Ordinal Analysis with an Introduction to Proof Theory [1st ed.] 9789811564581, 9789811564598

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Logic in Asia: Studia Logica Library Series Editors: Fenrong Liu · Hiroakira Ono · Kamal Lodaya

Toshiyasu Arai

Ordinal Analysis with an Introduction to Proof Theory

Logic in Asia: Studia Logica Library Series Editors Fenrong Liu, Tsinghua University and University of Amsterdam, Beijing, China Hiroakira Ono, Japan Advanced Institute of Science and Technology (JAIST), Ishikawa, Japan Kamal Lodaya, Bengaluru, India Editorial Board Natasha Alechina, University of Nottingham, Nottingham, UK Toshiyasu Arai, University of Tokyo, Tokyo, Japan Sergei Artemov, City University of New York, New York, NY, USA Mattias Baaz, Technical university of Vienna, Austria, Vietnam Lev Beklemishev, Institute of Russian Academy of Science, Russia Mihir Chakraborty, Jadavpur University, Kolkata, India Phan Minh Dung, Asian Institute of Technology, Thailand Amitabha Gupta, Indian Institute of Technology Bombay, Mumbai, India Christoph Harbsmeier, University of Oslo, Oslo, Norway Shier Ju, Sun Yat-sen University, Guangzhou, China Makoto Kanazawa, National Institute of Informatics, Tokyo, Japan Fangzhen Lin, Hong Kong University of Science and Technology, Hong Kong Jacek Malinowski, Polish Academy of Sciences, Warsaw, Poland Ram Ramanujam, Institute of Mathematical Sciences, Chennai, India Jeremy Seligman, University of Auckland, Auckland, New Zealand Kaile Su, Peking University and Griffith University, Peking, China Johan van Benthem, University of Amsterdam and Stanford University, The Netherlands Hans van Ditmarsch, Laboratoire Lorrain de Recherche en Informatique et ses Applications, France Dag Westerstahl, Stockholm University, Stockholm, Sweden Yue Yang, Singapore National University, Singapore Syraya Chin-Mu Yang, National Taiwan University, Taipei, China

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Toshiyasu Arai

Ordinal Analysis with an Introduction to Proof Theory

123

Toshiyasu Arai University of Tokyo Tokyo, Japan

ISSN 2364-4613 ISSN 2364-4621 (electronic) Logic in Asia: Studia Logica Library ISBN 978-981-15-6458-1 ISBN 978-981-15-6459-8 (eBook) https://doi.org/10.1007/978-981-15-6459-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Ordinal analysis is a field in proof theory, in which ordinals play a central role. Let us expound how theories relate to ordinals. Let T be a computable theory comprising a small portion of arithmetic, where by a computable theory we mean the set of axioms in the theory is computable under a suitable encoding of syntax. We are supposing that the theory T is P11 -sound. Namely, for every P11 -sentence A on natural numbers, or in an equivalent form, if T proves the sentence A, then A is supposed to be true in the standard model N. On the other side, we see easily that the cut-free fragment of the x-logic with the following x-rule is P11 -complete.


C; AðnÞ


ðn 2 NÞ C; 8xAðxÞ Furthermore for every true P11 -sentence A, there exists a computable and cut-free derivation dA of A in x-logic. This means that there exists a computable, and well-founded tree treeðAÞ \x x for which there exists a computable decoration, i.e., a function on treeðAÞ such that treeðAÞ with the decoration is a cut-free xderivation of A (the decoration assigns a sequent, an inference rule, etc. to each node in treeðAÞ). Therefore, when T is P11 -sound, there must exist an ordinal aA \xCK 1 for each Tprovable P11 -sentence A such that the ordinal aA gives an upper bound for the depth of the well-founded tree treeðAÞ. Then there would exist an ordinal jTj\xCK 1 which is the supremum of the ordinals aA for T-provable P11 -sentences A. In ordinal analysis, we describe the ordinal jTj for a given theory T without assuming that T is P11 -sound. In many cases, the description of the ordinal jTj is done as follows. Some operations on ordinals are defined in such a way that jTj [ 0 is the least ordinal which is closed under the operations. For example, when T is the first-order arithmetic PA, two operations on ordinals are typically considered: the ordinal x and the exponential function a 7! 2a . The first epsilon number e0 is the least ordinal closed under these two. v

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Which operations on ordinals are associated with theories? This depends not only on theories, but also on the methods how the theories are expressed and analyzed. The main approaches in ordinal analysis are cut-elimination in sequent calculi (Gentzen, Schütte) and epsilon substitution method (Hilbert, Ackermann). In this monograph, we focus on the first, which has been the dominant one in ordinal analysis. The associated operations on ordinals correspond to or express a structural complexity of procedures in cut-elimination. We transform derivations to eliminate cut inferences in sequent calculi, and the operations are closely related to the transformations. C; :C C; C ðcutÞ C These operations warrant the executability of the procedures. Specifically let us assume that the associated operations on ordinals are well defined, by which we mean that an operation yields a well ordering from well orderings. Then we see that a cut-elimination procedure with these operations gives us a cut-free derivation. Once a cut-free derivation with a subformula property is gained, we can conclude that the end formula is true by transfinite induction on depths of the cut-free derivations, where the depth is bounded by the least ordinal closed under the associated operations. Thus, we comprehend what structure is involved in derivations of the theory T through the associated operations on ordinals as long as we are concerned with cut-elimination. Moreover, the P11 -soundness of T follows from the well-definedness of the operations. In this way, we obtain a deep understanding of T in terms of the ordinal analysis of T. The ordinal jTj is called proof-theoretic ordinal of the theory T, and it can be alternatively defined by the supremum of the order types of computable, strict partial orders such that T proves its well-foundedness. Although the definition of the ordinal jTj concerns only with provable P11 -sentences, it turns out in almost all cases that jTj characterizes some proof-theoretic limitations of T. For example, let f be a computable total function. We see that T proves the fact that f is a total function iff f is defined by a transfinite recursion on ordinals up to an ordinal\jTj. Note that the fact is written in a P02 -sentence. By the equivalence we could yield an independent P02 -statement. Let us mention the contents of the monograph. In Chap. 1, some prerequisites are explained such as one-sided sequent calculi, recursion-theoretic prerequisites, and the definition of the proof-theoretic ordinal. Our base theory, the elementary (recursive) arithmetic EA, is introduced, and the Kripke-Platek set theory KP is also defined. These materials are scattered in textbooks on logic.

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Though this monograph concerns mainly with ordinal analysis through cut-elimination in sequent calculi, Chaps. 2 and 3 are intended to be an introduction to proof theory. The reason for such an introduction is due to the fact that the ordinal analysis is based on proof-theoretic analyses of finite derivations. There are two topics in Chap. 2. First, the completeness of cut-free fragments is proved through a canonical proof-search method or refutation method. The sequent calculus for the classical first-order logic is best seen as a refutation procedure. Inference rules are regarded as searching step of a counter-model of the endsequent when we turn inferences upside down. Besides the classical first-order logic, the same method is employed to applied logic calculi, predicative second-order logic, and the x-logic. Since the calculi with the cut inference are sound, this yields the cut-elimination theorem (Hauptsatz). It is convenient for us to have cut-free derivations for a further analysis. Second, some consequences of the cut-elimination are presented. Herbrand’s theorems for formulas in prenex normal form are proved through the midsequent theorem. Moreover, witnesses for existential quantifiers are extracted from cut-free derivations. An example of the witnessing theorem is the Parikh’s theorem stating that if a R1 -formula is provable in a bounded theory, then there exists a term bounding the unbounded existential quantifier in the R1 -formula. The theorem is applied to bounded arithmetic and employed to characterize proof-theoretically the class of rudimentary functions in set theory. The second example of the witnessing theorem is on primitive recursion. In theories with induction schema for existential formulas, only a primitive recursive function is proved to be total. The Parsons-Mints-Takeuti theorem is the prominent example: provably computable functions in a fragment IR1 of arithmetic are primitive recursive functions. The proof of the theorem is based on a witnessing argument due to S. Buss and is modified to obtain a proof-theoretic proof of a theorem due to H. Friedman, which states that the second-order arithmetic WKL0 for the weak König’s lemma is P02 -conservative over IR1 . A proof-theoretic proof of the fact was first obtained by W. Sieg. Our proof deviates from Sieg’s in replacing the weak König’s lemma by an equivalent reduction property. Furthermore, the witnessing argument is employed to show a theorem due to M. Rathjen, which states that only the primitive recursive set functions in the sense of Jensen-Karp are provably total R1 -functions in a restricted Kripke-Platek set theory with R1 -induction schema. The proof of the cut-elimination theorem via a canonical proof search provides us no quantitative information. In Chap. 3, cut-elimination procedures are presented, by which we obtain a bound of the depth of the resulting cut-free derivations. The bound is given by an iteration of exponential functions of the depth of the given derivation. Logics dealt in the chapter are classical logic, intuitionistic logic, and fixed point logics in which a predicate constant Iu together with inference rules is added for positive formula u. The inference rules are for the axiom 8x½uðIu ; xÞ $ Iu ðxÞ stating that Iu is a fixed point of u. When the underlying logic is intuitionistic, u is a strictly positive formula. For the cut-elimination in the classical fixed point logic, an asymmetric interpretation is employed to replace the fixed point Iu by stages Iun of inductive definitions. The replacement is done

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according to contexts. As contrasted with the classical logic, the cut inferences from derivations in the intuitionistic fixed logic are eliminated quickly through a parallel elimination procedure. Chapter 4 is of central importance in ordinal analysis. The target theories in the chapter are theories with induction schemata, i.e., complete induction schema, transfinite induction schema, and foundation schema in set theory. Let PA þ TIð\KÞ denote an extension of the first-order arithmetic PA with transfinite induction schema for a standard and computable well-order \ of type K. Assuming that the ordinal K is an epsilon number, the proof-theoretic ordinal of the theory PA þ TIð\KÞ is shown to be equal to the ordinal K. First the fact is proved by eliminating cut inferences from derivations of transfinite induction TIðÞ for a computable strict partial order  in x-logic. We see that the ordinal K is an upper bound of the depths of the resulting cut-free derivations. On the other side, each cut-free derivation of TIðÞ is seen to express the well-foundedness of  directly. This means that the depth of the cut-free derivation is nearly equal to the order type of . Moreover, it is easy to embed finitary proofs in the theory PA þ TIð\KÞ into infinitary derivations with cut inferences with depth \K. Thus, we get an upper bound K of the proof-theoretic ordinal of PA þ TIð\KÞ, which is at the same time the least upper bound. By analyzing cut-free infinitary derivations, we show over a weak arithmetic that each P03 -theorem in PA þ TIð\KÞ follows from a weakly descending chain principle up to an ordinal\K. To obtain a finer analysis, i.e., not just the proof-theoretic ordinal, but a characterization of consequences in lower complexity such as P0i ði ¼ 1; 2Þ, it is convenient for us to analyze finitary proofs directly following Gentzen’s consistency proof. We show over a weak arithmetic that each P02 -theorem follows from a descending chain principle, and each P01 -theorem follows from an inference rule for a transfinite induction, each of which is up to an ordinal\K. For example, the class of provably computable functions in PA þ TIð\KÞ is shown to be equal to the class R\K of computable functions defined by transfinite recursion up to an ordinal\K. The importance of these results lies in the fact that when a stronger theory T is analyzed proof-theoretically, the analysis shows in many cases that T is proof-theoretically reducible to a PA þ TIð\KÞ. Specifically T is shown to be P11 conservative over PA þ TIð\KÞ. Next we examine closely the class R\e0 of provably computable functions in PA. First the descending chain principle up to e0 is shown to be equivalent to apparently weaker principle stating that there is no primitive recursive, slowly descending chain of ordinals\e0 . The equivalence is based on a representation of natural numbers using Grzegorczyk functions fFn gn , which dominates each primitive recursive function. This yields P02 -independence statements from PA: terminations of Goodstein sequences and Grzegorczyk sequences. Second we introduce fundamental sequences of ordinals\e0 and Hardy functions Ha ða\e0 Þ, a classic approach to the class R\e0 . We show a theorem stating that each provably computable function in PA is dominated by an Ha . Although the theorem is seen

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from a fact that each function in R\e0 is dominated by an Ha , we prove directly the theorem through a finer derivability relation k αc Γ to control numerical witnesses occurring in derivations. Though the approach with fundamental sequences is elegant and intuitively appealing, it would be complicated when we need to handle larger ordinals. For the obstacle, we introduce thirdly a syntactic approach called the norm-bounding recursion. Finally, we show that the intuitionistic fixed point theory FiX i ðTÞ over theories T ¼ PA, KP is a conservative extension of T. To illustrate an idea of a proof, let us consider the intuitionistic fixed point theory FiX i (HA) over the intuitionistic arithmetic HA. Note that this yields the fact that FiX i (PA) is a conservative extension of PA. Given a finitary proof of an arithmetic formula A in FiX i (HA), let us embed it to an infinitary derivation, and cut inferences are eliminated quickly as in Chap. 3. This results in an almost cut-free derivations in depth\e0 . By a partial truth definition we can conclude that the end formula A is true. By formalizing the proof in A. Although the above-sketched proof does work, it is cumHA, we obtain bersome in formalizing the proof through infinitary derivations. Instead of this we give a finitary analysis of the proofs in the intuitionistic fixed point theories, since a formalization of the finitary analysis is a trivial matter. Furthermore, the conservation theorem is utilized in showing that a theory T is P11 -conservative over, e.g., a theory PA þ TIð\KÞ, where K is the proof-theoretic ordinal of T as follows. T is analyzed proof-theoretically through infinitary derivations and cut-elimination. We need to handle a derivability relation αc Γ here. Suppose that the relation is definable as a fixed point. Then a proof of cut-elimination is carried out in an intuitionistic fixed point theory FiX i ðPA þ TIð\KÞÞ. Since FiX i ðPA þ TIð\KÞÞ is conservative over PA þ TIð\KÞ, we conclude that T is P11 -conservative over PA þ TIð\KÞ. Therefore, it is desirable to have a finitary analysis of intuitionistic fixed point theories. In Chap. 5, we discuss straightforward extensions of results in Chap. 4, which are obtained by iterating cut-elimination procedures for first-order arithmetic. In iterating procedures, the (binary) Veblen hierarchy fua ga\x1 appears, where each ua is a normal, i.e., strictly increasing and continuous function on x1 defined as follows. u0 ðbÞ ¼ xb , and for a [ 0, ua enumerates the common fixed points of functions uc ðc\aÞ. We establish equivalences between second-order arithmetics ATR0 ¼ FP0 ¼ WOPðkX:uX0Þ0 and the proof-theoretic ordinal of these is equal to the first strongly critical number also known as the Feferman-Schütte ordinal C0 , where ATR0 is a second-order arithmetic whose main axiom is the arithmetical transfinite recursion admitting a jump hierarchy of sets along a well ordering. FP0 is another second-order arithmetic whose main axiom states that a fixed point of a positive formula exists as a set. The main axiom of the second-order arithmetic WOPðkX:uX ð0ÞÞ0 is the well-ordering principle for the normal function a 7! ua ð0Þ, which states that if X is a well ordering, then a linear ordering uX ð0Þ is well founded. It is easy to see that FP0 comprises ATR0, and ATR0 proves the ⊥

⊥

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well-ordering principle WOPðkX:uX0Þ0 . We show by a proof-search method in Chap. 2, the fact that the well-ordering principle WOPðkX:uX0Þ0 implies FP0. For the proof-theoretic ordinal of these second-order arithmetic, FP0 is reduced to a theory ðP0 1  CAÞ\C0 of the jump hierarchy via asymmetric interpretations as for fixed point logic in Chap. 3, and proving a cut-elimination theorem for ðP0 1  CAÞ\C0 is a straightforward extension of the proof in Chap. 3. Finally, we extend proof-theoretic analyses for predicative theories to a metapredicative theory ATR0 þ , whose main axiom states that there exists an arbitrarily large countable coded x-model of ATR0. A theory is said to be metapredicative if its proof-theoretic ordinal is larger than the limit C0 of predicativity (over N), but we can analyze it through a straightforward extension of analyses for predicative theories. In Chap. 6, we analyze impredicative theories proof-theoretically. To extract computational (or countable) contents, we employ a technique, collapsings, by which uncountable infinitary derivations and uncountable ordinals are collapsed down to countable ones. This is done through Mostowski collapsings of Skolem hulls as in the Condensation lemma, which is a key to prove the fact that the GCH (generalized continuum hypothesis) holds in the constructible universe L. We introduce for each computable well order of type m, theories IDm for iterated positive elementary inductive definitions over natural numbers, and set theories KPm for iterated admissibility. The intended model of KPm is the constructible set LXm , where 0\a 7! Xa denotes the continuous closure of the function 0\a 7! sa for the a-th admissible ordinal sa . Next an iterated Skolem hull Ha ðXÞ of sets X of ordinals and a collapsing function ðr; aÞ 7! wr ðaÞ are introduced simultaneously, where r is a recursively regular ordinal. It turns out that HeXm þ 1 ðf0gÞ is isomorphic to a computable set of ordinal terms, thereby yielding a notation system of ordinals. The theory IDm is shown to prove the well-foundedness of the notation system up to each ordinal term. We consider the simplest case to illustrate the ideas in cut-elimination procedures for impredicative theories. Namely, a theory ID1O , which is ID1 with ordinals, is analyzed through the operator controlled derivations, and ID1 through the X-rule. Then an ordinal analysis of the set theory KPm is given. Finally, we give two applications of the method of the operator controlled derivations. The first one is a relativized ordinal analysis. An operator to control derivations is a relativized Skolem hull Hha relative to a function fh on ordinals. In the second, a set theory KPPN þ 1 for PN þ 1 -reflections is proof-theoretically reduced to a set theory of iterations of PN -recursively Mahlo operations. A transitive and well-founded model of KPx controls derivations in the second. In the end of each chapter except Chap. 1, we include a list of problems, Exercises. Some of them provides additional topics and applications of techniques developed in the texts. Answers of the selected problems are found in Appendix 7. In the final section, Notes, of each chapter, sources and credits are collected, and suggestions for further reading are given. We don’t intend by no means to give a complete list of references since it is practically impossible.

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For other topics not handled here, readers are suggested to read monographs by Takeuti, Schütte, Girard, Pohlers, and articles in the handbook. I’d like to thank to Prof. Hiroakira Ono for encouraging me to write this monograph, and to the anonymous reviewers and Dr. Kentaro Sato for helpful comments. Tokyo, Japan October 2019

Toshiyasu Arai

References Girard, J.-Y.: Proof Theory and Logical complexity, vol. 1. Bibliopolis, Napoli (1987) Handbook of Proof Theory. S. R. Buss (ed.) Elsevier (1998) Pohlers, W.: The First step Into Impredicativity. Universitext. Springer-Verlag, Berlin (2009) Schütte, K.: Proof Theory. Translated from the Revised German. In: Crossley J. N. (ed.) Grundlehren der Mathematischen Wissenschaften, Band 225. Springer-Verlag, Berlin-New York (1977) Takeuti, G.: Proof Theory, 2nd edn, North-Holland, Amsterdam (1987) reprinted from Dover Publications (2013)

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Sequent Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Subrecursive Functions and Weak Arithmetic 1.3.2 Incompleteness . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Jumps and Trees . . . . . . . . . . . . . . . . . . . . . 1.3.4 P11 -Completeness of W . . . . . . . . . . . . . . . . 1.3.5 Inductive Definitions . . . . . . . . . . . . . . . . . . 1.4 Proof-Theoretic Ordinals . . . . . . . . . . . . . . . . . . . . . 1.5 Kripke–Platek Set Theory . . . . . . . . . . . . . . . . . . . . 1.5.1 R-Reflection and Its Consequences . . . . . . . . 1.5.2 Constructible Hierarchy and Admissible Sets . 1.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Calculi for Classical Logic . . . . . . . . . . . . 2.1 Completeness of Cut-Free Fragments . . 2.1.1 Canonical Proof Search . . . . . . 2.1.2 Applied Calculi . . . . . . . . . . . . 2.1.3 Predicative Second Order Logic 2.1.4 Predicative Extensions . . . . . . . 2.1.5 x-Logic . . . . . . . . . . . . . . . . . 2.2 Consequences of Cut-Elimination . . . . 2.2.1 Midsequent Theorem . . . . . . . . 2.2.2 Interpolation Theorem . . . . . . . 2.2.3 Herbrand’s Theorems . . . . . . . . 2.2.4 Epsilon Theorems . . . . . . . . . . 2.2.5 Parikh’s Theorem . . . . . . . . . .

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2.2.6 Primitive Recursion 2.2.7 Primitive Recursion 2.3 Notes . . . . . . . . . . . . . . . . 2.3.1 Section 2.1 . . . . . . 2.3.2 Section 2.2 . . . . . . 2.3.3 Exercises . . . . . . . . References . . . . . . . . . . . . . . . .

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3 Cut-Elimination with Depths . . . . . . . . . . . . . . . . . . . . . . 3.1 Classical Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Elimination Procedure for Classical Calculus . . . 3.1.2 Predicative Second Order Logic . . . . . . . . . . . . 3.1.3 Propositional Logics . . . . . . . . . . . . . . . . . . . . 3.2 Fixed Point Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Intuitionistic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Elimination Procedure for Intuitionistic Calculus 3.3.2 Intutionistic Fixed Point Logic . . . . . . . . . . . . . 3.4 Epsilon Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Section 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Section 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Section 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Section 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Epsilon Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Provability of Transfinite Induction . . . . . . . . . . . . . . . . . 4.1.1 A Standard e0 -Order . . . . . . . . . . . . . . . . . . . . . . 4.2 Unprovability of Transfinite Induction . . . . . . . . . . . . . . . 4.2.1 Bounding Order Types of Provably Well-Founded Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 2-Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Consistency Strengths Calibrated with Ordinals . . . . . . . . 4.3.1 Consistency Proof . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 P2 -Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Intuitionistic Fixed Point Theories . . . . . . . . . . . . . . . . . . 4.4.1 Finitary Analysis of Intuitionistic Fixed Points Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Section 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Section 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.5.3 Section 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.5.4 Section 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5 Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Binary Veblen Functions and Club Sets . . . . . . . . . . . . . . . . . . 5.2 Axiom Schemata in Second Order Arithmetic . . . . . . . . . . . . . 5.3 Theories of Jump Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Transfinite Inductions in the Theories of Jump Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Predicative Cut Elimination . . . . . . . . . . . . . . . . . . . . . 5.4 Classical Fixed Point Theories . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Eliminating Fixed Points . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Proving the Fixed Point Axiom from the Well-Ordering Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Simulating Jump Hierarchies by Fixed Points . . . . . . . . 5.5 Proof-Theoretic Strengths of the Well-Ordering Principles . . . . 5.5.1 Derivative and Countable Coded x-Models . . . . . . . . . . 5.5.2 Elimination of the Inference for Well-Ordering Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Proof-Theoretic Ordinals of the Well-Ordering Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Section 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Section 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Section 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Section 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.5 Exercises 5.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Collapsings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Theories IDm , P11 -CA and KPm . . . . . . . . . . . . . . . . . . . . . 6.1.1 Kripke–Platek Set Theory for Iterated Admissibility 6.2 Higher Ordinals with Collapsing Functions . . . . . . . . . . . . 6.2.1 Recursive Notation System TðXm Þ of Ordinals . . . . . 6.3 Well-Foundedness Proof . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Impredicative Cut Elimination . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Operator Controlled Derivations . . . . . . . . . . . . . . . 6.4.2 X-Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Ordinal Analysis of Kripke–Platek Set Theory KPm . . . . . . 6.5.1 Ramified Set Theory . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.6 Applications of Operator Controlled Derivations . . . . . . . . . . . 6.6.1 A Relativized Ordinal Analysis . . . . . . . . . . . . . . . . . 6.6.2 Reducing Pn þ 1 -Reflections to Iterated Pn -Reflections . 6.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Section 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Section 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.3 Section 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.4 Section 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.5 Subsection 6.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.6 Subsection 6.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.7 Exercises 6.6.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Chapter 1

Introduction

In this chapter some prerequisites are explained such as one-sided sequent calculi, recursion-theoretic prerequisites, and the definition of the proof-theoretic ordinal. Our base theory, the elementary (recursive) arithmetic EA, is introduced, and the Kripke–Platek set theory KP is also defined. These materials are scattered in textbooks on logic.

1.1 Languages In this book, languages considered are restricted to be countable unless otherwise stated. Let L be a countable language for predicate logic without equality, which consists of function symbols and relation symbols. Each symbol receives a nonnegative integer called the arity of the symbol. 0-ary function symbols are individual constants denoted c. Terms in L or L -terms are generated from countable lists of free variables denoted a0 , a1 , . . . or a, b by applying function symbols. Closed terms are terms in which no (free) variable occurs. When we analyze classical logic or theories in classical logic proof-theoretically, we assume that each formula is in negation normal form. By a formula in negation normal form we mean a formula in which the implication → does not occur, and the negation ¬ is applied only to atomic formulas. Then the set of relation (predicate) symbols contains the complement R for each relation symbol R in the set. Thus the set of relation symbols R = {Ri , R i : i ∈ I }(I : a non-empty countable set), where R and R is the complement each other. Set (R) := R. By a literal we mean an atomic formula, i.e., one of formulas R(t1 , . . . , tn ), R(t1 , . . . , tn ) for terms t1 , . . . , tn and n-ary relation symbols R, R. Formulas in negation normal form are generated from literals by means of logical connectives, boolean (or propositional) connectives ∨, ∧, and quantifiers ∃, ∀ with bound variables denoted x0 , x1 , . . . or x, y. Let us specify quantifications. When A © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 T. Arai, Ordinal Analysis with an Introduction to Proof Theory, Logic in Asia: Studia Logica Library, https://doi.org/10.1007/978-981-15-6459-8_1

1

2

1 Introduction

is a formula and x a bound variable not occurring in A, ∃x A[x/a] and ∀x A[x/a] are formulas, where A[x/a] denotes the result of replacing every occurrence of a free variable a in A by x. We denote A(x) instead of A[x/a] when no confusion likely occurs. The negation ¬A of a formula A is defined recursively by de Morgan’s law and the elimination of double negations: 1. ¬L :≡ L¯ for a literal L. 2. ¬(A ∨ B) :≡ ¬A ∧ ¬B and ¬(A ∧ B) :≡ ¬A ∨ ¬B. 3. ¬∃x A(x) :≡ ∀x¬A(x) and ¬∀x A(x) :≡ ∃x¬A(x). Closed formulas (or sentences) are formulas in which no free variable occurs. Formulas are denoted A, B, C, . . . V ar (e) denotes the set of free variables occurring in terms and formulas e. A formula is said to be positive with respect to a predicate R, or R-positive if R does not occur in it. Alternatively we say that the predicate R occurs only positively in the formula. Similarly a formula is said to be R-negative if R does not occur in it. When we consider intuitionistic logics and theories in intuitionistic logic, the implication → is a primitive logical connective, and ⊥ is an atomic formula denoting the absurdity. Set ¬A :≡ (A → ⊥).

1.2 Sequent Calculi We adopt one-sided sequent calculi, when we analyze classical logic or theories in classical logic proof-theoretically. Structural rules (weakening or thinning, contraction and exchange) are not included in one-sided sequent calculi since these are admissible rules or implicit in inference rules. A finite set {A0 , . . . , An−1 } (n ≥ 0) of formulas is said to be a sequent or a cedent. Its intended meaning is the disjunc tion i 0} denotes a Definition 1.5 1. B = {zero, S, x −y, class of initial functions. The 0-ary function zero = 0, the successor function ˙ = max{x − y, 0}, the maximum S(x) = x + 1, the modified subtraction x −y k max(x, y) and projections Ii (x1 , . . . , xk ) = xi . 2. f is defined from h, g1 , . . . , gm by Composition if f (x) = h(g1 (x), . . . , gm (x)) for x = x1 , . . . , xk . 3. f is defined from g, h by Primitive Recursion if f (x, 0) = g(x) and f (x, y + 1) = h(x, y, f (x, y)).  4. f is  defined from g by Summation if f (x, y) = i Fn (x) for x ≥ 2. Proof Propositions 1.3.1, 1.3.2 and 1.3.3 are proved simultaneously by induction on n. The case n = 0 is easily seen. Let x > 0. Then Fn+1 (x) = Fn (Fn(x−1) (x)) ≥ Fn (x) > x by IH. Proposition 1.3.4 (y) is proved by Fn+1 (x) = Fn(x−1) (Fn (x)) > Fn (x) > 0. From this we see Fn+1 (x) ≥ x (x) by induction on y. Again by IH we obtain Fn+1 (x + 1) = Fn (Fn (x + 1)) > (y) (y) Fn (Fn(x) (x)) ≥ Fn(x) (x) = Fn+1 (x). From this we see Fn+1 (x + 1) > Fn+1 (x) by (y+1) (y) induction on y. Fn+1 (x) > Fn+1 (x) is seen from Proposition 1.3.1.  Lemma 1.4 For each primitive recursive function f (x), there exists an n for which f (x) ≤ Fn (max{x, 2}) holds for any x = x1 , . . . , xk . Proof We show the lemma by induction on the definitions of primitive recursive functions f . For the initial functions f (x) ≡ 0, f (x) = xi , f (x) = xi + 1, f (x) ≤ F0 (max{x, 2}) holds. Let f (x) = h(g1 (x), . . . , gm (x)) by Composition. By IH with Proposition 1.3.4 we have an n such that gi (x) ≤ Fn (max{x, 2}) for any x and any i ≤ m. Moreover h(y1 , . . . , ym ) ≤ Fn (max{y1 , . . . , ym , 2}) for any yi . Then Propositions 1.3.3 and

1.3 Arithmetic

7

1.3.1 yield f (x) ≤ Fn (max{g1 (x), . . . , gm (x), 2}) ≤ Fn (Fn (max{x, 2})) ≤ Fn+1 (max{x, 2}). Finally let f (x, 0) = g(x) and f (x, y + 1) = h(x, y, f (x, y)) by Primitive Recursion. By IH with Proposition 1.3.4 we have an n such that g(x) ≤ Fn (max{x, 2}) and h(x, y, z) ≤ Fn (max{x, y, z, 2}) for any x and any y, z. By induction on (y+1) (max{x, y, 2}). From this we see that for z = y we show that f (x, y) ≤ Fn (y+1) (2) (z) ≤ Fn(z+1) (z + 1) = Fn+1 (z + 1) ≤ Fn+1 (z) max{x, y, 2} ≥ 2, f (x, y) ≤ Fn ≤ Fn+2 (z). When y = 0, f (x, 0) = g(x) ≤ Fn (max{x, 2}) = Fn(1) (max{x, 2}). For y > 0, f (x, y) = h(x, y − 1, f (x, y − 1)) ≤ Fn (max{x, y − 1, f (x, y − 1), 2}) ≤ (y) (y+1) (max{x, y, 2}) by ProposiFn (max{x, y − 1, Fn (max{x, y − 1, 2}), 2}) ≤ Fn tions 1.3.3 and 1.3.1.  Corollary 1.1 E ω = PRIM. Proof E ω ⊂ PRIM is obvious. To see PRIM ⊂ E ω inductively, let f be a function defined from g, h ∈ E n+1 by Primitive Recursion. By Lemma 1.4 we can assume  that f (x) ≤ Fn (max{x, 2}). Proposition 1.2.5 yields f ∈ E n+1 . Definition 1.6 The elementary (recursive) arithmetic EA for E 3 is a fragment of the first-order arithmetic in the language L (EA) = {=, ≤, 0, S, +, ·, ex p}, where ex p(x) = 2x is the function symbol for the exponential. EA4 is another fragment in an expanded language L (EA4 ) = L (EA) ∪ {tower }, where tower (x) = 2x = ex p (x) (1) is the function symbol for the tower of exponentials. L (PRA) = {=, ≤} ∪ {f : f ∈ PRIM} denotes a language for PRA, primitive recursive arithmetic having function symbols f for each primitive recursive function f ∈ PRIM. The defining axioms of function symbols are: x + 0 = x, x + S(y) = S(x + y), x · 0 = 0, x · S(y) = x · y + x, ex p(0) = 1 := S(0), ex p(S(x)) = ex p(x) · 2, (2 := S(1)), tower (0) = 1, tower (S(x)) = ex p(tower (x)). If f is defined from h, g1 , . . . , gm by Composition, then f(x) = h(g1 (x), . . . , gm (x)) is its defining axiom. If f is defined from g, h by Primitive Recursion, then f(x, 0) = g(x) and f(x, S(y)) = h(x, y, f(x, y)) are its defining axioms. The axioms for the discrete order ≤ are 0 ≤ x, x ≤ S(y) ↔ x ≤ y ∨ x = S(y), x ≤ x, x ≤ y ≤ z → x ≤ z, x ≤ y ∨ y ≤ x, x ≤ y ≤ x → x = y. In the following L is a language with a binary relation symbol ≤. 1. A formula in L is said to be bounded if every quantifier occurring in it is bounded by a term in L , ∃x ≤ t, ∀x ≤ t, where ∃x ≤ t A(x) :≡ (∃x(x ≤ t ∧ A(x))) and similarly for bounded universal quantifier. 2. Δ0 (L ) = Σ0 (L ) = Π0 (L ) denotes the class of bounded formulas in the language L . 3. Σn (L ) denotes the class of Σn -formulas in L , each of which is of the form ∃x1 ∀x2 · · · Qxn B with B ∈ Δ0 (L ) and Q = ∀ when n is even, and Q = ∃ else. 4. Let {=, ≤, 0, S} ⊂ L ⊂ L (PRA). IΣn (L ) denotes a fragment of first-order arithmetic in the language L . Its axioms are the Peano axioms ∀x(S(x) = 0),

8

1 Introduction

∀x, y(S(x) = S(y) → x = y), the defining axioms for function symbols f for f ∈ L , the equality axioms, the axioms for the discrete order ≤, and the complete induction schema for Σn (L )-formulas A: A(0) ∧ ∀x(A(x) → A(S(x))) → ∀x A(x). The Peano arithmetic or the first-order arithmetic in the language L is defined  to be PA(L ) := IΣω (L ) := n 0 and classes {f : f ∈ E 3 } ⊂ L . Proposition 1.4 Let 0 < n ≤ ω. Then for any languages L , L  such that {f : f ∈ E 3 } ⊂ L ∩ L  and L ∪ L  ⊂ L (PRA), I Σn (L ) = I Σn (L  ) holds. Proof This is seen from the fact that each primitive recursive function is Σ1 (L ) definable in I Σ1 (L ) for L = {f : f ∈ E 3 }. Proposition 1.5 1. The partial truth definition for Σ0 (L (EA))-formulas is definable by a Σ0 (L (EA4 ))-formula provably in the arithmetic EA4 . 2. For each n > 0 a partial truth definition for Σn (L (EA))-formulas is definable by a Σn (L (EA))-formula Tr n provably in the arithmetic EA. Proof Note that EA is strong enough to encode finite sequences of natural numbers, cf. Proposition 1.2.4. Then an arithmetization of syntax and a partial truth definition are carried out in EA, cf. [6]. Proposition 1.5.1 is seen from the fact that each E 3 function f (x1 , . . . , xn ) is bounded by a function 2c (max{x1 , . . . , xn }) for a c, and there is an E 4 -function sending Gödel numbers t of closed terms t in the language  L (EA) to their values t N in the standard model N.

1.3.2 Incompleteness As said above, the language is assumed to be countable. We assume tacitly a suitable encoding of the syntax, such as terms, formulas, sequents. A is then the code (Gödel number) of formulas A. Furthermore we are concerned only with formal (first-order or second-order) theories T , which are elementary recursive. This means that the set of coeds of axioms in T is elementary recursive. Proposition 1.6 (Craig’s trick) Any r.e.(=recursively enumerable=computably enumerable) theory is axiomatizable by an elementary recursive set of sentences.

1.3 Arithmetic

9

Proof Let T h be an r.e. theory. We can assume that T h = ∅. Let e be a number such that ∀x(x ∈ T h ↔ ∃y T (e, x, y)) for the Kleene’s T -predicate. Pick an a ∈ T h. Then for inverses (n)i (i = 0, 1) of a bijective pairing function, let f (n) =  (n)0 if T (e, (n)0 , (n)1 ) . Since each of T and (n)i is elementary recursive, so is f . a otherwise Moreover it is clear that r ng( f ) = T h. Now let T h  = {A(n+1) : A = f (n), n ∈ N}, where for formulas A, A(1) ≡ A and A(n+1) ≡ (A ∧ A(n) ). Then T h  axiomatizes  the theory T h, and we see easily that T h  is elementary recursive. Let PrT (y) :≡ ∃xPrfT (x, y) be a standard Σ1 -provability predicate for an elementary recursive theory T ⊃ EA, where PrfT (x, y) is an elementary recursive relation codifying ‘x is a code of a proof of a formula y in T ’. Assume that PrT (y) enjoys the Löb’s derivability conditions, D1: T A ⇒ EA PrT (A) for formulas A, D2: EA PrT (A → B) → PrT (A) → PrT (B) for formulas A, B, and D3: EA PrT (A) → PrT (PrT (A)) for formulas A. Let Φ be a class of sentences. RfnΦ (T ) denotes the local reflection schema for T with respect to sentences in Φ: RfnΦ (T ) = {PrT (A) → A : A ∈ Φ}. RFNΣn (T ) denotes the uniform reflection schema for T with respect to Σn -sentences: RFNΣn (T ) = ∀A ∈ Sntn [PrT (A) → T rn (A)] where Sntn denotes the set of codes (Gödel numbers) of Σn -sentences, and T rn a Σn formula defining a partial truth definition for Σn -sentences. The uniform reflection schema RFNΠn (T ) for Πn -sentences is defined similarly. RFNΣn (T ) is said to be the n-consistency statement of T when n = 1, 2. It is easy to see that RFNΠn+1 (T ) is equivalent (over EA) to RFNΣn (T ) for each n < ω, and RFNΠ1 (T ) is equivalent to the consistency statement CON(T ) :⇔ ¬PrT (⊥), where ⊥ is a false sentence ⊥, e.g., 0 = 1. Gödel’s second incompleteness theorem states that T  CON(T ) unless T is inconsistent. Moreover RfnΣ1 (T ) is stronger than the consistency CON(T ), i.e., T + RfnΣ1 (T ) CON(T ), but T + CON(T )  RfnΣ1 (T ) unless T + CON(T ) is inconsistent. Moreover T + RfnΣ1 (T ) CON(T + CON(T )) is seen from the fact that CON(T ) is a Π1 -sentence. In general the following holds. Theorem 1.1 (Kreisel–Lévy Essential unboundedness theorem) Let T ⊃ EA be one of fragments in Definition 1.6, and n ≥ 0. 1. For any Σn+1 -sentence C, T + C RfnΠn+1 (T ) ⇒ T ¬C. 2. For any Πn+1 -sentence C, T + C RfnΣn+1 (T ) ⇒ T ¬C. Proof First consider the case n = 0 in Theorem 1.1.2. Suppose T + C RfnΣ1 (T ) for a Π1 -sentence C. An instance PrT (¬C) → ¬C of RfnΣ1 (T ) yields T + C ¬PrT (¬C), i.e., T + C CON(T + C). Hence T + C is inconsistent.

10

1 Introduction

Next we show Theorem 1.1.1. Theorem 1.1.2 for n > 0 is seen similarly. Assume T + C RfnΠn+1 (T ) for a Σn+1 -sentence C. We show T ¬C. Let A denote a Πn+1 -sentence such that EA A ↔ [C → ¬PrT (C → A)]

(1.1)

Since T + C RfnΠn+1 (T ), we obtain T + C PrT (C → A) → C → A. Hence T + C ¬A → ¬PrT (C → A). Thus (1.1) yields T + C ¬A → A. Therefore T + C A and T C → A. By the derivability condition (D1) we have  T + C PrT (C → A). By (1.1) we conclude T + C ¬A, and T ¬C.

1.3.3 Jumps and Trees Let T X (e, x, y) be Kleene’s T -predicate with an oracle X , which means that y is the code of a halting computation of a program P with its code e = P under the input x and the oracle X ⊂ N. For the result extracting function U , {e}X (x)  z :⇔ ∃y[T X (e, x, y) ∧ U (y) = z]. {e}X (x) ↓ :⇔ ∃y[{e}X (x)  y]. A set A of natural numbers is computable (recursive) in X if there exists an e such that A = {x ∈ N : {e}X (x)  0} and ∀x({e}X (x) ↓). Then the jump of the set X , denoted by X  is the set {x ∈ N : {x}X (x) ↓}. The jump operation X → X  can be iterated. Define X (n) recursively by X (0) = X and X (n+1) = (X (n) ) . It is well known that a set A of natural numbers is arithmetical in X , i.e., A = {x ∈ N : N |= B[X , x]} for a first-order formula B in the language L (EA) ∪ {X } iff there exists an n < ω such that A is computable in the jump X (n) . RCA0 and WKL0 are two of the big five of the subsystems of second order arithmetic. The axioms of RCA0 are purely universal ones which define 0, S, +, ·, ≤, the complete induction schema for Σ10 -formulas, and Δ01 -Comprehension Axiom Δ01 -CA: ∀n(A(n) ↔ ¬B(n)) → ∃X ∀n(X (n) ↔ A(n)) (A, B ∈ Σ10 ). In these axioms, by a Σ10 -formula we mean a formula ∃x R(x, a, X) with a bounded formula, i.e., Δ01 -formula R possibly having first-order parameters a, and second-order ones X. WKL0 is obtained from RCA0 by adding the weak König’s lemma, i.e., König’s lemma for binary trees. Definition 1.7 1. For strings σ, τ ∈