on ordinals 48, 146
power set 10, 62, 161
special transfinite 260
Plato 11
prewellordering 104
primitive recursion 19, 20, 27,
39-40, 404, 409
W k - , 302
recursion theorem
first 120
second 343
principle
abstraction 56
recursive functions
choice 126
partial 19, 244-45
construction 80
primitive 19, 20, 27, 39,
join 58
meta-loeb 373
40, 93, 101, 246, 303,
338, 348
reducibility 244
provably 40, 91, 101
reflection 224
representability of, 20
CL-reflection 106
recursor 19, 146, 260, 261
:~-reflection 371
reducible formula 317
process 390
m-reducible property 71
product
reducibility,
see
principle
cartesian 61
reduction l e m m a 295,318, 346
generalized 58
reduction relation 22, 23
progressive property 126, 227,
259, 354
proposition 51
infinitary 165, 166-169
reduction theorem 109
reflection 224
propositional function 51
~t~-reflection 371
provability interpretation 370
repetition rule 338
pseudo-well-ordering (pwo)
representable
126, 259
function 20
Index
449
internal abstraction 353
set 88
level transfinite induction 293
representation theorem for
local abstraction 222
extensional operations 117
CL-reflection 106
Ressayre 253
~verse mathematics 217, 238, 241
REFL + 237
Rice 44, 68, 70, 71, 72, 110, 116,
second-order comprehension
200, 239
122
Rice's theorem 71
soundness 54
Rice-Shapiro
Tarski's 53, 385
transfinite induction 259,
family 116
278, 354
theorem 110, 122
E-transfinite induction 139
Richter 230, 232, 253, 254, 255
Rogers 70, 71, 82, 258
transfinite recursion 227-229
Russell 56, 70, 90, 151, 162, 215,
Turner's 396
type-free abstraction 56
222, 223, 351
satisfaction 68, 69
Schfitte 7, 9, 217, 241, 243, 257,
Schellinx 34
269, 270, 271, 273, 278, 280,
schema
312,316, 319
ATR 240
Schwichtenberg 302,304, 312,335,
338, 402
bar induction 241
bounded collection 139
Scott 5, 13, 17, 28, 32, 33, 34, 37,
bounded complete induction
104, 114, 217, 410
139
bounded separation 139
choice 33
A-comprehension 108
A l-comprehension 240
H~-comprehension 241
Nl-dependent choice 240
elementary choice 130
Scott's extension theorem, 32
Scott topology 28, 104, 114
section 90
Seldin 18, 21, 22, 34
selection 127
semi-inductive
definition 390
model 394
sense 422
separation 98, 108,
CL-, 107
elementary comprehension 58
elementary dependent choice
sequent calculus, 286, 303-304,
314, 361
130
explicit abstraction 60
extended abstraction 222
set
admissible 125, 137
generalized coinduction 207
arithmetical 240
generalized induction 87
bounded (of ordinals) 269
Herbrand 372
closed (of ordinals) 269
N-induction 15, 50
F-closed 47
=t( + )-N-induction 27
90-closed 357
Index
450
coinductive 89
supervaluation model 357
complete 47
Suslin 80, 98
consistent 47
~-definable 262
Tait 286, 289, 314, 320, 324,
El- , Al-definable 231
A-dense 393
Tait's 2nd cut elimination 320
333, 348
F-dense 179
Takeuti 9, 238, 243
O0-dense 357
Talcott 402
02-dense 387
Tarski 37, 44, 53, 65, 70, 113, 125,
hyperelementary 89
201, 215,352,358, 385,412,
inductive 89
424
iterative 63
tautology lemma 292, 295,320, 363
representable 88
term model,
Shapiro 110, 116, 117, 122
see
model
theorem
Shepherdson 103, 109, 112
approximation 99
Shoenfield 14, 246, 376
boundedness 147
Sierpinski 243
cardinality 191
Simpson 217, 238, 241, 243, 262
characterization 376
Smullyan 109
conservation 101,335
soundness 153
decomposition 393
formalized :}r_, 370
embedding 190
positive, negative 220
A + - 221
fixed point for operations 16
~c~-' 160
splitting pair 195
stabilization theorem 391
stably inside 390
stably outside 390
n-stage 313
Stewart 243
Strahm 27
subsequence relation 244, 341
substitution
closure 289
instance 289
lemma 292,316, 344
substitutivity 292, 295,320
subsystems of second order
arithmetic 238-239
sum
direct 61
generalized 58
fixed point for predicates 63
generalized induction 87
internal N-induction 368
Kleene basis 246
Knaster-Tarski 113
Levy absoluteness 206
Myhill-Shepherdson 112
perfect set 206
periodicity 392
reduction 107
representation 119
Rice 71
Rice-Shapiro 110
separation 107, 108
stabilization 391
Suslin-Kleene 80, 98
transfinite induction 354
tree 246
uniform ordinal
comparison 94
upper bound 328
Index
451
for set 198, 239
theory
admissible set 139
Vaught 201
minimal frame MF 50
Vauzeilles 253
of operations OP 15
Veblen 269, 270
Visser 194, 390
prewellordering P W 106
t r u t h with levels TL 219
VF 356
topology
class 114
positive information 28
RS-topology 117
translation 140, 174, 201, 242,
247, 305, 372,413, 417
lemma 247
transpose 60
tree, recursive wellfounded 245
Troelstra 27
T-rules 361
truth
reflective, self-referential 2,
5, 6, 7, 43, 44, 50, 51, 85,
103, 104, 120, 125, 151, 177,
178, 180, 196, 198, 206, 215,
216, 217, 218, 220, 223, 230,
249, 257, 258, 285, 286, 303,
311,351,379, 394, 401,403,
412, 420, 422,423, 424, 425
stable 394, 395, 396, 399
Turner 380, 394, 396
type 402, 403, 406
finite 64, 65, 75
Ulm 243
uniform ordinal comparison 94
ungrounded element 204
union 71
generalized 61
universe 61, 226
unparadoxical element 204
variable 9
individual 14
for levels 218
weakening 292, 316, 345, 363
well-founded,
tree
see
well-ordering 258-259
predicative 269, 277
Weyl 83, 213, 215, 217, 225, 229,
257, 260
Weyl's principle
229
Zorn's lemma 187
This Page Intentionally Left Blank
LIST OF SYMBOLS
Part I lists the abbreviations designating formal systems, arranged in order
of appearance. Part II contains abbreviations for axioms, axiom schemata
and rules, while Part III contains basic abbreviations and symbols.
In parts II-III, the list is arranged per chapters and, within each chapter, in
order of appearance. We give the page number of the first occurrence of the
each symbol we consider.
I. Formal Systems
PC, I, 15
OP, I, 15
classical predicate logic
theory of operation
O P - , I, 16
OPA-, I, 17
CL, I, 21
PA, I, 27, 40
PRA, I, 27, 39
....
without N-induction
O P - b a s e d on A-calculus
pure combinatory logic
Peano arithmetic
primitive recursive arithmetic
PAl, I, 4O
M F - , II, 43, 50
MF, II, 5O
NMF, II, 54
ID1, II, 66
Peano arithmetic based on El-induction
minimal framework without N-induction
MF with full induction
neutral minimal framework
fixed point theory of elementary inductive
definitions
MF with class N-induction
MF with property N-induction
pure property theory
MFc, II, 67
MFp, II, 67
PT, II, 77
PW c ( P W - , PWp), IV, 104, 105
KPU(op), V, 139
Ec~, VI, 158
F ~- n t, VI, 158
BLc, VI, 163
MFS-, VII, 199
TL ( T L - ) , VIII, 219, 220
TLR, VIII, 224
T L R - , VIII, 224
TLR*, 250
ATR0, VIII, 241
MF c ( M F - , MFp) +approximation axioms
admissible set theory above combinatory logic
Myhill's system with levels of implication
formal deducibility with levels of implication
Behmann's logic with class-N-induction
minimal framework with sets
theory of truth with levels (without
N-induction)
theory of truth with levels and reflection
TLR without N-induction reflection
TLR plus axioms ONT+BLQ
arithmetic transfinite recursion
Symbols
454
a L C A o, ZLAC o
1-Ii-CA o, a~2-ca0 , VIII, 241
basic subsystems of 2nd order arithmetic
MFR(p), IX, 278
STLR, X, 289
MF c plus RAM(a, p) for each a < F 0
sequent calculus for truth with levels
STLR ~176X, 294
infinitary STLR
STLRn, X, 297
sequent calculus for truth up to level < n
with bounded level quantifiers
X, 297
I T n~176X, 304
union over STLRn, n E
infinitary sequent calculus for truth up
RSn, XI, 314
to level n
ramified system for truth of level n
STLR,
OP ~176XI, 327
OP based on w-logic
VF , XII, 352
basic non-reductive theory for self-referential
VF c (VFp), XII, 352
V F - + class (resp. property) N-induction
VF0, XII, 355
V F - in the language of pure combinatory logic
truth without N-induction
ID 1 (acc)' XII, 356
theory of accessibility inductive definitions
V F H - , VFHc~ VFH p, XII, 372
IL, XIII, 38O
extensions of V F -
FSL, XlII, 384
IFT, XIII, 388
F T ( F T - ) , XIII, 388
LIS, XIII, 396
LES, XlII, 400
internal non-reductive T-logic
Friedman-Sheard system
internal Fitch's logic
Fitch's theory (without N-induction)
internal axioms for stable truth
LTCw, XIV, 411
external axioms for stable truth
logical theory of constructions (without
proposition and truth predicates)
LTC with propositions and truth of
LPCR, XIV, 423
lowest predicate calculus with reflection
LTC0,
XlV, 4os
arbitrary finite level
II. Axioms, rules and other symbols
Chapter 1
COMB, 15
combinatory logic
PAIR, 15
NAT, 15
pairing
natural numbers
NIND, 15
number-theoretic induction schema
Ext op' 16
extensionality for operations
MS.I-MS.4, 17
Meyer-Scott axioms
CT, 2O
Church's thesis
EA, 27
enumeration axiom
Symbols
NIND for positive existential formulas
3(+)-NIND, 27
ACN, 33
axiom of choice restricted to N
ACN! , 33
comprehension for operations on N
EI-IND , 39
NIND for El-formulas
Chapter 2
T.1-T.5, 49
axioms for reflective truth
RES, 49
CONS, 49
COMP, 54
restriction axiom
consistency axiom
AP, 56
abstraction principle
completeness axiom
EC, 58
elementary comprehension
J, 58
join principle
P-NIND, 66
property N-induction
CL-NIND, 66
class N-induction
CP, 8O
construction principle
Chapter 3
GID, 87
generalized induction schema
lr, 98
approximation operation
HAX, 100
7r-axioms (or approximation axioms)
Chapter 5
choice axiom for operations on V
extensional choice axiom
extensional dependent choice axiom
ACv(oP) , 128
Ext-AC, 129
Ext-DC, 129
EAC, 130
EDC, 130
elementary choice schema
elementary dependent choice schema
Chapter 6
Hyp/Tnd, }
Lift, D, N
axioms for Myhill's system
159
Eq, K, S
IA, EA, I~A, E~A,
/
I V, E v , 1--1v , E-~ v , Red~ 159-160
logical rules for Myhi11's system
/
V v ; IV, EV, I~V, E~V
I n D , E n D , 160
I n ' D , E n i D , 160
E.I-E.7, 164
)
rules for level n implication
rules for level n negated implication
axioms for Behmann's logic
455
Symbols
456
Chapter 7
Set.l-Set.3, 199
set axioms
R, 199
"anti-cantorian axiom"
GID ^, 2O7
generalized coinduction principle
Chapter 8
LIND, 219
local N-induction
PRO J, 219
projectibility axiom
REF, 224
reflection principle
LIM, 226
limit axiom for universes
WP, 229
Weyl's principle
~ - C A , 239
~-comprehension schema for analysis
a~-DC, 240
~-dependent choice schema
BI, 241
bar induction
RPC, 244
reducibility principle for classes
ONT, 25O
ontological axiom
BLQ, 250
bounded level quantifier axiom
Chapter 9
RAM(p, a),
IU(~), 278
278
existence axiom of bounded ramified hierarchy
transfinite induction for classes of U up to /~
Chapter 10
TI(lev),
290, 293
Level induction
( ^ ), ( v ), 290
logical rules
(Vx), (3x), (Vi), (3j), 291
quantifier rules
cut rule
(Cut), 291
(w), (3~), 294
(v)b, (~)b, 297
infinitary level quantifier rules
(N), ( ~ N ) , 305
rules for N
bounded level quantifier rules
Chapter 11
LOG, 314
OPER, 314
PERSij , 314
CONSi, 314
FIX i, 314
INIn, 315
logical axioms
(T a + l ) ( - ~ T a + l ) , 315
ramified successor rules for T of level n
operational axioms
persistence axioms
level i consistency
fixed point axioms for level i
initial axioms for level n
Symbols
(T-LIMa), (F-LIMa), 315
ramified limit rules for T of level n
Chapter 12
T-elem, T-out,
}
T-univ, T-log, T-imp 352
T-rep, T-cons
WF( -~ ).1, WF( ~ ).2, 356
T-Herb, 372
I-CL-NIND, 372
I-NIND, 372
T+-elem, T+-elim, ~ }
T+-univ, T+-log
372
T+-imp, T+-rep
T+-cons, T(T---*) ~ 372
T-schemata
axioms for the largest -~-wellfounded
part
Herbrand's T-schema
internal class-N-induction
internal N-induction
strengthened T-axioms and rules
strengthened T-axioms and rules
Tax-imp, T-uniVax )
Chapter 13
T-intro, T-elim,
~T-intro, ~T-elim} 388
T-negT, 388
T+-negT, 389
T-Rcomp }
T-S4comp 389, 396
T-S5comp
T-rules
T~TA---~TA
T(T~TA--~TA)
Turner's schemata
Chapter 14
SE.1-2, 409
LE.1-LE.3, 409
NEV.1-NEV.2, 409
PT.I-PT.4ij, 411
special axioms
lazy evaluation axioms
number evaluation axioms
axioms for propositions and truth
457
Symbols
458
III. O t h e r S y m b o l s
Introduction
E[x " - t], F V ( E ) , 9
~, W, 9
~(X), CZ, 10
S I- A, lO
Jtt~l:A, 10
F I X ( r , ~ ) , 48
0(.;1~), 48
O(21~, ~), 48
Prop(x), 51
A ^, 54
a:::~b, 54
aC:~b, 54
Chapter 1
{ x ' A } , 55
77,77,55
N,14
K , S , 14
Cl(x),55
(--,--),14
(--)1'(--)2
14
t + l , 14
( . . . ) , (...)k, 15
'
)~xt, 15
n,15
FP, 16
R N, 19
Vn, 3n, 19
V n < m , 3n<m, 2o
CR, 23
R E D , 23
~-- n' 23
C T M , T M , 26
en, 28
Pw, 28
RE, 28
FUN(a), 28
CL, 55
V xriy, 3xrly, 5s
E ( a , f ) , 58
II(a, f ) , 58
N,-,59
N, 59
a - - b, 60
a - e b , 6o
[a---,b], 61
V, 61
a|
61
I x y A ( x , y), I(A), 63
IA, 66, 67
Sat(x, y), 69
Chapter 3
C l o s A ( - ), 87
GRAPH(F), 2s
ENV(~,S),
it]]p, 30
HYP(~I~, S), 89
I N D ( .flt~, S ) , s 9
D M, 34
Doo, 34
Chapter 2
[A], 45
ID, N E G , T R , 45
ALL, A N D , OR, 45
F ( S ) , 47
SEC(.~, S),
s8
90
lal,92
7tax, 98
Chapter 4
x < zY, x < zY, 105
~a, 105
RD(x, y), lO7
Symbols
CSP(x, y), 107
SEP(x, y), 108
ER(y, z), 109
459
Chapter 7
COMP(Jtt~),
b is extensional2, 110
f is extensional I , 110
/ ( f ) , 112
E ( b ) , 113
V(e), 114
Cl-Zf, 114
ECL-OPEN, 116
R S , 116
EFF, 118
ECL, 118
179
C O N S ( ~ ) , 179
F I X ( ~ ) , 179
FIXcs(Jfi~ ), 180
FIXcp(A[~ ), 180
]~I', 180
tg~, 180
S d, 181
UP(S), DOWN(S),
D(~), 181
[.JC,
I-It,182
INT(.A,),
186
P(.At), 186
Chapter 5
MAX(Jfi~), 187
Gr(31~), 204
P a r a d o x ( . . & ) , 204
-~ w' 126
5(a, ~ ~), 12r
Ext2( f ), 128
2-Ext( f ), 129
AD(U), 132
U-AT, 132
U-SET, 132
I n t r ( J ~ ) , 204
Chapter 8
"~V, 218
i0, i 1, . . . . 218
LT, 218
- - U , ~ U, 133
m
x C uY, x C uY,
134
Vx C y, 3x C y, 138
R e ( f ) , 147
~,--/,218
Tit, Fit, 218
tr]is, t-~is, 218
Cli(t),218
A +, 218
Chapter 6
R, R(i), 222
FFRa,
153
x~:~i y, 222
Adr(R),
153
Univ(y),
TR(y, A, -~ w, z),
rt
A-
B, 162
Funcl(f),
:::~, 166
RT(r), 167
LR, 16s
--~ 169
:::>c~' 169
C~ ~
226
T I ( -~ w, b), 22r
RD,154
D , 157
166
227
La, 230
~1 (L~), A l(Lc~), 231
/~-~-, 232
Ta, 232
t , 232
IN, 232
v(~),
235
Ct, 237
181
Symbols
460
NF(A),
"~2' 239
A0, ~10, II 1, 239
~,
293
C~#j3 , 295 (see ch.ll)
II~, II1, ~ , 239
S T L R ~ 1 7 I6- ~ F, 294
"~'n* ' 297
T Pe , 245
A[m,n], rim, n],
W(X),245
c~ F, 299
~Jc~(P), 245
LevPar(F), 299
Chapter 9
f: CL~CL,
[ r I, 299
ffi~( w k ) , 302
259
aj~a, 262
POS n and NEGn,
W~k, 262
Kn(A),303
en, 306
GO(E),
264
U"(z)
267
Cg
Chapter ll
f" ~ ~ ~, 269
E X , 269
~'n, r' 312
fix(f),
T~(t), Fa(
269
Lev(B), 313
Stn(A),313
270
Rn(A), 313
Ac~ , 313
CO, 27O
r o, F ~ , 270
~'Y, 270
f ( t l , s 1) . . . . .
CN,
RSn~
f ( t n,sn), 272
272
, 272
t[,~], 276
Good(%P(z)), 278
IU(fl),
, 313
toC~s, t~us , 313
E f ix( f ), f' ,269
r
303
278
Chapter 10
~V+~ 288
A+,E,
II, A o, 287
Lc( A ), 288
rk(A), 288
Ti_Clause(t), Fi-Clause(t ), 289
p F~ 316
~#j3, 318
A[fl,7], 324
O P ~ F- p
F~ 327
TI(~), 335
TI(
< 7), 335
T I o p ( < ol), 335
[ E] , 336
Dimk(d, [A1, ~r), 336
Truen(FA]), 336
[e](x), 338
RF(f) , END(f), 339
LAT(I), A F ( / ) , 339
DEP(I), 339
CODE, 340
Symbols
OT*,
341
Length(s),
C h a p t e r 14
341
DEn(f),342
OB, 403
BOOL , 403
LC(DER(f)),342
f t - pc~ F, 344
CF( f ), 347
(~----~fl), ( O q . . . O~n)---+Ol , 403
_J_, 403
~TC,
9
C h a p t e r 12
IMPLY, 352
WF( -~ ), 354
XII-A, 357
SENT(JtI~), 357
F I X o ( . A g ), 357
Ate(a),
Eats(a),
Sentcc(a),
359
ON, 360
Ord(x), 360
AD, 360
TI(ON, B),
E-
E ' : or, 4o5
[E]o. , 406
m o . , 406
N ( x ) , 409
360
Nat(x), 410
EqNat(x, y), 41o
C h a p t e r 13
383
T h oo(.J~ ),
403
(xy)f(x, y), 404
LEV, 404
NEV, 404
Pi(x), T, 404
Pa, Deeidenat, 404
Spread, Decide, 404
inl(x), inr(x), Ind, 404
-- o" 406
}__p
a F=~/k, 361
J(X),
461
383
ko(A), 383
Th(,At~), 383
Diag(~), 386
.At-NOR, 3s6
X I I - 2 A, 387
(I)2 ( X ) , 387
N~(.AI~), 387
limin f , 39o
In(A, X ) , In(X),
390
Out(A, X), Out(X),
Stab(A, X), 390
Unstab(A, X), 390
Con f (X), 393
Cycle(X), 393
Init(X), 393
390