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
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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 _
(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
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.
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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,
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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);
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(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]
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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)}).
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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:
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-~(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}.
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[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),
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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-:
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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)).
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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").
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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).
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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).
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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);
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(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
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[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
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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 -~(
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);
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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 .
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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
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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~)).
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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<
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
]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
1.3
b E O ( ~ ) and c is not in p-form b
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
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
b
]S,~] b is in S-form and c is in ~-form: (Vc'<< c)(PRD(b) <_c').
b
b
(vb' << b)(3c' << c)(b' < c'). b
(3b' << b)(Vc' << c)(b' < c') b
(vb'<< b)(Vc' << c)(b' < c') b
(~b' << b)(~c' << c)(b' < c') b
(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)),
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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
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[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,
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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
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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.
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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
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[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
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;
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[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:
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[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
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[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
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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))).
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[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;
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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;
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(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 _
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
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[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).
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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)
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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).
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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
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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)).
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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.
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[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).
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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.
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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.
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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))).
(,)
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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
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)).
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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)
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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=(
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.
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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 _
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,).
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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].
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(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};
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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).
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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).
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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;
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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
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(.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
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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
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(.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
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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:
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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
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.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
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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).
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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
.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
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[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).
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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
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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
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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.
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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.
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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.
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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-;
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(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
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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.
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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
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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);
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(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
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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-
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(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).
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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.
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[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
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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).
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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)
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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
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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
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[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).
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[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 ,
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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).
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[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
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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.
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[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).
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(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.
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(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
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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
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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:
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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
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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
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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:
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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 _
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
For a full proof, the reader is sent to Richter-Aczel (1974), while a sketch is
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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~
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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 ;
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( 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
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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
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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.
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[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.
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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)),
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[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.
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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.
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[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
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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
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[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
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{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
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[Ch.8
R - {n E w" (VG E ww)(3m E w)(
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 --
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
(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
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[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
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[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 "_
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
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[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~
(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,~
~.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.
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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
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-,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
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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
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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
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( # 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
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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.
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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.
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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
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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 _ 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).
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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
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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
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 ?
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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- <
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 _
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
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
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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
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[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).
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[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
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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
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[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.
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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
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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).
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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
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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.
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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
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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
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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
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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
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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~).
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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
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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~;
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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
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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;
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(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~
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
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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
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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~' 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.
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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
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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).
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[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
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[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.
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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.
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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;
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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.
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(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
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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
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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.
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(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
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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
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
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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.
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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
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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)};
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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).
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Conversely, if a E X(a+r there is some ~ < or+CA, such that a E X(~) for every /3 satisfying ~
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:
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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
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(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).
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(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
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
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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
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
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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:
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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.
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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.
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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
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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);
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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:
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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.
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(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
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(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.
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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
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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);
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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:
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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).
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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));
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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.>; (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);
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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:
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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
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[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
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)"
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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
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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.
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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
