EFFICIENT METAMATHEMATICS @- Ml 1%'2:7 M QB L13 'Mir/"'\'L'§':@°7"":-\"/" itMW'i. .1.J Laurina Christina. Verbrugge

EFFICIENT METAMATHEMATICS ACADEMISCH PROEFSCHRIFT Ter verkrijging van de graad van Doctor aan de Universiteit van Amsterdam, op Gezag van de Rector Magnificus Prof. Dr. P.W.M. de Meijer in het openbaar te verdedigen in de Aula der Universiteit (Oude Lutherse Kerk, ingang Singel 411, hoek Spui) op dinsdag 14 september 1993 te 15.00 uur. door Laurina Christina Verbrugge geboren te Amsterdam Promotor: Prof. dr A.S. Troelstra. Co-promotores: Dr D.H.J. de Jongh Dr A. Visser Faculteit Wiskunde en Informatica Universiteit van Amsterdam Plantage Muidergracht 24 ILLC Dissertation Series, 1993, no. 3 ISBN 90-800769-8-8 Cover illustration by Saul Steinberg ©1954 The New Yorker Magazine, Inc. Druk: Haveka B.V., Alblasserdam Ter nagedachtenis aan Laurina Verbrugge-de Graag (1901-1993)

Contents Dankwoord ix I Introduction and background 1 1 Introduction 3 1.1 What is efliciency? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 What is metamathematics? 5 1.3 Interpretability and its logics . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Metamathematics of efficient mathematics . . . . . . . . . . . . . . . . . . 6 1.5 Efficient metamathematics of inefficient mathematics 6 1.6 What to expect from the rest of the dissertation? 7 2 Background 9 2.1 Theories of arithmetic 9 2.2 Provability logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Bounded arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 IAO + Q, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Buss' systems of bounded arithmetic and the polynomial hierarchy 13 2.3.3 Metamathematics for bounded arithmetic 17 2.4 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Interpretability logic 23 2.6 Definable cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Cuts may help to characterize interpretability 29 2.8 Between IAO and 113.0+ EXP . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.8.1 IAO+ EXP proves restricted consistency statements . . . . . . . . 32 2.8.2 Conservativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.8.3 Non-conservativity and incompleteness . . . . . . . . . . . . . . . . 37 II Metamathematics for Bounded Arithmetic 41 3 A small reflection principle for bounded arithmetic 43 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Dcompleteness and the NP = co-N P problem . . . . . . . . . . . . . . . . 45 3.3 The small reflection principle . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Injection of small (but not too small) inconsistency proofs 60 vii 4 Provable completeness for E1-sentences implies something funny, even if it fails to smash the polynomial hierarchy 65 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 If S1}proves completeness for all Z1-sentences, then NP (7co-NP = P . . . 66 5 On the provability logic of bounded arithmetic 67 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Arithmetical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.3 Trees of undecidable sentences . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.4 Upper bounds on PLQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.5 Disjunction property 76 III Metamathematics for Peano Arithmetic 79 6 Feasible interpretability 81 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2 Feasible interpretations in various settings . . . . . . . . . . . . . . . . . . 82 6.3 Soundness of ILM for feasible interpretability over PA . . . . . . . . . . . 90 6.4 Interpretability does not imply feasible interpretability 95 6.5 ILM is the logic of feasible interpretability over PA . . . . . . . . . . . . . 99 7 The complexity of feasible interpretability 109 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.2 Preliminaries and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.3 Characterizations of feasible interpretability . . . . . . . . . . . . . . . . . 112 7.4 The set of Elf-axiomatized theories feasibly interpretable over PA is 22- complete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.5 Lindstr6m's general lemmas polynomialized . . . . . . . . . . . . . . . . . 118 7.6 Feasible interpretability is 22-complete . . . . . . . . . . . . . . . . . . . . 121 Bibliography 125 Samenvatting 131 viii Dankwoord Om te beginnen wil ik mijn oprechte dank betuigen aan het triumviraat van promotor en co-promotores. Mijn promotor, professor A.S. Troelstra, dank ik voor zijn commen- taar en voor zijn relativerende opmerkingen. Dick de Jongh deed nuttige en doordachte aanbevelingen naar aanleiding van vele eerdere versies van dit proefschrift. Albert Visser tenslotte wil ik bedanken voor zijn aanstekelijke enthousiasme en de flexibiliteit waarmee hij op een moeilijk moment onmiddellijk een werkplek in Utrecht voor me verzorgde. Tijdens de eerste paar jaar van mijn promotie-onderzoek vormden de discussies met Edith Spaan een bron van inspiratie, waarvoor ik haar zeer erkentelijk ben. I have had the good fortune to be supported by NWO (project 611-308-004) and the Tempus Project, which enabled me to visit Prague, Siena and Florence several times. I gratefully thank my colleagues Petr Hajek, Vitézslav Svejdar and Franco Montagna for their hospitality and stimulating discussions. A special thanks goes to Alessandro Berarducci, who has been a wonderful co-author and mentor. In fact, I learnt most of my knowledge about weak arithmetics from his "scatola magica" of index cards crammed with elegant proofs. I also want to thank him and Silvia Cantelli for their generosity and for their fascinating stories, told in a mixture of Italian, German and English. De laatste tijd heb ik op allerlei manieren veel steun van mijn familie, vriendinnen, vrienden en collega's gekregen. Ik denk bijvoorbeeld aan mijn kamergenoot Domenico Zambella, die me voorzag van krentenbollen en levendige gesprekken over begrensde rekenkunde. Volodya Shavrukov las blijkens zijn rake commentaar het manuscript van dit proefschrift op nauwgezettere wijze dan wie ook, en maakte bovendien een mooie macro. Mijn twee paranimfen, Marjanne de Jong en Andreja Prijatelj, wil ik bedanken voor de humor en creativiteit waarmee ze me voortdurend hebben gemotiveerd. Joke Hendriks heeft me tijdens onze wekelijkse bijeenkomsten moed ingesproken en is tegelijkertijd een streng extern geweten voor me geweest -die combinatie werkte perfect. Maarten de Rijke wil ik bedanken voor zijn effectieve hulp bij IATEX-en printproblemen, en Marjet Denijs voor het vormgeven van de voorkant van dit proefschrift. I would like to thank all my colleagues in logic and theoretical computer science for providing fruitful discussions and a pleasant atmosphere; specifically I thank Lev Bek- lemishev, Harry Buhrman, Marc Jumelet, Marianne Kalsbeek, Karen Kwast, Alexander Razborov, Dirk Roorda, Victor Sanchez, Martijn Spaan, and Yde Venema. Tot slot dank ik mijn ouders, die mijn keuzen soms met opgetrokken wenkbrauwen begroetten, maar altijd achter me stonden. Rineke Verbrugge Amsterdam, juli

Part I Introduction and background

Chapter 1 Introduction Verde embeloso de la vida humana, loca esperanza, frenesi dorado, suefio de los despiertos intrincado, como de sueiios, de tesoros vana; alma del mundo, senectud lozana, decrépito verdor imaginado; el hoy de los dichosos esperado y de los desdichados el mafiana: sigan tu sombra en busca de tu dia los que, con verdes vidrios por anteojos, todo lo ven pintado a su deseo; que yo, mas cuerda en la fortuna mia, tengo en entrambas manos ambos ojos y solamente lo que toco veo. (Sor Juana Inés de la Cruz, seventeenth century, Mexico) 1.1 For more than a century now, mathematicians have been concerned with ways to transfer proofs in which the existence of objects satisfying certain properties is claimed in such a way as to enable us to find particular objects satisfying those properties. Stated in modern terms, they wanted to find witnesses for provable existential statements in a constructive way. They invented new constructive logics to accomodate their concern. Thus, for example, if some statement of the form 'v'a:ElyA(:z:,y) ("for all :1:there exists a y such that A(:z:,y) holds") is provable in intuitionistic Heyting Arithmetic, then its proof provides a recursive algorithm to find for every :1:a witness y such that A(:z:,y) holds. Moreover, we can assign a numerical code e to the recursive algorithm, for which Heyting Arithmetic proves \7':z:3yT1(e,2:,y). Here T1 is Kleene's primitive recursive T-predicate for What is efficiency? Green allurement of our human life, mad Hope, wild frenzy gold-encrusted, sleep of the waking full of twists and turns for neither dreams nor treasures to be trusted; soul of the world, new burgeoning of the old, fantasy of blighted greenery, day awaited by the happy few, morrow which the hapless long to see: let those pursue your shadow's beckoning who put green lenses in their spectacles and see the world in colors that appeal. Myself, I'l1act more wisely toward the world: I'll place my eyes right at my fingertips and only see what my two hands can feel. (translation Alan S. Trueblood) 3 4 CHAPTER 1. INTRODUCTION recursive functions of one variable; T1(e,:z:,y) stands for "y is the numerical code of a computation of the value of the function with associated code e on input 2:". If all quantifiers of A are bounded, then truth of 'v':z:3yA(:c,y)already provides a recursive algorithm e for finding witnesses. As in the case of Heyting Arithmetic, we have that if 'v':1:ElyA(:c,y)is not only true but also provable in Peano Arithmetic, then Peano Arithmetic proves V:c3yT1(e,a:,y) (see [Kr 51, Kr 52, Kr 58]). Parsons and Mints independently proved a similar result for IE1, the subsystem of Peano Arithmetic in which induction is only allowed for formulas of the form ElzB(z) where all quantifiers in B are bounded. If IE1 proves V:c3yA(:c,y) for bounded A, then there is even a primitive recursive function that provides the witnesses (see [Pa 72, Mi 71]). Since the rise of computer science the desire for constructivity has been growing more and more stringent. It is no longer sufficient that the constructions to find the witnesses are given by recursive, or even primitive recursive algorithms. The algorithms have to be eflicient. For example algorithms that need a computation time exponential in the length of the input are ruled out. Unfortunately, the precise mathematical meaning of the noun "efficiency" is hard to pinpoint. The dictionary definition does not offer much guidance: efficiency [F., from L., efiiciens, -ntem, pres. p. of ejfficere, to EFFECT], n. Adequate fitness; power to produce a desired result; (Eng) the ratio of the output of energy to the input of energy [HS 84]. If we want to classify algorithms as to their efficiency in a mathematically fruitful way, we should abstract as much as possible from highly specific factors like the programming language and the size, kind and operating speed of the computer on which the program is run. Cobham gave just such a classification in [Co 64]. He classified algorithms in terms of the number of steps taken by a Turing Machine to complete the computation as a function of the length of the input. For example, an algorithm runs in polynomial time if there are fixed integers c and k such that for all n, the computation on inputs of length n is completed in at most c 71''steps. Efficiency is thus a relative measure, not an absolute one. When I do abuse language by using the adjective "efficient" in an absolute sense in this dissertation, it refers to algorithms which run in (deterministic or sometimes non-deterministic) polynomial time. In the literature, the word "fea.sz'ble"is used a little more precisely than "efficient"; a feasible algorithm should run in time polynomial in the input. However, even for this relatively young word non-standard uses abound (see e.g. de volume [BS 90]). In 1986, Buss introduced systems of arithmetic that cater to the need for efficient algorithms [Bu 86]. For example, if the best-knownof his systems proves a statement of the form V:i:ElyA(:L',y)where A defines a predicate in NP (i.e. computable by a non- deterministic polynomial time Turing machine), then there is a polynomial time algorithm f that computes for every 3: a witness f(1:) such that A(:z:,f(1:)) holds. For syntactic reasons Buss called his hierarchy of systems Bounded Arithmetic. Induction is not allowed for all first order formulas, as in the standard system of arithmetic, Peano Arithmetic. Instead, every system allows induction only for a specific class of bounded formulas, in which all quantifiers are bounded by a term in the language of Bounded Arithmetic. Because of the concern for efficiency behind his systems we can see them as an example of efficient mathematics, and his most well-known system S; even as feasible mathematics. 1.2. WHAT IS METAMATHEMATICS? 5 1.2 What is metamathematics? In 1931, Godel proved his First and Second Incompleteness Theorems (see [G6 31, Ho 79]). They are theorems about systems of mathematics. For example, the Second Incompleteness Theorem says that every consistent theory that can, by some coding mech- anism, express enough information about its syntax, cannot prove its own consistency. Theories like Peano Arithmetic (PA) prove a formalization of the Second Incomplete- ness Theorem, namely the arithmetical sentence that expresses "If PA is consistent, then PA does not prove that PA is consistent". In general, metamathematics is the study of mathematical theories by mathematical methods like formalization. Lob [L6 55] listed the properties of the formalized provability predicate that are sufficient for proving the formalized version of G6del's Second Incompleteness Theorem. In the seventies, the conditions listed by L6b came into their own as the modal logic of provability, where DA is read as "A is provable" (see e.g. [MS 73]). This provability logic, which we call L after Lob, has proved to be very useful in the study of provability predicates for theories like Peano Arithmetic. The outstanding result in this area was attained by Solovay, who proved in 1976 that L exactly captures the modal properties of the provability predicate of Peano Arithmetic. More precisely, he showed that for any modal formula A, L proves A if and only if Peano Arithmetic proves all translations of A in the language of arithmetic that interpret D as the formalized provability predicate [So 76]. 1.3 Interpretability and its logics As early as the 19th century, mathematicians sought a relative consistency proof in order to show that the consistency of Euclidean geometry implies the consistency of Hyperbolic geometry. A positive answer would show that the parallel postulate is not provable from the other Euclidean axioms. One of the earliest proofs, by Poincaré, made use of an interpretation, even though that concept was not formally defined at the time. We do not give a precise definition here. Intuitively, the theory U interprets the theory V if there exists a translation of the language of V into the language of U that enables us to "see" a model of V inside every model of U. Interpretations have been used for various purposes, at first mainly for relative consistency proofs as in Poincaré's example (see also [Hi 1899]), and later for proving theories to be undecidable (see e.g. [TMR 53]). The study of interpretability picked up momentum in the seventies with papers by Hajek and Solovay (see e.g. [Ha 71, Ha 72, So 76b]). Their results, which include the Orey-Hajekcharacterization for interpretability over theories like Peano Arithmetic, gave the study of interpretability its proper place as a part of metamathematics. The eighties saw the development of modal logics for interpretability, first introduced in [Sv 83] and [Vi 89]. Here D is a binary modal operator corresponding to interpretabil- ity over a base theory T. In contrast to the case of provability logic, there are several interpretability logics around, capturing the principles that govern interpretability over various kinds of base theories (see e.g. [JV 90, Vi 90a]). For example, Berarducci and Shavrukov independently proved by modified Solovayconstructions that the interpretabil- ity logic ILM is the logic of interpretability over Peano Arithmetic (see [Ber 90, Sh 88]). 6 CHAPTER 1. INTRODUCTION 1.4 Metamathematics of efficient mathematics If we want to prove an analog of the formalized version of Godel's Second Incompleteness Theorem for systems of efficient mathematics like Bounded Arithmetic, we are forced to make our metamathematics efficient, too. First of all, we need efficient numerals. This is because S. . . S 0 has length exponentialg',._/ 1: times in the length of 1:written in binary; so we cannot prove in Bounded Arithmetic the totality of the function that sends natural numbers It to the code of S . . . S 0. Thus we use numerals that are based on the binary expansion of k and are of lerigtfhfllinearin the logarithm of k. Once we have made this move, it is not difficult to prove that L6b's logic L is sound with respect to Bounded Arithmetic (see subsection 2.33). Thus Godel's Second Incom- pleteness Theorem holds, and its formalized version is provable. Rosser strengthened G6del's First Incompleteness Theorem by constructing an arith- metical sentence R that is independent of Peano Arithmetic, provided that Peano Arith- metic is consistent [Ros 36]. In chapter 3, we prove the formalized version of Rosser's Theorem in Bounded Arithmetic. In more technical terms, we prove the arithmetical sen- tence expressing "if Bounded Arithmetic is consistent, then neither R nor -R are provable from it". Here R is constructed by G6del's method of diagonalization; informally, H says "there is a proof of my negation which is smaller than any proof of myself". The proof of the formalized version of Rosser's Theorem does not come cheaply. We use almost the whole chapter in order to prove a "small reflection principle" on which the proof is based. However, the real trouble only starts when we want to prove Solovay's Complete- ness Theorem for Bounded Arithmetic. As we mentioned, it is easy to prove that L is sound with respect to Bounded Arithmetic, but as far as we know the provability logic of Bounded Arithmetic might well be a proper extension of L. The problems we encounter when we want to adapt Solovay's construction to Bounded Arithmetic are discussed in chapter 3 and chapter 4. They are related to open problems in complexity theory. In chapter 5, we show that formulas having models on suitably simple Kripke trees can be translated into arithmetical sentences that are consistent with Bounded Arithmetic. We also show that the provability logic of Bounded Arithmetic cannot be the modal theory of a class of Kripke trees. The general question 'What is the provability logic of Bounded Arithmetic' is still left open for future research. Because interpretability logics always include a provability logic (-=A > _Lbeing equiv- alent to DA), we do not yet have enough material to find the interpretability logic of Bounded Arithmetic. 1.5 Efficient metamathematics of inefficient mathe- matics What happens when the techniques from eflicient metamathematics of efficient mathe- matics are turned loose on normal mathematics? We may study interpretability between extensions of inefficient theories like Peano Arithmetic and Zermelo Fraenkel set theory. For example, it has been known for a long time that in every model of Zermelo Fraenkel set theory, we can see a model of Peano Arithmetic. Now we might ask: can we see it in an efficient way? In other words, is there a translation from the language of arithmetic 1.6. WHAT TO EXPECT FROM THE REST OF THE DISSERTATION? 7 into the language of set theory such that set theory proves all axioms of Peano arithmetic by proofs that are easy to compute given the original axioms? The answer turns out to be yes. In chapter 6 we ask a more general question: are all interpretations that we know feasible? The answer is: yes and no! Yes, because well-known interpretations like that of ZF plus the negation of the continuum hypothesis into ZF are feasible. No, because we can use tricks like diagonalization to make some theories U, V such that U interprets V, but not by any feasible interpretation. We then restict our attention to feasible interpretability between finite extensions of Peano Arithmetic. What is the logic of feasible interpretability over such theories? It turns out to be ILM, the same interpretability logic that is arithmetically sound and complete with respect to normal interpretability over Peano Arithmetic. Finally, in chapter 7, we establish the intrinsic complexity of the formula "PA + A feasibly interprets PA+B". Feasible interpretability over PA turns out to be E3-complete, contrasting with the fact that standard interpretability over PA is I13-complete. We also prove that the formula "PA interprets PA + A but not by any feasible interpretation" is I13-complete. 1.6 What to expect from the rest of the disserta- tion? The remainder of part I contains some preliminaries needed to read parts II and III of the dissertation, as well as some material not covered in those chapters but interesting in its own right. Part II Chapter 3 A small refiection principle for bounded arithmetic. This is based on the paper [VV] written jointly with Albert Visser; which in its turn is based on [Ve 88] and [Ve 89]. Chapter 4 Provable completenessfor Eysentences implies something funny, even if it fails to smash the polynomial hierarchy. This is based on unpublished work with Alexander Razborov. It is reproduced here with his permission. Chapter 5 On the provability logic of bounded arithmetic. This is based on the paper [BV 91]. Preliminary results can be found in [Ve 88]. Part III Chapter 6 Feasible interpretability. This is based on [Ve93]. Chapter 7 The complexity of feasible interpretability. A previous version of this chapter has been submitted to the book Feasible Mathematics 11,edited by J. Remmel et al., to appear with Birkhauser.

Chapter 2 Background 2.1 Theories of arithmetic Definition 2.1.1 The language of arithmetic contains 0, S, +, -, = and 3. Definition 2.1.2 Robinson's Arithmetic Q is a theory in the language of arithmetic given by the following axioms: Q1 \/2:(S:z:¢ 0); Q2 V:I:,y(S:1: = Sy ->:1:= y); Q3 \7':I:(:1:75 0 -+3y 1: = Sy); Q4 Va:(:c + 0 = 1:); Q5 V33,y(:I: + Sy = S(:1: + 31)); Q6 V:1:(:I:0 = 0); Q7 V:r,y(:r ~53/ = (1: y) + Iv); Q3 VI)!/(33 S y H 3Z(Z + I = 31)); Definition 2.1.3 Peano Arithmetic PA contains the theory Q plus the induction scheme <p(0) /\ V~'c(<p(-'6)-> <p(5$)) -> V:w(1'), where cpmay be any first-orderformula in the language of arithmetic. Remark 2.1.4 We presuppose familiarity with the arithmetical hierarchy (see a textbook on recursion theory, e.g. [So 87]). We remind the reader that a formula in the language of arithmetic is A8 = 28 = H8 if all its quantifiers are bounded, which means that they are of the form V1:3 t where t is any term not involving :12.Z3?,+,-formulashave the form Elxcpfor some cpin H2. Dually, H9,"-formulas have the form Vzccpfor some cp in 29,. A8-formulasdefine primitive recursive relations of natural numbers, but not all prim- itive recursive relations are A8 definable: only those from the so-called linear time hier- archy (see [HP 93, Definition V.2.10, Theorem V.2.16]). X?-formulas define recursively enumerable (r.e) relations, whereas l'I','-formulasdefine co-r.e. relations. We usually use A0 for A8, 2,, for E9, and IL, for H91. 10 CHAPTER 2. BACKGROUND Definition 2.1.5 Let F be a class of formulas in the language of arithmetic. Then If' contains the theory Q and additionally the induction scheme <p(0) A V=v(<.0(9~')-> sp(5-73)) -> V2=<p(-'6) for cp E F. Remark 2.1.6 Note that the induction axioms for A0-formulas can be written in H1- form, namely as V1/(<P(0) /\ V93 S 3/(<P(='~')-* <p(5$)) -* V1: S 1/<P(17=')l fOI' (P 6 Ag. Definition 2.1.7 0 LU1(0) = O; 0 w1(:z:)= 12"'for x > 0; here = flog2(S:1:)l. Definition 2.1.8 0 23 = 1:; I 2':+1= We sometimes write ea:p(:c)instead of 2'. Many mathematicians, most notably Bennett, Paris, and Pudlak [Ben 62, Di 80, Pu 83] have constructed A0-formulas corresponding to the relation y = 2'. We refer the reader to [HP 93, Section V.3] for a clear description of the construction of such a formula. Similarly, there exist A0-formulas defining the relation 3/ = 2:', y = w1(a:) and even to the graphs of most functions that we introduce in section 2.3. For those functions we use "y = f(a':')" as shorthand for the appropriate Ao-formula. In IA0 we can prove the recursive clauses for these functions, as well as some other useful facts, for example that 2' grows faster than wl. Definition 2.1.9 0 EXP := V:c3y("y= 2"'); 0 SUPEXP := 'v':1:3y("y=2'"). We have the following hierarchy of theories: QQIAO§IAo+EXP§IAo+SUPEXP§I21§IZ2§...§PA. 2.2. PROVABILITY LOGIC 11 2.2 Provability logic Definition 2.2.1 The language of modal logic contains a countable set of propositional variables, the propositional constant J_, boolean connectives n, A, -+,and the unary modal operator D. The modal provability logic L is axiomatized by all formulas having the form of propositional tautologies (including those containing the D-operator) plus the following axiom schemes: 1. Cl(A ->B) -» (DA ->DB) 2. D(ClA -->A) -+ DA 3. CIA -vDU/l The rules of inference are: 1. If lA ->B and lA, then lB (modus ponens) 2. If lA, then IDA (necessitation) We suppose that the reader is familiar with Kripke frames and forcing relations on them. For L, finite Kripke trees give a good semantics. To make this precise, we state a definition and a theorem. Definition 2.2.2 A Kripke tree is a frame (K, 4) in which 0 4 is a strict partial ordering, i.e. it is transitive and asymmetric; o for every element of K, the set of its predecessors is finite and linearly ordered by 4; 0 there is one root which precedes all other elements. L6b's logic L is modally complete with respect to finite Kripke trees: Theorem 2.2.3 For every modal sentence A, the following are equivalent: 1. L lA; 2. for all finite trees (K, <) and points I: E K and for allforcing relations II-on (K, <), we have It Il-A; 3. for all finite trees (K, -<) and for all forcing relations H-on (K, <), we have kg H-A where k0 is the root of(K, 4). Proof. See [Sm 85, Theorem 2.2.3] QED Definition 2.2.4 Let T be a theory in the language of arithmetic. A T-interpretation * is a function which assigns to each modal formula A a sentence A' in the language of T, and which satisfies the following requirements: 1. .L" is the sentence 0 = 1. 2. ' distributes over the boolean connectives, i.e. (A ->B)' = A' ->B', etc. 12 CHAPTER 2. BACKGROUND 3. (DA)'' = Prov1~('A*"). Here ProvT("/1") is the formalization of "A' is provable from T" (see subsection 2.3.3). Clearly ' is uniquely determined by its restriction to the propositional variables. The presence in the modal language of the propositional constant .1.allows us to consider closed modal formulas, i.e. modal formulas containing no propositional variables. If A is closed, then A' does not depend on ", e.g. (DJ.)* is the arithmetical sentence Provq~("O= 1"). For arithmetical recursively enumerable theories T with IAO+ EXP Q T Q PA we have for all modal formulas A: L lA <=> for all T-interpretations ', T lA'. The soundness direction (=>) is not difficult to prove. It hinges on the observation that for T Q IAO+ EXP the formalized version of L6b's Theorem is provable: T |Prov1~("Provq~('-go") ->cp") ->Prov1~("cp_'). Incidentally, the formalized version of L6b's Theorem immediately implies the formal- ized version of G6del's Second Incompleteness Theorem by taking _Lfor cp: T lProv1~("ProvT('-J_") ->_L") -+ProvT("_l_"). Solovay proved the completeness direction (<=) for PA in his landmark paper [So 76]. It was pushed down to IAO+ EXP in [JMM 91]. 2.3 Bounded arithmetic In this section we introduce classical arithmetical theories that are strictly weaker than IAO+ EXP. It turns out that there are two salient theories of this kind: Paris and Wilkie's IAO + 91 [WP 87] and Buss' S21[Bu 86], both of them satisfying L6b's logic. We will not state all the interesting results that appeared in the standard references to the area of weak arithmetics (see [Bu 86, WP 87] and Chapter V of [HP 93]). Instead we quickly review those concepts that we need in the sequel. The principal feature distinguishing various theories of Bounded Arithmetic from Peano Arithmetic is that in the former induction is restricted to bounded formulas. 2.3.1 IAO+ 91 Definition 2.3.1 The language of IAo+Q1 as introduced in [WP 87] contains 0, S, +, -, = and 3, and additionally the logical symbols -=,->and V, and variables 121,222,. . .. With regard to logical axioms, we use a Hilbert-type system as in [WP 87], but other choices are reasonable too. For example, a Gentzen style sequent calculus with cut rule or natural deduction would do. However, we do not use a logic in which only direct proofs (i.e tableau proofs or cut-free proofs) are allowed. As non-logical axioms we consider a set containing the following: o a finite number of universal formulas defining the basic properties of the function and predicate symbols of the language: 1. ogo/\-.(50go); 2.3. BOUNDED ARITHMETIC 13 .V:1:(:r+O=:z:/\:£-0=O/\a:-S0=a:); . \7':z:Vy(S:z:=S'y->:z:=y); V:vVy(-"BS 53/"" (-72SyV:I== 3.74)); 'v':z:'v'y(:c+Sy=S(1:+y)); V2:Vy(a:-Sy=(1:-y)+x);.°°.°'r"'°°'° o a formula V:1:Ely<p(x,y), where cpis the A0-formula defining the relation 3/= w1(1:) (= :1:"l;see definition 2.1.7); o the scheme of induction for A0-formulas. 2.3.2 Buss' systems of bounded arithmetic and the polynomial hierarchy Definition 2.3.2 The language of Buss' bounded arithmetic consists of O, S, +, -, =, 3, (= l'log2(:z:+ 1)], the length of the binary representation of 2:), and :1:#y (= 2l"'l""l,the smash function). Remark 2.3.3 Note that the smash function # allowsus to express terms approximately equal to 2P(l""l)for any polynomial P. More precisely, for every 72,1: 2 2 the following holds: 2'='" < a:# . . . #1; < 22"='"'2,_ _ n times as is easily proved by induction. This property of # is useful when we want to define polynomial time functions. Definition 2.3.4 The hierarchy of bounded arithmetic formulas is defined as follows: 1. 28 = N3 = A8 is the set of formulas with only sharply bounded quantifiers V1:3 |t|, 3:1:3 |t| (where t is any term not involving 3:) 2. 231+, is defined inductively by: 0 2?," I_>Hi, and is closed under A, 32: 3 t and V2:3 |t|; 0 if B 6 1-1:", then fiB E 2%". 3. Iii" is defined inductively by: 0 Hi" Q 2:, and is closed under /\, 'v':c3 t and 33: 3 |t|; 0 if B E 2):", then fiB 6 Hz". 4. ELL, and Hf," are the smallest sets which satisfy 2,3. Definition 2.3.5 If R is a theory and A a formula, we say that A is AL, with respect to R iff there are formulas B 6 232+, and C' 6 UL, such that R lA <-+B and R lA H C. 14 CHAPTER 2. BACKGROUND We never leave out the superscripts b from the levels Elf, and Hf',of Buss' bounded arithmetical hierarchy, so our use of 2,, for E9, and IL, for H9, should not give rise to confusion. The hierarchy of bounded arithmetic formulas is constructed in such a way that all levels H? and 2? except E3 correspond to levels of the polynomial hierarchy, which is well-known from structural complexity theory. Without defining all the basic notions of complexity theory, for which the reader may turn to [BDG 87], we give one of the standard definitions. Definition 2.3.6 The polynomial hierarchy is defined as follows: 1. P = N1'is the set of predicates on the natural numbers which are recognized by a deterministic polynomial time Turing machine; 2. NP = Elf is the set of predicates on the natural numbers which are recognized by a nondeterministic polynomial time Turing machine; 3. {If is the set of predicates Q such that there is an R E A? and a polynomial P, such that for all :3, Q(a':') <=> 3y 3 2P('3')R(:i:',y). 4. H? is the set of predicates Q such that there is an R E Elf, so that for all if, Q(:E) <=> -R(:i~'). 5. A?" is the set of predicates which are recognized by a deterministic polynomial time Turing machine with some oracle from Elf. As usual we use the name co-NP for Hf. There are many open questions about the polynomial hierarchy. The most important one is: is there a Is:such that 27; = E',';+1, in which case the hierarchy collapses? More particularly, does NP = co-NP? Or even P = NP? It is also unknown whether for any k, A' = Zfiflllfi, and in particular whether P = NP 0 co-NP. Definition 2.3.7 A is polynomially reducible to B if there is a polynomial time com- putable function f such that V:r(:z:E A <-+f(:t:) E B). Note that polynomial reducibility is analogous to many-one reducibility from ordinary recursion theory. Definition 2.3.8 B is NP-complete if all A E NP are polynomially reducible to B. Similarly, B is co-NP-complete if all A E co-N P are polynomially reducible to B. Remark 2.3.9 It is easy to see that for every N P-complete set B, the following holds: 0 If B E Co-NP, then NP = co-NP; o IfB€P,thenP=NP. Remark 2.3.10 From results of Stockmeyer, Wrathall, and Kent and Hodgson [St 76, Wr 76, KH 82] it follows that the bounded arithmetical hierarchy is related to the poly- nomial hierarchy in the following way: XiH is the class of predicates which are defined by formulas in 25',+1. In particular, NP is the class of predicates which are defined by 2'1'-formulas; similarly co-NP is the class of predicates defined by II'{-formulas. We refer the reader to [Bu 86, Chapter 1] for proofs of these correspondences. 2.3. BOUNDED ARITHMETIC 15 Definition 2.3.11 The theory S; consists of BASIC, a finite list of axioms defining the basic properties of symbols in the language of bounded arithmetic, plus the following induction scheme PIND(2f): A(0) /\ vx(A(L:xj) -»Am) -»'v':rA(:z:) for A E 2?. Definition 2.3.12 S2 := U,S;. Definition 2.3.13 The theory T; consists of BASIC plus the followinginduction scheme: A(0) /\ 'v'x(A(:1:)-> A(S:1:)) ->V:rA(:1:) for A E Elf. Definition 2.3.14 T2 := U,«T;. Buss proves that for each 2',S;" F T; (see [Bu 86, Corollary 2.21]). It is clear that also for each 2'Z 1, T; lS;. Thus, T2 = S2. One of the most important theorems about bounded arithmetic is Parikh's Theorem. It implies that every A0-definable provably total function of S2 can increase the length of its input only polynomially. Parikh originally proved his theorem for IA0, for which the A0-definable provably total functions are even more severely limited than for S2: they can increase the length of the input only linearly. We state a version of Parikh's Theorem for Buss' theories S;. Theorem 2.3.15 (Parikh's Theorem) Lett 2 0. Suppose that «,0is a boundedformula and that S; l- 'v'1:Elyc,0(x,y).Then there is a term t(a:) such that S; l- \:/:z:Ely3 t(:c)cp(:z:,y). Proof. Buss gives a proof-theoretic proof (see [Bu 86, Theorem 4.11]). However we prefer to give a model-theoreticproof, because it is easier to understand and much shorter. So suppose that there is a bounded formula upsuch that S; l- 'v':z:Elynp(a:,y), but for every term t(:1:), S; l7''v':r:3y5 t(a:)cp(:1:,y). Now if c is a fresh constant, the set of formulas 5$+{Vy S c#...#c-=s0(c.y) I /66w} \ is finitely satisfiable. Thus by the Compactness Theorem there is a model M l= 5%+ {Vy S c#...#c-'<p(c.y) I /9E w}-\__.',:/ 1: times Suppose that a is the interpretation of c in this model. Next, take the submodel M' of M defined by: M' := {b€M | 3nEw(b§c#...#c)}. n times It is easy to check that M' is closed under 0, S, +, -, S, |:z:|, [;:z:j, and #. Moreover the induction axioms of S; can be written in H1-form, so they still hold in .M's initial segment ./V1'.Therefore M' l: S;, but M' bé 33/<p(a,y),contradicting our first assumption that S; l- 'v':z:3ycp(:c,y). QED 16 CHAPTER 2. BACKGROUND Definition 2.3.16 121','-functionsare those computable by a polynomial time Turing ma- chine. For i > 1, Elf'contains those functions computable by a polynomial time Turing machine from finitely many oracles in 2','-'_1. Buss proved that the provably total 2'1'-definablefunctions of S; are exactly the func- tions computable by a polynomial time Turing machine. More precisely, and at the same time more generally, we have the following two theorems: Theorem 2.3.17 Let i 2 1. Let g be an m-aryElf-function. Let t(a':') be a term so that for all 5:'E cum, g(:z':')3 t(a':'). Then there is a E?-formula A such that: 1. S; lV:i:'3y3 tA(:i:',y); 2. S; lV/:i:',y,z(A(:i:',y) A A(:i:', z) ->y = z); 3. For all 5:'E w"', A(;i:',g(;i:')) is true. Proof. See [Bu 86, Theorem 3.1] QED Theorem 2.3.18 (Buss' Main Theorem) Leti 2 1. Suppose S; lV:z':'ElyA(x,y)where A(:c,y) is a Zf-formula with only :i:',yfree. Then there is a term t(:i:'), a Sf-formula B and a function g in Df' such that 1. S; lVa':'Vy(B(.7':',y) ->A(:z':',y)),' 2. S; lV:i:"v'y,z(B(:f,y) A B(:E, z) ->y = z); 3. S; l- 'v':i:'3y3 tB(:i:',y); 4. For all ii, to ]= B(fi,g(n')). Proof. See [Bu 86]. Buss uses methods well-known from proof theory. We give a short sketch. Suppose S; lV:i:'ElyA(:r,y),by a proof p. Then we can apply cut elimination to obtain a term t and an S; proof p' of 'V/2':'3y3 tA(:z:,y) that only cuts Z?-formulas. Thus, p' contains only 2;' and I1?-formulas. Next we can directly extract from p' a Bf-algorithm for computing a function g such that for all n',w ]=A(r'i, For an elegant model-theoretical argument, which is inspired by Visser's unpublished proof of Parson's and Mints' theorem [Pa 72, Mi 71] that the primitive recursive functions are exactly the provably total functions of 121, see [Za 93]. QED Remark 2.3.19 Note that, if for some Z?-formula A and some term t(:1':'), \7':i:'3y3 tA(:ic',y) is true but not necessarily provable in S;, then we know only that there is a witnessing function in Bf". Corollary 2.3.20 Let A(a') be a formula such that S; proves that A is equivalent to a E','and to a H';-formula. Then A is a polynomial time predicate. In other words, if S; proves that some predicate is in NP 0 co-NP, then it is already in P. We remind the reader that it is an open question whether NP F1co-NP = P. Even though we do not need it in the sequel of the dissertation, we cannot resist the temptation to end this introduction to relations between complexity theory and bounded arithmetic with the statement of a beautiful result by Krajicek, Pudlak and Takeuti. Theorem 2.3.21 Fori 2 0, if T; F S;", then 2?" = Hf". Proof. For the very ingenious argument, see [KPT 89]. QED 2.3. BOUNDED ARITHMETIC 17 2.3.3 Metamathematics for bounded arithmetic In order to prove G6del's Incompleteness Theorems for bounded arithmetic, Buss arith- metized the usual notions of metamathematics (see [Bu 86, Chapter It turns out that most predicates needed can be All'-defined (or sometimes ElA'1'-defined)in 5;. Moreover, these definitions are intensionally correct in the sense of [Fe 60], which means that the usual connections between them can be proved in S2}. Here follows a list of predicates used in the sequel. 0 Seq(w) for "w encodes a sequence''; 0 Len(w) = a for "if w encodes a sequence, then the length of that sequence is a; otherwise a = 0"; 0 Term(v) for "v is the Godel number of a term"; 0 Fmla(v) for "v is the Godel number of a formula"; 0 PrfT(u, v) for Fmla(v)/\ "u is the Godel number of a proof in T of the formula with Godel number 22";when T is clear from the context, we drop the subscript. o ProvT(v) := 3uPrfT(u,v); we sometimes abbreviate Prov(rg0") as Clap. The predicates Seq, Len, Term, and Fmla are A'{-definablein S, and so is Prfo, where the formula a is A'; with respect to 521. The condition on a is not a severe restriction. To any recursively enumerable set one can associate a polynomial time function having that set as its range, therefore one can suitably axiomatize any theory T which has a recursively enumerable set of axioms including BASIC. Notation 2.3.22 Instead of the usual numerals S"Oof Peano Arithmetic, we use canon- ical numerals /icdefined inductively by: 0 U = 0; . 27:71 = 216+ (50); . §k'+i2 = (SSO) - (16191). Note that the length of the term It is linear in the length of the binary representation of k, a property that the S'°Oobviously do not satisfy. The shortness of canonical terms plays a crucial role in many proofs, for example in Buss' proof that S; enjoys provable Z3';-completeness (see [Bu 86, Theorem 7.4]). S; can ZI','-definea function Num(:z:) such that Num(:c) stands for the Godel number of the term ':5.For ease of reading, we will however abuse notation; thus if A(:z:)is a formula with free variable 1:we write 'A(a)' instead of Sub('/1", ':c",Num(a)). Sometimes we are even more sloppy and leave out the numeral dashes altogether. In those cases the context should provide enough material for the reader to know what is meant. Lemma 2.3.23 (cf. Lemma 7.5 of [Bu 86]) Let t be any term with free variables a1, . . . ,a,,. Then S21l- 'v'a1,...,akProv("t(E,...,Zi;;) = t(Tczl1T,i...-,T(1_,,-)_'). 18 CHAPTER 2. BACKGROUND Proof. We use induction on the complexity of t. o If t is 0, then the equality axioms immediately give us S; lProv('(7 = U"). o Ift is a variable symbol :12,then the equality axioms give us S; lV:::Prov('":E= E"). 0 If t = Sr, then we have by induction hypothesis S; l- 'v'a1,...,a;.Prov(rr(fi,...,cfi) = r(a1, . . . ,ak)'). Therefore it suffices to show that S21lVbProv("Sb = Q"). In order to be able to apply the formalized version of PIND(Z3'1')in S2', we need to find some fixed polynomial P such that the proofs of Sb = ST)are of length 3 P(|b|), because then there is a term t(:1:)for which we can prove S2'l- \7'bEly3 t(b)Prf(y, "Sb = Sb"). The polynomial that we need will be quadratic. We leave the exact computation to the reader. Informally, the reasoning inside S; is as follows: - We clearly have Prov('SU = S_O'). -Suppose that b > 0 is even. By the definition of efficient numerals and the BASIC axioms we immediately have proofs of length linear in |b| of Sb = 5+ 30 = 37:. -Suppose that b >Li_s odd. Thenle_have a proof from BASIC of length linear in |b| of Sb: 2Lgbl+2 =2-Slgbj. By the induction hypothesis and th_e__BASI_C__ax_iomswe have a proof of length quadratic in = |b| -1 of =g_S[$bj.Combined,these twogivea proof of length quadratic in |b| of Sb = Sb. 0 We leave the cases for +, -,#, and | | to the reader. QED Theorem 2.3.24 (Provable 2'1'-completeness, Buss) Let A be any Z'1'-form'ula.Let a1, . . .,ak be all the free variables of A. Then there is a term t(a1, . . . , ak) such that S; l- 'v'a1,...,ak(A(a1,...,a;,) -+Elw3 tPrf(w, rA(cT1',. . . ,a7;)')). Proof. We give only a small hint: the reader may look up the full proof in [Bu 86, Theorem 7.4]. We use induction on the complexity of A. The most difficult step is the one for the bounded universal quantifier. So suppose that A is Va:3 |s|B(a1, . . . ,a;,,:r:), and that for all b 3 |s| we have proofs of length polynomial in max(|a1|, . . . , |a,,|, of B(a1, . . . , a;., b). Then we can combine these |s| + 1 short proofs in order to construct a proof of length polynomial in max(|a1|, . . . , |a;.|, |s(a1, . . . ,ak)|) of the formula V3 3 |s|B(a1, . . . ,ak,:1:). QED Using theorem 2.3.24, it is easy to see that L6b's logic (see definition 2.2.1) is arith- metically sound with respect to S2}.In particular this means that we can, in the standard way, prove Godel's Second Incompleteness Theorem and its formalized version for S21. Theorem 2.3.25 (cf. Theorem 7.10 of [Bu 86]) S; if -vProv('J_") and S; lProv('-Prov(".l_") ->_L") ->Pr0v(r.L"). 2.3. BOUNDED ARITHMETIC 19 Proof. We leave the well-known proofs to the reader. QED We refer the reader to section 2.6 for the proof of a much stronger result: IAo+EXP I7' Con(Q). By the way, Rosser's strengthening of G6del's Second Incompleteness Theorem is also provable for theories like S2}and IAo+Q1. However, we use the "small reflection theorem" to prove it. Thus the reader will have to wait for theorem 3.3.24 of Chapter 3 in order to find Rosser's Theorem as a corollary. Sometimes, we will use the name IAO+ 91 for Buss' theory S2 (see Definition 2.3.12), in which induction for formulas from the hierarchy of bounded arithmetic formulas in a language containing # and | | is allowed. Because S2 is a conservative extension of IAO+ 91, the name change has no repercussions on results that do not hinge on the details of formalization. More formally, we have the following: Lemma 2.3.26 There is a A0-formula 1l2(:z:,y,z)such that w I: :z:#y = z 4-»1/2(a:,y,z), IA0 + 01 proves the BASIC properties of# for zp, and 1. IAO + 91 lV:1:,y3zw(a:,y, z) and 2IA0 + Q1l_ V-73.3/aZ1aZ2(1l'(17a3/all) A 7"-73:?!»Z2) -+Z1: 22)' Similarly there is a A0-formula X(a:,y) defining | | in IA0 + 91, and there is a A0- formula {(;z:,y)defining in IAO+ Q1. Definition 2.3.27 A Ao(f1,...,f,,)-formula is a bounded formula in the language of arithmetic to which the function symbols f1,. . . ,f,, are added. Lemma 2.3.28 130 + 01+ V:ryz(($#y = 2 H w(1=,y,z))/\(|r| = y H x(1=.y))/\(l;';IBl = y <->{(13.1/)) + IA0(#, | |, Lgxj) /\ "BASIC". Proof. As [Bu 86, Theorem 2.2 and Corollary 2.3]. See also [PD 82] for a more general lemma. QED Definition 2.3.29 (see [Pu 85]). A theory T is sequential if it is a theory with equal- ity, there is a distinguished provably non-empty domain N(:z:) that satisfies Robinson's Arithmetic Q, and there exists a formula fi(t, w, z) ("t is the w-thelement of the sequence coded by 2'') such that: T l- \7':I:,y,vElz'v't,w [N(v) /\ w 3 v ->(B(t,w,z) H ((B(t,w,:r) Aw < 12)V (t = y Aw = Examples of sequential theories are IAO,S; for i 2 1, PA, ZF, and GB. 20 CHAPTER 2. BACKGROUND 2.4 Interpretations Tarski introduced the formal notion of interpretability in [TMR 53]. We give a variant of his definition here. Let U, V be two E';-axiomatized theories in languages containing finitely many non- logical symbols. Let the axioms of V be given by the 2'1'-formulaay. An interpretation K of V into U is given by: 0 a formula 6(3) of LU defining the universe, such that U lEl:2:6(:z:); o a function from the relation symbols of Ly to formulas of LU, respecting the original arities; 0 a function from the function symbols f of Ly to formulas wf of LU, such that if f is k-ary, then 1/), has k + 1 free variables and U l6(J:1) A . . . A 6(:c;.) -+3!y(6(y) /\ 1pf(:r1,. . . ,:1:;,,y)). We warn the reader that the image of = need not be =. We can extend K in the obvious way to map all formulas cpof Ly into formulas cpKof LU. To do this we relativize all quantifiers to 6 while we respect the Boolean connectives. In fact we can, in an intensionally correct way, A';-define in IAO + 01 a function K corresponding to this mapping. For ease of reading, we will write a" even if a is a Godel number. Thus U D V can be defined in as follows: U D V :<->ElK("Kis an interpretation" /\ 'v'a(ay(a) ->3pPrfU(p,a.K))). (2.1) (By abuse of notation we denote by D both the arithmetization of the interpretability predicate and the corresponding modal operator that we will introduce in section 2.5.) It would be nice to be able to prove in the theories that we are interested in such as IAO+ 91 that interpretability gives rise to relative consistency. However it seems that one cannot do this straight-away,but one needs a collection principle: Definition 2.4.1 B21 2: IA0 plus the scheme Vu(\7'a: < u3y<p(:z:,y) -->3v'v':c < u3y < vcp(:z:,y)) for «,0E 21. Indeed we have BZ1+ Q1lU D V ->(Con(U) ->Con(V)). We leave the proof to the reader. However, we prefer to make a definitional move. For the remainder of the dissertation, we define U D V as smooth interpretability (discussed in [Vi 91a]): U DV :<-->ElK("Kis an z'nterpr."/\'v'u3v'v'a< uflp < v(ay(a) ->PrfU(p,aK))).(2.2) Now we do have IA0+Q1lU D V -+(Con(U) ->C'on(V)). 2.4. INTERPRETATIONS 21 For theories containing B21 + 01 the definitions of standard interpretability (2.1) and smooth interpretability (2.2) collapse. In Part III of this dissertation, we are concerned mostly with extensions of PA Z_> B21 + 91, thus we may freely use the standard definition. Moreover, for extensions of PA one can, and we will, without loss of generality assume the image of = to be =. This restriction makes life easier, although it is not essential for most results. We can view interpretability in a semantic way. An interpretation K of V into U determines, in every model M of U, a new model MK with underlying set {a E M | M ]= 6(a)}. The reader may check that for every a1, . . . , a;. E M", we have the following: MK ]=<p(a1,...,a;,) <=> M ]=cp(a1,...,a,,)K. For finitely axiomatized theories V, Montague in [M0 65] was the first to explicitly relate the syntactic and semantic definitions of U D V. It is interesting to note that many famous relative consistency proofs in the mathe- matical literature arise from interpretations. Thus we have both ZF D ZF + CH and ZFC DZFC + -CH. For arithmetical theories extending PA, Hajek [Ha 71]gave an elegant characterization of interpretability. Because the characterization was implicit already in Orey's [Or 61], many authors refer to it as the Orey-Hajek characterization. In order to describe it we first need two definitions and a lemma. Definition 2.4.2 Prov,,,~_r('A")stands for "there is a proof of A from T in which only axioms with Godel number 5 k are used". C'on,,(T) := nProv;,,T('_L"). Definition 2.4.3 A theory T with T Q 121 in the language of arithmetic is essentially reflexive if for all sentences A and for all k, T lProv;.,T('A") ->A. Remark 2.4.4 In the literature, different definitions of essential reflexivity abound. For example, [HP 93, Definition III-2.33] is as follows: A theory T with T Q 121 in the language of arithmetic is essentially reflexive if for each theory T' Q T and for all k, T' lCon;.(T'). Lemma 2.4.5 IfT is an extension of PA by a primitive recursive set of axioms in the language of PA, then T is essentially reflexive. Proof. See [Ber 90, Theorem 2.6]. A feasible version can be found as our lemma 7.2.5. QED Theorem 2.4.6 (Orey-Héjek, [Or 61], [Ha 71]) Let U and V be primitive recursive extensions of PA in the language of PA. Then the following holds: PA lU D V 4-»'V/lcProvu("Con,.( V)"). Proof. See [Ber 90, Theorem 2.9 and Remark 2.10]. For a feasible version, see lemma 7.3.1 of this dissertation. QED Definition 2.4.7 A theory V is II,-conservative over U if for all H1-sentences 7r, V l7r é U l7r. 22 CHAPTER 2. BACKGROUND We abbreviate the formalization of "V is H1-conservative over U" as U l>n, V. Theorem 2.4.8 Let U and V beprimitive recursive extensions of PA in the language of PA. Then the following holds: PA lU l>n, V 4-»VkProvU("C'on,.( V)"). Proof. See [HP 93, Theorem III-2.40]. A feasible version appears as lemma 7.3.2. QED Theorem 2.4.9 Let U and V be primitive recursive extensions of PA in the same lan- guage. Then the following holds: PAl-UI>V<-»UI>n,V. Proof. Immediately from theorem 2.4.6 and theorem 2.4.8. See corollary 7.3.3 for a feasible version. QED Although the modal principle M (see section 2.5) corresponding to the following the- orem was baptized 'Montagna's Principle' in the eighties, the unformalized version of the underlying theorem was proved by Lindstréim already in the seventies. Theorem 2.4.10 (Lindstriim, [Li 79]) Let U and V beprimitive recursive extensions of PA in the language of PA. Let S be a primitive recursive set of Z?-sentences. Then the following holds: PAl-Ul>V-»U+Sl>V+8. Proof. We remind the reader that for interpretations between theories extending PA, we take the image of = to be =. A precise formal proof can be gleaned from the proof of theorem 6.3.10 in chapter 6. Here we give a sketch with a more model-theoretic flavor. It is easy to see that the following fact implies our theorem: Let M be a model of U, and let K be an interpretation of V into the theory of M; we call the interpreted structure MK. Then M can be embedded as an initial segment of MK. In order to prove this fact, we define pism(s) for "s is a partial isomorphism" and the relation G(x,y) as follows: pism(s) := seq(s) /\ (s)o = OKA Vi < lh(s) -1((s),-+1 = SK(s),~) G(j, y) := 3s(pism(s) /\ lh(s) = j + 1 A (s),- = y) By induction it follows that for every x E M there is a unique y E MK such that M l= G'(x,y). Therefore, there is a function g corresponding to G. It is easy to see that g is an embedding into MK and that it preserves 0, S, + and -. Now we need only show that the image of M is an initial segment of MK. But because V 2 PA, and M l: Vx(g(Sx) = SK(g(x))), this is not difficult: we have M I: 'v'x'v'u(6(u)/\ u <K g(Sx) --+u <K g(x) V u = g(x)). Now by induction on x E M we find that for every u 6 MK such that M I: u <K g(x), there is a y E M such that Ml=y<xandMl=u=g(y). QED 2.5. INTERPRETABILI TY LOGIC 23 For finitely axiomatized theories the situation is different. As reflexivity for such theories would be in contradiction with G6del's Second Incompleteness Theorem, the Orey-Hajek characterization is not applicable (see however section 2.7). Instead, we have Friedman's characterization. Readers unfamiliar with tableau prov- ability may find a description in definition 2.8.2. We define Tabprovp(rcp") := -1Tabcon(T + mp). Theorem 2.4.11 (Friedman's characterization) SupposeU and V are sequentialthe- ones and V is finitely axiomatized. Then IA0 + EXP lU D V 4-» Tabprov1A0+EXp(" Tabcon(U) ->Tabcon(V)-'). Proof. See [Vi 90a]. QED It is also clear that for finitely axiomatized theories U, V, U D V is 3271',so 1A0+o,+U >1/-.nQ(U DV). Examples of finitely axiomatizable sequential theories are IAO+ EXP (even verifiably in IAO + 01; see lemma 2.8.22), IAO + SUPEXP, and S._§_and III, for 2' 2 1. At the moment of writing, as far as we heard nobody knows whether IA0 and 1A0 + 01 are finitely axiomatizable. 2.5 Interpretability logic Interpretability logic extends provability logic. The modal formulas A D B correspond to arithmetical formulas T + A' D T + B', where T is an arithmetical theory. Definition 2.5.1 IL contains the provability logic L (see definition 2.2.1) plus the fol- lowing five axiom schemes: J1 C1(A->B) ->(A DB); J2 (A DB)/'\(B [>C)--+(ADC); J3 (A DC)/\(B I>C)-+(AVB DC); J4 (A D B) ->(<>A -»<>B); J5 <>A D A. Definition 2.5.2 ILM = IL+M, where M is the axiom (A DB) ->(AADC DB/\ElC'). Definition 2.5.3 [LP = IL + P, where P is the axiom (A D B) ->C1(AD B). Definition 2.5.4 An IL-frameis a frame (W, R), where R is a transitive conversely well- founded relation on W, with additional relations S,,, for each 111E W, having the following properties: 0 Su, is a relation on {w' E W | wRw'}; 0 S.,, is reflexive and transitive; 24 CHAPTER 2. BACKGROUND 0 if wRw', wRw" and w'Rw", then w'Sww". Definition 2.5.5 An I LM-frame is an IL-frame satisfying the followingextra condition: 0 if uSwvRz, then uRz. Definition 2.5.6 A simplified ILM-frame is a frame (W,R), where R is a transitive conversely well-founded relation on W, with root b say, with one additional binary relation S such that 0 S is reflexive and transitive and R Q S; o if uSvRz, then uRz. Definition 2.5.7 An IL-modelis given by an IL-frame (W, R, {Sw | w E W}) combined with a forcing relation satisfying the following clauses: o u H-DA if and only if Vv(uRv => 22|l~A); o u II~A [> B if and only if Vv(uRv and v If A => 3w(vSuw and 21)Il- In [JV 90], de Jongh and Veltman prove that IL is modally sound and complete with respect to IL-models, and that ILM is modally sound and complete with respect to IL-models on ILM-frames. Visser showed that ILM is already complete with respect to models on simplified ILM-frames; for a proof see [Ber 90]. Definition 2.5.8 A T-interpretation is a map " which assigns to every propositional variable p a sentence p' of the language of T, and which is extended to all modal formulas as follows: 1. (A l>B)*=T+A" I>T+B" 2. (ElA)* = Pr0vT(A*) 3. * commutes with the propositional connectives. Here l> abbreviates the formalization of (smooth) interpretability. IL is arithmetically sound with respect to sequential theories extending IAO+ Q1. Smoryriski and Visser proved that ILP is arithmetically sound and complete with respect to the finitely axiomatized theories GB and ACAO. Next Visser generalized the result and proved arithmetical soundness of [LP with respect to finitely axiomatized sequential theories extending IAO+ SUPEXP (see [Vi 90a]). This means that for such theories T and for all modal formulas A, ILP lA <=> for all T-interpretations ", T lA'. Berarducci [Ber 90] and Shavrukov [Sh 88] independently proved that ILM is arith- metically complete with respect to interpretability over PA. It is also arithmetically sound (see [Vi 90a]). 2.6. DEFINABLE CUTS 25 2.6 Definable cuts Because PA proves induction for all first order formulas, no proper cuts of models of PA can be defined by formulas. In the context of weaker theories where induction is restricted to a proper subset of all formulas, on the contrary, definable cuts have proved to be highly useful tools. For example, in weak theories like IA0, fast growing functions such as exp are not provably total. For results that one normally derives using the totality of such functions, one can find analogs in weak theories by constructing small cuts of numbers for which some of the fast growing functions in question do have the necessary values. The reader will find a formalization of this intuition in lemma 2.6.9, lemma 2.6.11 and theorem 2.6.12. Moreover definable cuts provide very natural interpretations in which the domain is restricted, but the original operations are left intact. We give examples of interpretations by definable cuts in lemma 2.6.14 and theorem 2.6.16. Such interpretations in turn give rise to relative consistency results which are provable in theories as weak as IAO+ 91. It is time to give some formal definitions. Definition 2.6.1 Let T Q Q be a XI'1'-axiomatizedtheory. A T-cut is a formula I such that: 1. T r 1(0), 2. T lV:z:Vy(I(y) /\ :1:3 y ->I(:c)), 3. T lV;z:(I(:1:) ->I(S:c)). Definition 2.6.2 Let T Q Q be a ZI'1'-axiomatizedtheory. A T-initial segment is a formula J such that: 1. TlJ(0), 2. T lV.7:'v'y(J(y) /\ as g y -»J(:z:)), 3. T lV1:'v'y(J(:c)/\ J(y) --+(J(S2:) /\ J(:I: + y) /\ J(:I: Remark 2.6.3 Note that if J is a T-initial segment in an arithmetic language, it de- termines an initial substructure in every model of T. Because J is T-provably closed under 0, S, + and and because the induction axioms of IA0 can be written in H?-form (see remark 2.1.6), these substructures will themselves be models of IA0. Thus T-initial segments provide interpretations of IA0 into T. Remark 2.6.4 For cuts 1, wefrequentlywrite 1:E I instead of The word cut is not used uniformly in the literature. For example, IAO+ Q1-cut often refers to a IAO + Q1-initial segment which is even provably closed under wl (see e.g. [Vi 90a]). The reason that in many applications such confusion is not harmful is provided by lemma 2.6.6 and lemma 2.6.10. Lemma 2.6.5 Suppose that T Q IAO and let I be a T-cut. Then there is a formula J such that 1. T lV:z:(J(:1:) ->I(:1:)); 26 CHAPTER 2. BACKGROUND 2. J is a T-cut; 3. T lV:cVy(J(:z:)/\ J(y) -+J(a: + y)), i.e. J is closed under +. Proof. Take J@)**KflAVMKw-*K$+wl It is easy to see that T lV:z:(J(a:)->I(2:)) and that J is a T-cut. For closure under +, reason in IA0 and suppose that $1,1226 J and that y E I. Then by definition of J we have, first, 1:1+ :52 E I. Also y + 2:1 E I, thus y + ($1 + 1:2) = (y + :1:1)+ 2:2 6 I. We may conclude that 171+ 21:26 J. QED Lemma 2.6.6 (Solovay's shortening lemma, [So 76b]) Suppose that T Q IAOand let I be a T-cut. Then there is a formula K such that 1. T lV:z:(K(:z:) ->I(:1:)); 2. K is a T-initial segment; Proof. First construct J from I as in lemma 2.6.5. Next, define K@%HMflAWUm-Jwwfl We leave it to the reader to prove that K is indeed the desired T-initial segment. QED The following lemma 2.6.8 is used in almost all applications of cuts. Note that it is essential that we use the efficient numerals 2':which are based on the binary expansion of 2:. First we introduce a notational convention. Notation 2.6.7 As in Pudlak's papers [Pu 86], we use the following notation: TP¢ to denote that there exists a proof of cpin T whose length (to which the length of proof lines contributes) is 3 n. Furthermore we use Tgww mm to denote that for some polynomial P we have for all n, T l-)- <p(n). Par abus de langage, we also use these abbreviations in formalized contexts whenever we think that their use will not confuse the reader. Lemma 2.6.8 (Pudlak) Suppose J is a T-initial segment. Then T l-|'_'-| Also we have IAO + 01 l- \7'1:Prov1~('J(:f)'). Proof. We give only a sketch, and leave the formal details to the reader. Essentially, in the proof of J(:T:), we follow the |x| steps it takes to build E from 5. At every step we instantiate either the proof of Vy(J(y) ->J(Sy)) or the proof of Vy(J(y) -->J(SSO -y)) with the appropriate efficient numeral. By using Modus Ponens a total of |:z;|times, we finally derive .](T). The length of the proof can evidently be bounded by a polynomial in By inspection of the proof we see that it can be formalized to get IAo + 91 l- 'v'a:ProvT('J(:f)'). Also it is useful to remark that in the proofs of J(:'zf),only formulas of a fixed complexity depending only on J are used. QED 2.6. DEFINABLE CUTS 27 Lemma 2.6.9 Suppose that T Q IA0. For every k and every T-cutI, there exists an T-initial segment J such that T lVa:(J(a:) -+I(2§)). (We use functional notation for brevity, but we remind the reader that there are appropriate equivalents using the A0- formulas that correspond to 21'= y.) Proof. We define Io, . . . , 1;, and J0, . . . , J,. by recursion. 0 10(3) 5"" 1(3); o for every i 5 lc, J,is constructed from I,by lemma 2.6.5; ° i+1($) 5" Ji(2I)- We prove by induction on i that every I, is a T-cut such that T lV:c(I,-(:5)-->I(2f)). For i = 0 this is clear. Suppose as induction hypothesis that it holds for i, and reason in T. First we show that \7':z:(I,-+1(:z:)->I(2f+1)). Suppose I E L", then by definition 2' E J,-, so because J,E L we have 2?' = 2?" E 1. Next we show that I,-+1 is a T-cut. Again, suppose that 1: E L", thus 2' 6 J,-. Since J, is closed under +, we also have 2' + 2* = 2=+1e J,-, thus :1:+1 e 1.11. To find the desired J, simply close Jk under by lemma 2.6.6. (Note that we do not have IA0 lV:c(J(:z:) ->J(2fi).) QED We remind the reader that col is defined in definition 2.1.7. Lemma 2.6.10 Suppose that T 2 IAO and let I be a T-cut. Then there is a formula J such that 1. T lV:r(K(:z:) ->I(1:)); 2. J is a T-initial segment; 3. T l- \7':c(J(a:) ->J(w1(a:))). Proof. First take J2 as defined in the proof of lemma 2.6.9, and close it off under by lemma 2.6.6 to get a T-initial segment K. Next define ~/(I) =+->31./(K(y) A as S 22"). We leave it to the reader to show that J is a T-initial segment such that T l- 'v':z:(J(:r:)-> K(:c)). For closure under wl, we reason in T and we use the fact, provable by induction, that for n > 1, w1(22") 3 224'". Now suppose that 2: E J. Then for some y E K (where we may take y > 1 without loss of generality), w1(:c) 3 w1(22") 3 224'". But because K is closed under +, we have 4 y E K, so by definition w1(:z:)E J. QED Lemma 2.6.11 Let cpE A0(ea:p). Suppose IAo+ EXP |-V;2:cp(:1:).Then there is a lc such that IAO l- 'v'x(2fi 1-» go(2:)). 28 CHAPTER 2. BACKGRO UND Proof. The proof is reminiscent of Parikh's Theorem (see theorem 2.3.15). Suppose, in order to derive a contradiction, that there is no I: such that IAO lV:c(2fi 1-» <p(:c)). Then, for a fresh constant c and for all k, IA0 I7'2;', 1-+<p(c). By the compactness theorem, there is a model M l: IAO+ {2,°,1 | k E w} + -cp(c). Next, we define M' := {b E M | Elk(M |= b < 2fi)}. Now M' l: IAO + EXP, so M' l= \7':z:cp(a:),in particular M' )= <p(c).But M' 9, M, so M l: cp(c). There we have our contradiction. QED Wilkie and Paris proved that the 1'11-consequencesof IAO+ EXP can be characterized using IA0-initial segments. Theorem 2.6.12 (Wilkie and Paris [WP 87], Corollary 8.8) Let (,0E A0(e:cp). Then the following two statements are equivalent: 1. IAO + EXP lVa:<,0(:z:) 2. There is an IAO-initial segment J such that IA0 l- 'v':z:(J(:c)-> Proof. 1 -+2 Suppose IAO + EXP l- 'v':z:c,o(:c).By lemma 2.6.11, there is a k such that IAO l- Va:(2','j1-» cp(:::)). By lemma 2.6.9, there is an IA0-initial segment such that IAO l- 'v':c(J(:z:)->2'; 1). Combining these two facts, we derive the desired conclusion IAOlV:z:(J(:1:)-+ 2 ->1 For the other direction, we need lemma 2.8.10. Thus we refer the reader to Corol- lary 2.8.11. QED Lemma 2.6.13 Let cpE Ao(e:z:p).If IAO+ EXP l- \7':1:<p(:c),then IAO}@ Proof. Suppose IA0+ EXP l- \7':c<p(:c).By theorem 2.6.12, there is an IAO-initial segment such that IA0 r 'v':1:(J(a:)-»<p(:1:)).Moreover by lemma 2.6.8, we have IA0 H-1"J(n). Thus IAO H-1".p(»r:). QED We give the most famous examples of interpretations provided by initial segments. One of them has an almost trivial proof, while the second one, on account of the weakness of Robinson's Arithmetic Q, needs a complicated argument. Lemma 2.6.14 IAO l> IAO + 91 on an IAo-z'nz'tz'alsegment. Proof. This is a particularly easy application of cuts. By lemma 2.6.10, we simply construct an IAO-initial segment closed under wl. Remember that IAO is a U1 theory, so IA0 l- (IAO+ 91)' (see remark 2.6.3). QED In order to prove that Q [>IAo on a Q-initial segment, we need a A0 "truth definition" (with one extra parameter) for A0-formulas. Such a definition was provided by Paris and Dimitracopoulos. 2.7. CUTS MAY HELP TO CHARACTERIZE INTERPRETABILITY 29 Lemma 2.6.15 (Paris and Dimitracopoulos) Thereis a A0-formulaI'(a:,z,u) (":3is satisfied by the sequence of numbers 2, with bound u) and a constant k such that: IA0 lZ (max(z) + 2)""' -+ " F(:r, z,u) satisfies Tarski's conditions for 2:6 A0" Proof. See [PD 82] or [HP 93, Theorem V.5.4] QED Theorem 2.6.16 (Wilkie) Q D IAOon a Q-initial segment. Proof. See [HP 93, Theorem V.5.7]. We give only the skeleton of the proof. There are three steps: 1. Let Q' be Q with three additional axioms: associativity of + and -, and left- distributivity (i.e. 2: - (y + z) = I y + :0 2). Nelson has shown that Q D Q' on a Q-initial segment [Ne 86]. 2. Every finite fragment of IA0 can be interpreted in Q' via a Q'-initial segment. 3. We take a finite fragment T of IAOwhich is so strong that: o T proves the properties of the exponentiation relation; 0 T proves Tarski's conditions for the satisfaction formula I' for Ao-formulas; o T proves the least number principle for F(:r, (y,p),u) with y as induction pa- rameter. Now to be able to prove the least number principle for all A0-formulas on an initial segment, it is sufficient to construct by lemma 2.6.9 a T-initial segment J which is so short that T lV:z:(J(:z:)->22' 1), so that we can replace the scheme by a single axiom. Finally, we combine all three interpretations. QED 2.7 Cuts may help to characterize interpretability In this dissertation many theories that we consider are not extensions of PA. On the other hand, they are almost all sequential. For such sequential theories U and V that extend Robinson's arithmetic Q, we can still prove analogs of theorem 2.4.6 and theorem 2.4.9 using definable initial segments. First we need a definition. Definition 2.7.1 For V a theory the axioms of which are defined by the Ell'-formula at/(y), let V[:z:]be the theory axiomatized by the formula av(y) /\ y 3 1:. Now we can define local interpretability as follows: U Dioc V : <=> 'v':z:3K[U D V[2:] by interpretation K]. Of course, if V is finitely axiomatized, U D V and U DamV are synonimic. 30 CHAPTER 2. BACKGROUND For the sake of legibility we will use quasi-modal abbreviations DU and 0;] for Provo," and Conau, where ag is the 2'1'-formuladefining the axioms of U. The following lemma has a proof analogous to the proof of theorem 2.4.10. Let Q+ be Q plus the axioms expressing that _<_is a linear order. Lemma 2.7.2 (Pudlék's Theorem on cuts, see [Pu 85]) Suppose U is sequential, V extends Q+, and U D V by interpretation K. Then for every V-initial segment I there is a U-initial segment J such that there is a definable initial embeddingfrom J into [K . Proof. See [Vi 90a] and [Vi 93]. There it is also stated that the theorem is verifiable in IAO+ 01. QED In the sequel of this section we use some abbreviations to improve ease of reading. 3J E U-cuts stands for the formalization of "there is a U-initial segment J such that". By <>j'UTwe abbreviate the formalization of "J does not contain a proof of .1.using only U-axioms with Godel number < x". Lemma 2.7.3 Let U be any sequential theory. Then we have IAO+ EXP l- \7'x3J e U-cutsDU<>j_,,T. Proof. See [Vi 93]. Partial truth predicates for formulas of limited complexity (see section 3.3 and [Pu 86, Pu 87]) play a crucial role in the argument. QED Theorem 2.7.4 (Visser, [Vi 93]) Let U and V be sequential theories extending 62*. Then we have IA0 + EXP lU >;.,,_.V 4-»'v'x3J E U-cuts C1U<>i,vT. Proof. ->Work inside IAO + EXP and suppose U l>;ocV. By lemma 2.7.3, we know that Vx':lI E V-cuts DVOLVT. For every x, we can apply theorem 2.7.2 to find a U- initial segment J such that C1U<>i,VT.Thus we derive V:z:3J E U-cuts ClU<>j_VT,as desired. 4Work inside IAO + EXP and suppose 'V/xElJE U-cuts ClU<>,J_.,VT.For every x we can now carry out a Henkin-construction giving us the desired local interpretation. QED Theorem 2.7.5 (Visser, [Vi 93]) Let U and V be sequential theories extending Q+. Then we have IAO + EXP lU l>;o,_.V 4-»VP 6 l'I1(3I E V-cuts Cl;/P' -->EIJ E U-cuts EIUPJ). Proof. ->This follows by theorem 2.7.2 formalized in IAO+ EXP. 2.7. CUTS MAY HELP TO CHARACTERIZE INTERPRETABILITY 31 +-This follows immediately from the 4--direction of theorem 2.7.4, when we note that O,_vT is a H1-formula. QED In the next theorem, the quantifier 3K stands for "there is an interpretation K". Theorem 2.7.6 Suppose U and V are sequential theories extending Q+. We have the following scheme of relationships (provable in IAo+EX P ) betweenthe various definitions of interpretability. Both arrows pointing down are strict. EU 6 U-cuts Va:DU<>i,VT <=> 3KV:z:DUO§VT U U D V -U U D;.,c V <=> 'v':c3J E U-cuts CIUOJKHVT <=> 'v':r3KE1U<>§vT Proof. o The =>-direction of SJ 6 U-cuts Va:ClUOi'vT <=> ElKV:cDU<>,'fVTis clear, be- cause every U-initialsegment provides an interpretation; the 4:-direction follows from theorem 2.7.2 as formalized in IAO+ EXP. c To prove 3J E U-cuts \7'1:ElU<>iVT=> U D V, one uses a formalized Henkin con- struction for a Feferman proof predicate; the argument is analogous to the proof of theorem 6.5.11. 0 By definition we have U D V => U D;,,,:V. c U D;,,,_.V <=> V/:z:3J E U-cuts DU<>;B',vTis just theorem 2.7.4. o V:c3J E U-cuts ElU<>i'VT <=> V:z:3KDU<>§vT is proved again by the fact that initial segments provide interpretations and by theorem 2.7.2. o The arrow EU 6 U-cuts V:cDU<>ivT => U D V is strict. For take U = V = IAo+Q1 and let J be any IAo+Q1-initial segment. Then we have of course IAo+Q1 DIAo+Q1, but for big enough lc (with respect to IAo+Q1 and J) we have DIAo+Q1Oi'IA0+nlT-» DIA0+m_l_,by Visser's adaptation of L6b's Theorem (see [Vi 93, Corollary 4.4]). o The arrow U D V => U D;.,,_.V is strict. For take U = IAO + 91 and V = IAO + (21+ {<>,,,UT | n 6 to}. Then U Dr" V, but by [Vi 93, Corollary 4.5], U does not interpret V. QED 32 CHAPTER 2. BACKGROUND 2.8 Between IA0and IA0+ EXP The results of this section are not needed in the subsequent chapters of the dissertation. Instead they appear here to give the flavor of the model-theoreticmethods used by Wilkie and Paris in [WP 87]. Also we hope that the reader will gain some understanding of the difference in strength between the theories IA0 and IAO+ EXP. For example, whereas IA0 interprets IAO+ 01, it does not interpret IAo+ EXP. Thus IAO+ EXP is much stronger that 1A0 and IAO+ 01. Another advertisement for the strength of IAO+ EXP is the fact that it proves tableauconsistency of IAO+ 91. On the other hand, perhaps surprisingly, IAO+ EXP does not even prove consistency of the extremely weak theory Q. This is caused essentially by the failure of IAo + EXP to prove a formalization of Gentzen's cut-elimination theorem. Another interesting question is the following: when is EXP necessary to prove some true H1-statement? A partial answer is given in subsection 2.8.2: in IAO+ EXP-proofs of sufficiently simple H1-sentences, namely those of VII';-form,one can get by without EXP and replace it by restricted consistency statements. 2.8.1 IA0+ EXP proves restricted consistency statements Definition 2.8.1 A k-formula is a formula with g is:logical connectives. (Note that a Ic-formulamay be arbitrarily long due to the presence of non-standard terms.) A k-proof is a proof in which only k-formulas appear. Pr0vT(<p,It) means that there is a k-proof of «,0from T. Similarly, Con(T,k) means that there is no k-proof of a contradiction from T. Definition 2.8.2 Let T be a set of sentences. We say that a sequence of sets of sets of formulas F0, . . . , F, is a tableau proof of an inconsistency from T if the following conditions hold: 0 For each X E I',, there is an atomic 6 such that 0 6 X and n0 6 X. o X E To implies X Q T U the set of logical equality axioms. o For each X E I", with 2'< 3 one of the following holds: 1X E F.'+1, 2. X U {0} E I",-+1for some w-19 E X, 3. X U {fi01},X U {02} E 1"," for some (01 -+02) E X, 4. X U {61, -02} E 1",-+1for some -r(01 ->62) E X, 5 . X U {6(t)} E I',-+1for some V:r0(:z:)E X and some term t which is free for :1:in 0(3), 6. X U {-=0(y)} E 1"," for some 4:/:z:6(:::)E X and some variable y which does not occur in any formula in X. o For each Y E I',-+1with 2'< .9 there is an X E T', such that Y is obtained from X by one of the rules 1-6. We write Tabcon(T) if there is no tableau proof of an inconsistency from T. 2.8. BETWEEN IA0 AND IAO+ EXP 33 Definition 2.8.3 Let L' be the language of arithmetic where successor, addition and multiplication are relation symbols. The only terms of L' are variables and U. IA5 is the reformulation of IA0 in L', with extra axioms expressing the totality of successor, addition and multiplication. Similarly any formula cp has an obvious L*-translation go' with the same unbounded quantifier complexity. Lemma 2.8.4 Suppose M l: IA0 + EXP. There is a A0(ea:p)formula Tr(a,y) such that for every sentence (p E L' and every a E M, a I: cp4:?» M l= Tr(a,r<p_'). (Here we have identified a with the substructure ofM that has universe {cc6 M I M l: 1: g a}. This presents no problem when we work in a relational language.) Proof. See [PD 82], and cf. lemma 2.6.15. QED Lemma 2.8.5 mo + EXP r vaviab [b= w§')(a)]. (Here w§i)(a) is defined informally as col appliedi times to a.) Proof. Remember that there is a A0-formula cp(a,i, b) which expresses b = w§i)(a). The lemma then follows easily by A0(ezp) induction, using an appropriate bound on w§')(a). QED Theorem 2.8.6 (Wilkie and Paris [WP 87], Lemma 8.10) 1. IAO + EXP lTabcon(IA5 + Q','). 2. Ifo E 22, then IA0 + EXP + 0 |'Tabcon(IA5+ Q; + 0'). Proof. We prove the second, more general, part of the theorem. Suppose that o = 3:z:Vy6(:z:,y), where 6 is a A0-formula, and reason in M l: IAO+ EXP+ 0. Let a be such that Vy6(a,y). Suppose that F0, . . . ,1", is a tableau proof in M of a contradiction from IA5 + Q';+ 0". Take b = w§'+1)(a+ 2), as is justified by lemma 2.8.5. Let hp,' I= F,' D ZHIUEI. Define 4-»3f: Var(l',-) r--+{u I u < w§i+1)(a+ 2)} such that b|= /\ Vcpf. X91'; 6/DEX Using the appropriate truth definition, P(i) can be expressed by a Ao(e$p) formula. We will use Ao(ea:p)induction (which is available in IAo+EXP) to prove that Vi 3 sP(i). This contradicts the fact that I', contains both 0 and n6 for some atomic 0. 34 CHAPTER 2. BACKGROUND 0 The base step relies on the observation that if B is a H1 axiom of IAO then b l: fl. Suppose for example that fl is the induction axiom V-'6,z(zb(=v,0) A Vy S z(w(9-'.y) -»w(=r. 511)) -* Vy S 2 ¢(-'v,y))- Let fl*(:2:o)be obtained from B by replacing 2/2with ':50l: 7,l2(:c,y)'(using the truth definition from Lemma 2.8.4). Then M l: 'v':cofi'(:z:o),so M |= fi*(b), hence b l: (3. o The induction step from P(i) to P(i + 1) hinges on the fact that the only time an unbounded quantifier 3 (i.e. wV-1)is to be eliminated in the tableau proof is on a subformula beginning with 3 of IA; + Q';+ 0'; but for the formula 3:cVy6(:z:,y) we already know that b l: 'v'y6(a,y); also the formulas 3y(y = 1: + 1),Ely(y = :z:1+:z:2),3y(y = 1:1-2:2),and 33/(y = w1(a:)) present no problem, because by induction hypothesis their free variables can be instantiated by parameters < w§i+1)(a+ 2). QED Domenico Zambella found the following generalization of Theorem 8.2 of [WP 87]. Theorem 2.8.7 Suppose T is a sentence, i Z 1 and M is a countable model satisfying 1. M l: IAO "l' Q1, and 2. for all k and for all Hf formulas (p with parameters a1, . . . ,a,, from M, M l: Va1,. . . ,a,.(Prou1A0+,('cp(m,. . . ,E,{)",k) ->cp(a1, . . . ,a,,)). Then there is a model M' |= IA0 + T such that M -<29M'. Proof. It is sufficient to find a model M' such that M' l: Diag£i.(M) + IAO+ T, where in Diaggi. new constants c, are used for elements a E M. So, in order to derive a contradiction, suppose that Diag,3e_.(M)+IAo+Tis inconsistent. Then there is a 2?-formula (p and there are a1, . . . ,a,. E M such that on the one hand M l: cp(a1, . . . , an), but on the other hand IAO+ T l- -:go(c,,1,.. . ,c,,n) by a proof in which all formulas have complexity is, for some standard k. This contradicts assumption QED Corollary 2.8.8 (Wilkie and Paris [WP 87], Theorem 8.2) Suppose T is a sentence and M is a countable model satisfying 1. M l: IAO+ 91, and 2. for all k, M l: Con(IA0 + T,k). Then there is a model M' l: IAO+ T such that M <21;M'. 2.8. BETWEEN IA0 AND IAO+ EXP 35 Proof. In order to be able to apply theorem 2.8.7, it is sufficient to prove that assump- tion 2 of theorem 2.8.7 is implied by the assumption that for all k, M l: C'on(IAa +'r, 1:). So suppose in order to derive a contradiction that for some kl and for some Hf'formula (p with parameters a1, . . . ,a,, from M, M l= Pr0vIAo+-r(r(p(fia - - - ,6Tn)l.k1)a (2-3) but M l: -1<p(a1,. . . ,a.,,). (2.4) Clearly mp(a1, . . . , an) is a 2','-formula. By inspection of the proof of 2'1'-completeness in IAo + 01 (cf. theorem 2.3.24), we conclude from (2.4) that there is a kg such that M I: Proz21A0+,('-=cp(fij,. . . ,Ei:)",k2). Together with (2.3), this implies the existence of a legsuch that M l: nCon(IAg + T,kg), contradicting the assumption that for all k, M l= Con(IAg + 'r,k). QED Remark 2.8.9 In assumption (2) of theorem 2.8.7, we may replace the formula Prov1Ao+r(r¢P(f1T, . . . , E17)",k) by Tabprov1Ao+,('cp(cT1, . . . , cT,,')"). Similarly we may replace Con(IA0, k) in assumption (2) of corollary 2.8.8 by Tabcon(IA0). (However, for the use of corollary 2.8.8 and theorem 2.8.7 in theorem 2.8.13 and remark 2.8.14, we need the original formulation.) Also, the assumption that M l: IAO+ 01 is not needed for theorem 2.8.7, although it is essential for its corollary 2.8.8. Lemma 2.8.10 [fa is a E2-sentence, then We(IAO+ EXP + 0 lC'on(IA0 + {Z1+ 0, Proof. By theorem 2.8.6, we know that IAO+ EXP + 0 lTabcon(IAo+ 91+ 0). We remind the reader that cut-free proofs can easily be converted into tableau proofs by an algorithm that increases the length of the proofs only polynomially. Now take some is:E w. From the formalization of the cut-elimination theorem given in the appendix to [Vi 92], it follows that IAO+ EXP lTabcon(IAo + 91+ 0) ->C0n(IAg + .01 + 0, 1:). (Indeed every k-proof with code p can be converted into a cut-free proof, and thus also into a tableau proof, whose Godel number is bounded by 2):; see definition 2.1.8 for an inductive definition of 2fi.) We may conclude IAO+ EXP + 0 lCon(IAg + {Z1+ 0, k). QED The above lemma is a strengthening of [WP 87, Proposition 8.5]. There it was proved that if 0 is a I'll-sentence, then Vk (IA0 + EXP + 0 1C0n(IA0 + 0, k)). 36 CHAPTER 2. BACKGROUND Remark 2.8.11 At this point we can provide the postponed proof of one direction of theorem 2.6.12. Let (,0E Ao(e:rp), and suppose that there is an IAO-initial segment J such that IAOlV;'l:(J(.'L')-v (2.5) We want to prove IAO + EXP 1-'v'z:cp(:z:). When we inspect the usual proof of A0(e3:p)-completeness in IAO+ EXP, we note that there is a k1 such that: IAO + EXP lV:1:(-vcp($)->Prov1Ao(rficp(:f)", k1)). (2.6) Also, because J is an IAO-cut, there is a kgsuch that IAo+EXP lV1:Prov[A,,('J(f)", kg); thus by (2.5), there is a k3 such that IAO + EXP lV:z:Prov1Ao("<p(T)",k3). (2.7) Next, we combine (2.6 ) and (2.7) to find a k such that IAO + EXP l- \7'1:(-<p(:z:)-> -wCon(IA0, k). By lemma 2.8.10 we finally conclude that indeed IAO+ EXP 1-V:rcp(:c). (Note that we only needed the fact that J is an IA0-cut, not that it is an IA0-initial segment.) 2.8.2 Conservativity Wilkie and Paris characterize the VII';-consequencesof IAO+ EXP by providing a basis over IAO+ 01: if one adds the restricted consistency statements to IAO+ 01, one can already derive all VH1?-consequencesof IAO+ EXP. We first need a definition. Definition 2.8.12 A U,--formula is a formula of the form Vsccpwhere «,0is a Hf formula. Note that all consistency statements for 2'1'-axiomatized theories can be written in U1 form. Theorem 2.8.13 (Wilkie and Paris [WP 87], Theorem 8.6) Let T be a H1-sentence. Then the following two theories have the same U1 consequences:0T12 0 T2 : lAo+Q1+{Con(lAg+T,k)|k€w}. Proof. T1 lT; follows from lemma 2.8.10. For the U1-conservativity of T1 over T2, suppose that M )= IAO+ Q1+{Con(IA0 + 7',k)|k E w} + fi\7'rcp(:z:), where -wcp(x)E Elf. We want to find M' with M' l: IAO + EXP + T + -vV1:cp(:z:). First we construct, by corollary 2.8.8, a model M' >-Ea;M such that M' l: IAO + T + -wV:z:cp(:c). Then by a trick reminiscent of the proof of Parikh's theorem, we let M' := {aeM*|3kew3beMM*|=a<2§',}. Note that 2)',is defined in M' for all k E (.0and b E M. This depends on the following fact which can be proved using Solovay's cuts, more precisely by inspection of the proof of lemma 2.6.8 and by lemma 2.6.9: 2.8. BETWEEN IAOAND IAO+ EXP 37 For any standard Is there is a standard m such that IAO+ n, |vbo,,,,,,. (25;1). Now M Q M' and M' I: IAO+ EXP. Since M' is an initial segment of M', 7' is preserved downwards hence M' |= T. Moreover there is a d E M such that M I: -w<p(d),so by corollary 2.8.8 part 2, since M -<3: M', M' I: fi<p(d);and since M is an initial segment of M', M' |= ficp(d), i.e. M' )= fi'v':1:<p(a:).QED Remark 2.8.14 If we use the full strength of theorem 2.8.7 instead of its corollary 2.8.8, we find that if T is a I'll-sentence, the following two theories have the same U, consequences: 0 T1:=IAo+EXP+'rand . T2 := IA0 + 91 + {Pr0z)IAo+T(r(p(1Tl7' ' ' 11Tr1)-11k) -4 'P(~'51»-° 7"-En) I k E (')7 cp E H? with variables among 3:1, . . . ,:r:,,}. (In fact, by lemma 2.8.10, T1 |T2.) 2.8.3 Non-conservativity and incompleteness Theorem 2.8.15 (Wilkie and Paris [WP 87], Theorem 8.11) 1. IAo + EXP is not U1-conservative over IAO+ 01; 2. If 0 is a 22 sentence consistent with IAO+ EXP, then IAO+ EXP + 0 is not U1-conservative over IAO+ 91 + 0. Proof. We prove the second statement. Define T := IAO+ 01 + 0, and construct it by diagonalization such that IAO + 91 l112+->-aTabc0n(T + 112). It is easy to see that ~10 is U1. Now suppose that T l- -up. Then T F -Tabcon(T + 712),so by definition T l1,12,contradicting the consistency of T. Thus IAo+Q1+a|f-=1/2. On the other hand, 0 /\ zpis 22, so by lemma 2.8.6, 1A0 + EXP + 0 + zb|Tabcon(IAo + 91+ or+ 1,0). Therefore by definition of ab, IAo+EXP+al- -aw. QED Corollary 2.8.16 (Wilkie and Paris [WP 87], Corollary 8.13) 1. Vkiln IAO+ 91+ Con(IA0, 1:) I7'Con(IA0, n); 38 CHAPTER 2. BACKGRO UND 2. Suppose 1r is a U1 sentence consistent with IAO+ EXP. Then 'v'k3nIAO+ 01+ Con(IAg + 7r,1:)17'Con(IAg + 7r, Proof. By theorem 2.8.15, there is a U1sentence it such that IAO + EXP + 1r+ C'on(IAg + 7r,k) 1-1,0, but IAO + Q1+ 7r + C'on(IAg + 7r, 1:) 17'1,0. On the other hand IAO + EXP + 1r 1C'on(IA0 + 7r,1:), so IAO + EXP + 7r 1-112. By theorem 2.8.13, there is an n such that IAO+ 01+ C'on(IAo + 1r,n) 1-zp. Therefore IAO+ 01+ Con(IA0 + 7r,1:)17'Con(IAo + 7r, QED Corollary 2.8.17 (Wilkie and Paris [WP 87], Corollary 8.14) 1. IAO + EXP 17'C'on(IA0); 2. If7r is a U1 sentence, then IAO + EXP + 7r1/ C'on(IAg + 7r) Proof. We prove the second statement. Suppose that IAo+EXP+7r 1-Con(IA0 +7r), then by theorem 2.8.13, there is a k E no such that IAO + 91 + C'on(IAo + 7r,k) 1- Con(IAg + 7r), contradicting theorem 2.8.16. QED Corollary 2.8.18 IAO+ EXP 17'Con(Q), but IAO+ SUPEXP 1-Con(Q). Proof. By theorem 2.6.16, Q D IAO on an initial segment. This can be verified in IAO+ 91 (see [HP 93, Theorem V.5.12]). Therefore IAO+ Q1 1Con(Q) +->C'on(IAg), so by corollary 2.8.17, IAO+ EXP 17'Con(Q). The fact that 1A0+ SUPEXP 1C'on(Q) follows from theorem 2.8.6, the considera- tion that as far as IAo+EXP is concerned, tableau provability and out free provability are equivalent, and from the fact that the cut-elimination theorem for the predicate calculus can be proved in IAO+ SUPEXP. QED Lemma 2.8.19 Let U Q IAo+Q1. Then IAo+Q1 proves the following: ifU is consistent, then U does not interpret IA0 + 01 + Con( U). Proof. The argument is similar to the proof of [Vi 90a, Proposition 6.2.2.2] Reason in IAO+ 01 and suppose that U D IAO+ D1 + Con( U) by the interpretation M. Define Provw(:c) :<=> Prov;A0+g,+(;,,,,(U)(a:) A ProvU(:cM). Because Provw can be written as an 32';-formula, the principles of L can be verified for Dw. Moreover we have the IL-consequence W D W + -rC'on( W). 2.8. BETWEEN IAOAND IAO+ EXP 39 See section 2.5 for a definition of IL. The reader may enjoy to figure out how to prove IL 1T D D_L. Also we have by definition of Provw: Provw(wC0n( W) ->-vCon(U)), so W DW+-wCon(U). (2.8) But by definition of Provw, we have Provw(Con(U)), so (2.8) implies -Con( W). Again by definition of Provw, we conclude that -wC'on(U). QED Theorem 2.8.20 IAO+ 01+ C0n(IA0) I7'C0n(IAg + EXP). Proof. Assume that IAO+ 91+ C'on(IA0) 1-Con(IAg + EXP). In IAO+ EXP, one can prove that the set {:12| 2: 1} is closed under successor, so it can be shortened to an IAO+ EXP-cut J closed under wl. Now IAO+ EXP D IAO+ 01+ Con(IA0), using the interpretation provided by J. Thus by our starting assumption IAO+ EXP D IAO+ 01 + C'on(IA0 + EXP), contradicting lemma 2.8.19. QED Remark 2.8.21 Wilkie and Paris [WP 87]showin their Theorem 8.19 that the statement Con(IA0) is even more hopelessly weak than theorem 2.8.20 suggests, for adding EXP makes no difference. That is, IAO+ EXP + Con(IAg) +7'Con(IAg + EXP). For the next theorem, we need one lemma. Lemma 2.8.22 IAO+ EXP is finitely axiomatizable. Proof. See [HP 93, Theorem V.5.6]. The proof uses a A0 truth definition for A0- formulas (see lemma 2.6.15) and then follows the argument of step 3 from the proof sketch of theorem 2.6.16. QED Theorem 2.8.23 Q does not interpret IAO+ EXP. Proof. Let EXP' be the finitely axiomatized version of IAO+ EXP, which exists according to lemma 2.8.22. If Q D IAO+ EXP, then certainly Q D EXP'. Since both Q and EXP' are finitely axiomatized, we have IAO+ 91 1-Q D EXP', so IAO+ 91 lCon(Q) ->C'on(EXP'), and a fortiori IAO+ Q, 1Con(IA0) ->Con(EXP*), contradicting theorem 2.8.20 QED

Part II Metamathematics for Bounded Arithmetic

Chapter 3 A small reflection principle for bounded arithmetic "What a curious feeling!" said Alice, "I must be shutting up like a telescope!" And so it was indeed: she was now only ten inches high, and her face brightened up at the thought that she was now the right size for going through the little door into that lovelygarden. (Lewis Carroll, Alice in Wonderland) Abstract. We investigate the theory IAO+ 01, and strengthen [Bu 86, The- orem 8.6] to the following: if NP aé co-NP, then E-completeness for witness comparison formulas is not provable in bounded arithmetic, i.e. IAO + 01 l7''v'bVc (Ela(Prf(a, c) /\ V2 3 afiPrf(z, b)) -+Prov('3a(Prf(a,c) /\ V2 3 a-wPrf(z,5))")). Next, we study a "small reflection principle" in bounded arithmetic. We prove that for all sentences cp, IAO + 01 l- \7'1:Prov('-\/y3 T(Prf(y, ->90)") The proof hinges on the use of definable cuts and partial satisfaction pred- icates akin to those introduced by Pudlak in [Pu 86]. Finally we give some applications of the small reflection principle, showing that the principle can sometimes be invoked in order to circumvent the use of provable Z3-completeness for witness comparison formulas. 3.1 Introduction A striking feature of Solovay's Theorem that Lo'b's logic is complete for arithmetical in- terpretations is its amazing stability. If one sticks to the unimodal propositional language and standard arithmetical interpretations, the result holds (modulo a trivial variation) for any decently axiomatized extension of IAO+ EXP. Such stability is in some sense a 43 44 CHAPTER 3. A SMALL REFLECTION PRINCIPLE weakness: unimodal propositional logic combined with the standard interpretation cannot serve to classify or give information on specific theories in a broad range. Of course this weakness disappears when we extend the modal language, but that is not our subject here (however see [Vi 90a, Bek 91]; [Bek 89]). Is there life outside the broad range of arithmetical theories satisfying Solovay's Com- pleteness Theorem? Clearly the question is only sensible if the theories under considera- tion verify L6b's logic, or perhaps some still interesting weakening of it. Two directions of research come to mind. The first one is to weaken the logic of the arithmetical theory. Specifically one can study theories like Heyting Arithmetic (HA), the constructive version of Peano Arithmetic. It turns out that HA verifies the obvious constructive version of L6b's logic plus a wide variety of extra principles (see [Vi 81, Vi 82, Vi 85]). The only definitive information that we have is a characterization of the closed fragment of HA. For all we know the provability logic corresponding to HA itself could be H3-complete. Moreover, extensions of HA have quite different provability logics. Note by the way that provability logics need not be monotonic in their arithmetical theories. The second direction of research is simply to look at classical arithmetical theories that are strictly weaker than, or even incompatible with, IAO+ EXP. It turns out that there are two salient theories of this kind: Paris and Wilkie's IAO+01 and Buss' S21,both of them satisfying L6b's logic (see [WP 87, Bu 86]). Does Solovay's Theorem still hold for them? At present nobody knows -or to be precise, we haven't heard that anybody knows. This chapter is a first contribution to an understanding of the difficulties involved in proving or disproving Solovay's Theorem for theories like lAo+Q1 and S21.Solovay's proof involves Rosser methods. The problem for us resides in the instances of H';-completeness that occur in the proof. Two points are important. a We do not know whether the instances of H';-completeness used in Solovay's proof are provable in our target theories. Buss proved that provability of H';-completeness with parameters in 3%implies NP = co-NP (see [Bu 86]). In section 3.2 we elaborate on this theme. To be specific, we prove that if NP 76 co-NP, then Z-completeness for witness comparison formulas is not provable in bounded arithmetic, i.e. IAO + 01 l7'VbVc (3a(Prf(a, c) A V2 3 anPrf(z, b)) -+Prov("3a.(Prf(a, E) /\ V2 3 a-uPrf(z,5))')). o In many cases we can circumvent the use of instances of H';-completeness. Svejdar discovered the first alternative argument when he surprisingly provided a proof of Rosser's Theorem that genuinely differed from Rosser's own proof (see [SV83]). To this end he introduced a principle which we have dubbed Svejdar's principle. In section 3.3 we prove a "small refiection principle" in our target theories from which §vejdar's principle immediately follows. More precisely, we show that for all sentences cp, IAO+ 91 lV:cProv(r'v'y3 f(Prf(y, ->cp)"). Svejdar's principle is not sufficient to derive Solovay's Theorem. However, it has been fruitfully exploited in the dogged attempt to use Solovay-likemethods to embed larger and larger classes of Kripke models for L6b's logic in our weak arithmetical theories. The state of this dogged art can be found in chapter 5 and [BV 93]. We end section 3.3 with some other applications of the small refiection principle. 3.2. Z'-COMPLETENESS AND THE NP = co-NP PROBLEM 45 In section 3.4, we use the small reflection principle in order to extend Krajicek and Pud1a'.k'sresult on the injection of inconsistencies into models of IAO+ EXP. Theorem 3.2.7 and theorem 3.3.20, the main results of section 3.2 and section 3.3, were published previously in the technical report [Ve 89], which in turn is based on my master's thesis [Ve 88]. We assume that the reader is familiar with [Bu 86] and [WP 87]. However, most of the definitions we need can be found in section 2.3. Remark 3.1.1 In cases where confusion seems unlikely, we will sloppily leave out some numeral dashes, in particular deeper nested ones. 3.2 Z3-completenessand the NP=co-NP problem In this section we will prove that, under the assumption that NP ;£ co-NP, the following holds: IA0 + Q1 l-,'\7'bVc (3a(Prf(a, c) /\ V2 3 a-wPrf(z, b)) H -+Prov("3a(Prf(a,E)AV23 a-iPrf(z, In the proofs of the lemmas leading up to this result we will frequently, often without mention, make use of the following proposition and its corollary. Proposition 3.2.1 ( [Bu 86]) Suppose A is a closed, boundedformula in the language of S2}and let R be a consistent theory extending S21. Then R lA ijfw ]= A. Corollary 3.2.2 ( [Bu 86], Prop. 8.3) Suppose A(a') is a boundedformula in the lan- guage of S21,and let R be a consistent theory extending S]. If R lV:i:'A(:i:'),then w ]= V:i:'A(1':'). In this section, we will use the name IA0+Q1 for Buss' theory S2 (see Definition 2.3.12), in which induction for formulas from the hierarchy of bounded arithmetic formulas in a language containing I ], and # is allowed. Because S2 is conservative over IA0 + 91, the name change has no repercussions on the results of this section. (In the next section, where we need to construct formalized satisfaction predicates, we will be more careful.) In order to prove the main theorem of this section, we need to prove a few seemingly far- fetched lemmas. Their proofs borrow heavily from the formalization carried out in [Bu 86]. To make these lemmas understandable, we will give some details of the formalization of the predicate Prf. Buss uses a sequent calculus akin to Takeuti's (see [Ta 75]). He considers a proof to be formalized as a tree, of which the root corresponds to the end sequent, and the leaves to the initial sequents of the proof. Every node of the proof tree is labeled by an ordered pair (a, b). The second member of this pair codes a sequent, and the first member codes the rule of inference by which this sequent has been derived from the sequents corresponding to the children of the node in question. For leaves, the first member of the corresponding ordered pair codes the axiom of which the initial sequent is an instantiation. The only extra fact we need here is that logical axioms are all numbered 0; in particular, for all terms t, the tree containing just one node labeled <0," -> t = t_') is a proof of -> t = t. Because of a peculiarity in the encoding of trees, by which 0 and 1 are reserved as codes for brackets, Buss encodes the proof just mentioned by (0, F -->t = t") + 2. 46 CHAPTER 3. A SMALL REFLECTION PRINCIPLE In the sequel, we will sometimes abuse Buss' conventions in order to keep the formulas legible. Thus, we will write (0, " -> d = d") for Buss' (0, (0 * Arrow) an*('I,,' anEquals an *"Id")) + 2. Lemma 3.2.3 Let 1,l2(d,b) be the formula V2 3 (0, P -+ d = d")fiPrf(z, b). The predicate represented by 1/2is co-NP-complete. Proof. Straightforwardly, it is a H';-formula, hence it represents a co-NP predicate. For the other side, viz. co-NP-hardness, begin by supposing A(a1, . . .,a;.) E co-NP. We will polynomially reduce A to 1,b. (For definitions of the complexity theoretic concepts that we mention, see definition 2.3.7 and definition 2.3.8; and see remark 2.3.10 or [Bu 86, Thm 1.8]). By provable )3';-completeness (see theorem 2.3.24), there is a term r(c'i')such that IAO + 01 l- ~A(&') -»32 3 r(a')Prf(z, 'aA(a"1, . . . ,m)") and thus in )= -A(&') -+Elz3 r(a')Prf(z,'fiA(cT1,... ,a;)") Because r(a') 3 'r-(a'-)"3 (0,' -> r ft')= r 67)'), we also have in l: ~A(a) ->32 3 (O," -->7T1')= @")Pr_f(z,'-oA(a"1, . . .,a;)"). (3.1) On the other hand, by Proposition 3.2.1 and the consistency of IAO+ 01, we have w l= 3z S (07 -* 7% = 7' &'-)">P7'f'(z.'-'A(<'z'I,- - - ,aI)") -+ -v4(5)- (3-2) From (3.1) and (3.2), we conclude that w l: A(a') «-»Vz 3 <0," --+mt") = _(-a')")fiPrf(z, 'wA(aj, . . . ,aT,.)'). This means by the definition of it that w l= 24(5) <->¢(7'(5)ar"'A(fT1,---,<Tk)1)- As both "wA(a7, . . . , 6;)" and r(a') can be computed from 6.'by polynomial time func- tions, we have reduced the co-NP predicate A to w. QED Lemma 3.2.4 Let B(a1, . . . ,a;,) be a llll'-formula representing a co-NP complete predi- cate. If NP 79 co-NP, then IAO + 01 l7'Va'(B(a') -->Prov('B(cT1', . . .,fi)")). Proof. An application of Parikh's Theorem for IAO+ 91 (cf. theorem 2.3.15). We leave the details, which are similar to part of the proof of [Bu 86, Theorem 8.6], to the reader. QED Lemma 3.2.5 If NP 94 co-NP, then IAO+ 01 l/VbVd (V2 3 (O," -> d = d')-Prf(z, 5) ->Prov('\7'z 3 (0, F -->d = d")fiPrf(z, b)"). 3.2. 2-COMPLETENESSAND THE NP = co-NP PROBLEM 47 Proof. Directly from Lemma 3.2.3 and Lemma 3.2.4. QED Lemma 3.2.6 IA0 + 01 proves the following: \7'bVd (Prov(F3a.(P7'j(a., F --+cFl= 3-') A V2 3 a.-vPrf(z, l)))") ->Prov(F'v'z 3 (0,F -> = c_l")fiPrf(z,5)F'))3 Proof. It is not difficult to see that for Buss' formalization of Prf, we have the following: IA0 + Q1lVdVa(P7'fla,F -> c-l= 31) ->a _>_(O,F -> ti = 37)), and thus IAO + 01 l- \7'b'v'd(3a(Prf(a, F -> 3 = (T) A Vz 3 a-vPrf(z, b)) ->Vz 3 (0,F -> 3 = c_i")fiPrf(z,b)). This in turn immediately implies our lemma. QED Theorem 3.2.7 If NP 76co-NP, then IA0 + 01 I/VbVc (3a(Prf(a, c) /\ V2 3 amPrf(z, b)) -»Prov(F3a(Prf(a, C) A V2 3 anPrf(z,l-)))"')). Proof. Suppose that NP =;éco-NP, and suppose, in order to derive a contradiction, that IA0 + 91 l- 'v'bVc (3a(Prf(a, c) /\ V2 3 afiPrf(z, b)) ->Prov(F3a(Prf(a,E)/\ V23 a-~Prf(z, Then, in particular, 1A0 + 91+ Vb':/d (Prf((0,F --» 3 = Em," -» E1= 8") /\'v'z 3 (0,F --+3 = d7)fiPrf(z,b) -->Prov(F3a.(Pr_/'(a,F -> d = FdF')/\ V2 3 a-«Prf(z, (3.3) We know that IAO + Q1l- 'v'd(Prf((0,F --+if = T), F -> 3 = d")) Combined with (3.3), this implies the following: IAO + 91 lVb':/d (V2 3 (0, F -> 3 = c_l1)fiPrf(z, b) ->Prov(F3a(Prf(a.,F -->3 = T) /\ V2:3 a-vPrf(z, Now we apply Lemma 3.2.6 to derive IAQ + 91 l' Vb)/d ('v'z3 (0, F -> = T')'1P7'_/(Z, b) ->Pr0v(F\'/z 3 (0,F -->d = c_iF')~Prf(z,5)")), in contradiction with Lemma 3.2.5. QED 48 CHAPTER 3. A SMALL REFLECTION PRINCIPLE Remark 3.2.8 We can prove that provable Z3?-completenessfails already for a much simpler l'I'1'-formula)((a, b, c) defined as follows: x(a,b,c) := 'v'a:§cVy§c(a-:1:2+b-y;£c). The fact that Z3?-completenessfails for X follows immediately from Lemma 3.2.4 and the following lemma, to which A. Wilkie attracted our attention. Lemma 3.2.9 (Manders and Adleman, see [MA 78]) The set of equations of the form (a -122+ b y = c), solvable over the natural numbers, with a, b, c positive natural numbers, is NP-complete. Note that Lemma 3.2.9 implies that the formula Elcrg cily 3 c(a 3:2+ b y = c) represents an NP-complete predicate, and thus that X as defined above represents a co- NP complete predicate. 3.3 The small reflection principle In this section, we will present a proof of the fact that IAO+ 01 proves the small reflection principle, i.e. for all (,0: IAQ + Q1lV:CCl(Cl,_.;p->gp), where Dip is an abbreviation for Prou('cpI) and Clxcpis a formalization of the fact that qp has a proof in IAO+ 01 of Gbdel number 3 :12.In fact, all arguments that we use can be carried out already in Buss' S5, as the reader may check for him/herself. In the proof, we will use the existence of partial truth- (or satisfaction-) predicates Sat" for formulas of length 3 n. The intended meaning of Sat,,(:c,w) will be "the formula of length 3 n with Godel number I is satisfied by the assignment sequence coded by w". Pudlak [Pu 86] has constructed partial truth predicates much like the ones we need. (An analogous construction, where Satn is related to quantifier depth instead of length, can be found in [Pu 87].) However, our construction departs from Pudlé.k's in two ways. Firstly, whereas Pudlak presents his results for theories in relational languages, we allow function symbols. Secondly and more importantly, IAO+ Q1 is neither finitely nor sparsely axiomatized. Regrettably we cannot even apply to IA0 + 91 a trick of Pudlak's which turns some non- sparse theories like PA and ZF into sparse ones (see Theorem 5.5. of [Pu 86]). Therefore we introduce new satisfaction predicates Sat,,_A(:r,u2)with as intended meaning: "the A0-formula of length 5 n with Godel number :1:is satisfied by the assignment sequence coded by in". Using these satisfaction predicates, we will be able to prove by short proofs that the A0-induction axioms are true. In order to start the construction of short satisfaction predicates, we need a few more assumptions and definitions. First of all, when formalizing, we view IAo+Q1 in a restricted way more akin to Paris and Wilkie [WP 87] than to Buss [Bu 86]: see definition 2.3.1. For this system, we can define the appropriate Alf-predicates Term(v), Fmla(u), Sent(u), Prf(u,u) in S}. In Buss' formalization of sequences, atstands for a function which adds a new element to the end of a sequence; ** stands for a function which concatenates two sequences; and B(t, w) stands for the function giving the value of the t-th place in the sequence coded by w. 3.3. THE SMALL REFLECTION PRINCIPLE 49 In this chapter, we denote concatenation of sequences sloppily by juxtaposition, and we leave out some outer parentheses; thus, for example, 3;" -->"2 stands for Buss's (0 =I= LParen) at*(y atImplies) * *(z * RParen). Definition 3.3.1 We formally define four concepts that we need in order to construct truth predicates. o w =,w' := Len(w) = Len(w') A \7't(t 3 Len(w) A t 55 2'-->fl(t,w) = B(t,w')), i.e. the only possible difference between the sequences coded by w and w' is at the i-th value; 0 Fmla,,(v) := Fmla(v) A Len(v) 3 n, i.e. v is the Godel number of a formula of length 3 n; 0 Fmla,,,A(v) := Fmla,,(v) "and 22codes a Ao-formula"; 0 Evalseq(:c,w) will mean that the sequence coded by w is long enough to evaluate all variables appearing in 2:, i.e. Evalseq(:c, w) := Seq(w) A (Fmla(:r) V Term(:1:)) A Vz'("the variable 12,- occurs in the term or formula with Code! number 2:" ->Len(w) Z 2'). Furthermore, we introduce the following two abbreviations: 0 Evalseq,,(a:,w) := Fmla,,(:z:) A Evalseq(:z:,w); 0 E'ualseq,,,A(:z;,w) := Fmla,,,A(:1:) A Evalseq(:1:,w); Next we define, by Buss' method of p-inductive definitions, a function Valsuch that, if t(v,-,,...v,-n) is a term of the (restricted) language of IAO + 91 and w codes a se- quence evaluating all variables v,-1,.. .v,appearing in t, then Val('t",w) gives the value of t[fi(z'1,w), . . . ,fi(i,,, w)]. Definition 3.3.2 Let Valsatisfy the followingconditions: 71 0 -nTerm(t) V -=Evalseq(t,w) -+Val(t,w) = 0; o the p-inductive condition: Ter'm(t) A Evalseq(t, w) -> (t = '0" A Val(t,w) = 0) V 311< t(t = "v," A Val(t, w) = fl(z',w)) V 3t1,t2 < TCT'7Tl(t1)/\ Te1'm(t2) A ((t = "S'"t1 A Val(t,w) = S( Val(t1,w))) V (t = t1' + "t; A Val(t, w) = Val(t1, w) + Val(t2, 112)) V (t = t1" - "tg A Val(t,w) = Val(t1,w)Val(t2,w)))) By induction, we can show that t#w will be a bound for Val(t,w). Thus, by [Bu 86, Theorem 7.3], Valis Al,'-definable (thus provably total) in S2';furthermore, the definition of Val in S21is intensionally correct in that properties of Val can be proved in S21(and thus also in IAO+ 01) by the use of induction. 50 CHAPTER 3. A SMALL REFLECTION PRINCIPLE Remark 3.3.3 Note that we cannot construct in IAO+ 01 a correct valuation function Val for a language that contains #. For, to any a we can associate a formalized term f(a) given informally as 1#2# . . . #2 where the number of TS is |a|. A correctly defined Valshould give Val(f(a), w) = exp(exp(|a| + 1) 2) 2 exp(a) (cf. [Ta 88]). Therefore by Parikh's Theorem (cf. theorem 2.3.15), Valcould not be A0-definable and provably total in IAQ + Q1. In the sequel, we will freely make use of induction for Ao( Val)-formulas in IAO+ 01, as is justified by the IAO+ Q1-analogs of Buss' Theorem 2.2 and Corollary 2.3. We will especially need the following lemma. Lemma 3.3.4 There exists a constant c such that for every term t with free variables among v,-,,...,v,-m and for every n with Len('t") 3 n, we can prove the following by proofs of length 3 c n: IA0 + 91 lEvalseq(rt', w) ->Val('t1, w) = t[fl(i1, w), . . . ,B(im, Proof. Straightforward by induction on the construction of t. QED For the definition of satisfaction predicates, we need one more definition. Definition 3.3.5 We formally define the following: s(i,a:, w) := (Subseq(w, 1, i) =o=:13)4:*Subseq(w,i+1,Len(w)+1). Thus, if w is a sequence of length 2 i, s(i, 1:,w) denotes the sequence which is identical to w, except that asappears in the i-th place. Definition 3.3.6 We say that Sat,.(:r,w) is a partial definition of truth for formulas of length 3 n in IAO + 01 iff IAO + 01 lEvalseq,,(:r,w) -> {S'at,,(;z:,w) 4-» [3t, t' < a:(Term(t) A Term(t') A :1:= tr = 't' A Val(t,w) = Val(t', w)) V Elt,t' < :I:(Term(t) A Term(t') A :1:= t" 3 't' A Val(t,w) 3 Val(t',w)) V 3y < :I:(:z:= 'F-'y A -wSat,,(y, w)) V 3y, 2 < ;7:(:c= yr --+"z A (Sat,,(y,w) ->Sat,,(z,w))) V Ely,i < :r:(:z:= 'Vv,~"y A 'v'w'(w =,w' ->Sat,,(y,w'))) V 3y,i,t < :1:(Term(t) A 1: = '('v'v,-3 "t")"yA Vw' 3 s(i, Val(t,w),w)(w =,~w' A B(i,w) 3 Val(t,w) ->Sat,,(y,w')))]} We denote the part between brackets on the right hand side of the equivalence by Z(Sat,,; :z:,w);note that these are just Tarski's conditions. Similarly, we say that S'at,,,A(:c,w) is a partial definition of truth for A0-formulas of length 3 n in IAO + 91 iff IAO + Q1 lEvalseq,,,A(:1:, w) -> {Sat,,_A(;r,w) 4-+ [3t, t' < :c(Term(t) A Term(t') A :1:= t" = 't' A Val(t,w) = Val(t',w)) V Elt,t' < :1:(Term(t) A Term(t') A :1:= t" 3 't' A Val(t,w) 3 Val(t',w)) V Ely < :1:(a:= "--"yA -Sat,,,A(y, w)) V 3y, z < :1:(:r= y' ->"2 A (Sat,,,A(y,w) ->Sat,,,A(z,w))) V 3y,i,t < a:(Term(t) A :2:= "(Vv,-3 "t')"yA Vw' 3 s(i, Val(t,w),w)(w =,w' A fi(i,w) 3 Val(t,w) ->Sat,,_A(y,w')))]} We denote the part between brackets [ ] on the right hand side of the equivalence by Z3A(Sat,.,A;2:,w). Note that the only difference between 2(Sat,,; :3,w) and Z3A(Sat,,,A;2:,w) is that in the latter, the disjunct for the unbounded quantifier V is left out. 3.3. THE SMALL REFLECTION PRINCIPLE 51 In the proof of the main theorem of this section, we will reason inside IAo+Q1, and we will need the existence of Godel numbers representing formulas Sat" that provably satisfy the conditions of the preceding definition. Therefore, in the unformalized proofs below, we take care that the formulas Sat,, and the proofs that they have the right properties be bounded by suitable terms. The following lemmas provide us with such formulas. In [Pu 86, Pu 87] Pudlak proves similar lemmas for a language without function symbols. Below, we sketch the adaptation of his method to our case. The parallel construction of a Ao( Val, | |, |_,1,_:c_|,#)-formula .S'at,,,Awhich works for A0-formulas is particular to this dissertation. We use the formula Sat,,,A only in our proof that S'at,, preserves the A0- induction axioms, but there its use is essential. Lemma 3.3.7 There exist formulas Sat,,(:1r:,'w)for n = 0,1,2,... of length linear in n, and such that, by a proof of length linear in n, IA0 + Q1 lEvalseq,,+1(a:,w) ->(Sat,,+1(:1:,w) 4-»Z(Sat,,;:r,w)). Proof. Sat." is constructed by recursion. We can define Sato arbitrarily, as there are no formulas of length 3 0. If we have the formula Satk, we obtain Sat;,+1 by substituting Satk for Sat" in the formula ZI(Sat.,,;a:,w) defined in Definition 3.3.6. Remember that we have to ensure that the length of the formula Sat" grows linearly in n. However, if we straightforwardly used E(Sat,,; 1:,w) as defined above, the length of Satn would grow exponentially in n, because E(Sat,,;:z:,w) contains more than one occurrence of Sat,,. Ferrante and Rackoff (in [PR 79, Chapter 7]) describe a general technique for writ- ing short formulas, due to Fischer and Rabin. Using these techniques, one can replace E(Sat,,; 1:,w) by a formula Z3'(S'at,,;1:,in) which contains only one occurrence of Sat,,, and which is equivalent to Z3(Sat,,;:c,w) in a very weak theory say predicate logic plus the axiom SO aé 0. Ferrante and Rackoff use the inclusion of +->in the language of the theory in an essential way. However, Solovay sent us a different construction of short formulas which circumvents the use of H. With his kind permission, we present a sketch of his proof. Solovay'sbasic idea is to shift attention from sets to characteristic functions. Without restriction of generality, we may assume that we work with unary predicates Sat,,(:c) instead of S'at,,(;z:,w). Let F,,(2:, y) := (y = 50 A Sat,,(:z:)) V (y = 0 /\ -Sat,,(:z:)). If we can find a formula H,, equivalent to F,, of length proportional to n, it will be easy to define using this formula our desired formula Sat,,+1. Let L be the language of IAO+ 91 enriched with a new binary predicate letter C'. We can find a formula <1)of L in prenex normal form, having only the variables 1: and y free, such that if G is interpreted as F,,, then <1)is interpreted as F,,+1. We show how to find a formula \IIwhich is equivalent to CDand which has only one occurrence of G. Assume that <I>starts with the string of quantifiers (Qlxl) . . . (Q,:1:,), and that there are k occurrences of G in the matrix of (D,say G(t1,m1), . . . , G(t,,,m;,). The formula 'I1will have the form (Q1221)---(Qr$r)(3y1)"'(3yk)lMA Sl- Here yl, . . . ,y,, are fresh variables (for the moment -in the final definition we will be less liberal with variables). The formula M is obtained from the matrix of (Dby replacing each 52 CHAPTER 3. A SMALL REFLECTION PRINCIPLE occurrence of G'(t,-,m,-) by m,- = y,-. 5'' job is to ensure that the y,-'sare chosen correctly. It is defined as follows. I: S I: \7'u213w2[G(w1,w2)/\ A('l.U1=t,'-* 'U)2= i=1 If we define H,,+1 from H,, using 'II,we get a formula of length proportional to nlog n, because at every step we introduce fresh variables in order to avoid clashes. There are however tricks to get by with a finite set of variables, as the reader may enjoy to figure out (or to look up in [PR 79, Chapter We will write 2'(Sat,,;:z:,w) for the equivalent of 2(Sat,,;a:,w) resulting from an ap- plication of the techniques described above. The length of Sat,, thus constructed via iterated application of E3'to Sat) is indeed linear in n. Moreover, for all n, the shape of the proof of )3(Sat,,;a:,u2) 4-»ZI'(Sat,,;:z:,u2)is the same for all n. Thus, the proofs of 2(Sat,,; :5,w) +-+E'(Sat,,;a:, w) grow linearly in n. Hence, as Sat,,+1(:c,w) E 2'(Sat,,; 11:,in), we have the following by proofs of length linear in n: IAO + 91 lSat,,+1(1:,u2) +->E(Sat,,;:z:,u2) (3.4) QED Lemma 3.3.8 IAO+ 01 proves by a proof of length of the order of n2 that the formula Satn as constructed in Lemma 3.3.7 is a partial definition of truth for formulas of length 3 n. Proof. We want short proofs showing that Sat" is a partial definition of truth for formulas of length 3 n in 1A0+ 01, i.e. IAO + Q1 lEualseq,,(:z:,u2) ->(Sat,,(:1:,u2) +->Z(Sat,,;;z:,w)). By (3.4), it suffices to show that, by proofs of length of the order n2, IAO + 01 lEualseq,,(:r,w) -->(Sat,,(:1:,u2) <->Sat,,+1(:z:,u2)). This can be proved by external induction on n. In fact, when we define 4),, := 'v':rVw(Evalseq,,(:z:,w) ->(Sat,,(a:,w) 4-»Sat,,+1(:1:,w))), the proofs of <I>,,-+<I>,,+1in IAO + 01 will have a shape which does not depend on n. (We refer those readers who seek elucidation by examples to [Pu 86, Lemma 5.1].) We can observe that every proof in IAO + 91 of <I>,,-+<I>,,+1is the instantiation of a single proof scheme. Thus, the length of the proofs of <I>,,-><I>,,+1increases only linearly in n, so that the length of the proof in IAO+ Q1 of \7':z:Vw(Eualseq,,(:z:,w) -+(Sat,,(:c,w) 4-»Sat,,+1(:z:,u2))), is of the order n2. QED Lemma 3.3.9 There exist formulas Sat,._,A(:z:,u2)for n = 0,1,2,... of lengths linear in n, and such that IAO + 01 proves by proofs of length linear in n that Sat,,+1,A(:z:,u2) +-+ EA(Sat,,,A;a:, w). The resulting formulas Sat,,,A(:t:,uJ) are Ao( Val)-formulas. 3.3. THE SMALL REFLECTION PRINCIPLE 53 Proof. Completely analogous to the proof of Lemma 3.3.7. Because 2A(Sat,,,A; :c,w) contains only bounded quantifiers, and because all quantifiers introduced by the Solovay method can be bounded, the resulting formulas are indeed Ao( Val). QED Lemma 3.3.10 IAO+ 01 proves by a proof of length of the order of n2 that the formula Sat,,_A(2:,w) as constructed in Lemma 3.3.9 is a partial definition of truth for A0-formulas of length 3 n. Proof. We adapt the proof of Lemma 3.3.8, incorporating the fact that we are con- cerned with A0-formulas only. Thus instead of <I>,,,we define <I>,,,A := 'v':z:Vw(Evalseq,,_A(:c,w) -+(S'at,,_A(a:,w) <-+Sat,,+1,A(:z:,w))). The proof of <I>,,,A-><I>,,+1,Aruns along the same lines as the proof of <I>,,-+<I>,,+1,using the extra fact that if :3 = y" -+"z and Fmla,,+1,A(a:), then Fmla,,,A(y) and Fmla,,_A(z), etc. QED We now show that the partial definitions of truth can, by proofs of quadratic length, be proven to satisfy Tarski's conditions, which justifies their name. Lemma 3.3.11 (cf. [Pu 86, Pu 87]) There exists a constant c such that for everyfor- mula cp with free variables among v,-1,. . . ,v,-m and for every n with Len('<p") 3 n, we can prove the following by proofs of length 3 c n2: IAO + 01 lVw(Evalseq('<p",w) -->(S'at,,("<p",w) H ap[B(i1,w), . . . ,fi(im,w)])) (3.5) and, ifcp is a A0-formula, we can also prove the following by proofs of length 3 c -n2: IA0+Q1 lVw(Evalseq(r<p',w) -»(Sat,,,A("<p",w) <->g0[B(i1,w), . . . ,B(i,,,,w)]))(3.6) Proof. By cases. If «,0is an atomic formula t 3 t' of length 3 n and with free variables among v,-1,. . . ,v,-m, Lemma 3.3.8 implies that we can prove the following by proofs of length linear in n: IAO+ 01 lVw (Evalseq("t 3 t",w) ->(Sat,,("t 3 t'",w) 4-» Val("t",w) 3 Val("t'",w))) By Lemma 3.3.4, we can then conclude that we can prove the following by proofs of length linear in n: IAO+ 01 lVw (Evalseq('t 3 t'",w) -+(Sat,,("t 3 t'",w) 4-»(t 3 t')[fi(i1,w),. .. ,fi(i,,,,w)])) The case for t = t' is analogous. For the non-atomic cases, we define \I1;,(z,b):= Vw(Evalseq(rw_',w) -->(S'at,,(r1/2",w)4-»w[fi(i1,w), . . . ,fi(i,,,,w)])). Every formula (,0of length 3 n is constructed from atomic formulas in at most n steps. Therefore, we would like to prove the following in IA0 + 01 by proofs of length linear in k: 1. \I';,_1(w) ->\II;,(fiw) for Len('-aw") 3 k; 54 CHAPTER 3. A SMALL REFLECTION PRINCIPLE 2- '1'k-1(¢)A 'pk-1(X) -*'I'k(¢ -' X) for L€n(r¢ -' X") S k; 3. 'I1,,_1(w) ->\I1;.(Vv,-1,0)for Len('Vv,-7,0") 3 k; 4. 'Il,,-1(w) -v 'Il;.(('v'v.-< t)i,b) for Len("('v'v,-3 t)w") 3 k. If we can find these short proofs, then we have for every formula goof length 3 n a proof of 'I',,(cp)of length of the order of n2, and we are done. We will leave the easy proofs of the four cases to the reader. QED Lemma 3.3.12 IAo+Q1 proves by a proof of length of the order ofnz that Sat" preserves the logical rules (Modus Ponens and Generalization) for formulas of length 3 n, i.e. IAO + (21 lEvalseq,,(y" -->"z,w) /\ Sat,,(y, w) /\ Sat,,(y'- ->"2, w) ->S'at,,(z,w) and IAO + Q1 lEvalseqn("'v'v,-"y,w) /\ Vw'(w =,w' ->Sat,,(y,w')) ->Sat,,("\7'v,-"y,w) Proof. Immediately from Lemma 3.3.8. QED Lemma 3.3.13 IAo+Q1 proves by a proof of length of the order ofn2 that Sat" preserves the logical axioms and the equality axioms for formulas of length 3 n, e.g. axiom scheme (1) of [WP 87]: PW1 IA0 + 01 lEvalseq,,(y" ->("z' -+"y")",w) -->Sat,,(y' -->("zr -+"y")",w) Similarly, the other propositional schemes (2) and (3) are preserved. Corresponding to axiom schemes (4), (5), and (6) we have the following: PW4 (Corresponding to axiom (4) of [WP 87]) IAO + Q1 lEvalseqn("'v'v,-"y ->Sub(y, "vi", t), w) /\ SubOK(y, "vi", t) ->Sat,.("'v'v,~"y ->Sub(y,"v,-", t),w), where Sub0K(y, rv,-1,t) is Buss'formalization of "the term with Godel numbert is free for the variable v,in the (term or) formula with G'o'delnumber y". PW5 (Corresponding to axiom (5) of [WP 87/) IA0 + 01 lEvalseqn("Vv,-("yr ->72") -->("yr ->Vv,~"z')",w) A "v,~does not appear free in the formula with Gddel nr. 3)" ->Sat,,("'v'v,-("yr ->"zr) ->("yr -+Vv,-'z")",w). PW 6 (Corresponding to axiom (6) of [WP 87]) IAO + 91 lEvalseqn(v1" = "v1,w) ->Sat,,(v1r = "'v1,w) and IAO + 91 lEvalseq,.(v,-F = "v,-r -+("yr ->"z')l,w) /\ Sub0K(y,"v,7,'vJ-1)/\ Somesub(z, y, "vi", rvj") _' Satn(,UI_r'= fivjr' _*(1yr __' '1zr')'I,,w), where Somesub(z, y, "v,-1,"v,-") is the formalization of "the formula with Godel num- ber z is the result of substituting the term vj for some of the occurrences of v,-in the formula with Godel number y". 3.3. THE SMALL REFLECTION PRINCIPLE 55 Proof. For the propositional axiom schemes (PW1), (PW2) and (PW3), the results follow almost immediately from Lemma 3.3.8. For (PW4), we need proofs in IAO+ 01 of length of the order of n2 of the following "call by name = call by value" lemma: Evalseqn("\7'v,-"y ->Sub(y, "vi", t), w) A Sub0K(y, "v,-", t) ->S'at,.(Sub(y,"v,-", t),w) <-+Sat,.(y, s(i, Val(t,w),w)). This can be proved by induction on n, in a way similar to the proof of Lemma 3.3.8. The rest of (PW4) then follows by Lemma 3.3.8 itself. For (PW5), we need proofs in IA0 + 01 of length of the order of n2 of the following: Evalseq,,("Vv,-("yr ->12") ->("y' ->Vv,-12')", iv) A "v,does not appear free in the formula with Go'del number y" A w =,w -* [5atn(y.w) H 5atn(y.w')l- I This can also be proved by induction on n; again, the rest of (PW5) follows by Lemma 3.3.8. The first equality axiom of (PW6) is proved immediately by Lemma 3.3.8. The second one has a proof similar to that of (PW4). QED Lemma 3.3.14 IA0+Q1 proves by a proof of length of the order ofnz that Satn preserves the basic non-logical axioms for formulas of length 3 n, e.g. IAO + 91 lEvalseqn(rO 3 0 A -150 3 0",u2) ->Sat,,(r0 3 0 A p50 3 0",u2). Similarly for the other five basic axioms relating the symbols 0,S,+, and 3 of the language. Proof. Immediately by Lemma 3.3.8 and Lemma 3.3.4. QED Lemma 3.3.15 IAO+ 01 proves by a proof of length of the order of n2 that Sat,,,A agrees with Sat" on A0-formulas of length 3 n, i.e. Evalseqn,A(x,w) ->[Sat,,,A(x,u2) <->Sat,,(x,w)]. Proof. By induction on n as in the proof of Lemma 3.3.10. Here, we take <I>,, := 'v'x'v'w(Evalseqn'A(x,u2) -+(Sat,,_A(x,w) <-+Sat,,(x,w))). As in Lemma 3.3.10, we use the fact that if x = y" ->"z and Fmla,,+1,A(x), then Fmla,,_A(y) and Fmla,,,A(z), etc. QED Lemma 3.3.16 IAo+Q1 proves by a proof of length of the order of n2 that Satn preserves the A0-induction axioms of length 3 n, i.e. Fmla,.,A(y) A Evalseq,,(Sub(y, "v1", 0)" A Vv1("y' -->"Sub(y, "v1", Sv1)') ->\7'v1"y,u2) ->Sat,,(S'ub(y,"v1',0)" A 'v'v1("y"-v"Sub(y,'v1",Sv1)') ->'v'v1"y,w). 56 CHAPTER 3. A SMALL REFLECTION PRINCIPLE Proof. We work in IAO+ 01 and assume Fmla,,,A(y) A Evalseq,,(Sub(y,rv1",0)" /\ 'v'v1("yr ->1Sab(y, "U1",Sv1)r) -->'v'v1"y,w). Because Sat,, is a partial satisfaction predicate for formulas of length 3 n, we can, by a proof of length of the order of n2, prove that the formula Sat,,(Sub(y,"v1_', O)" A Vv1("y" ->"Sab(y, F121",Sv1)") ->'v'v1"y,w) is equivalent to the following formula: Sat,,(Sub(y,rv1", ),w) /\\7'w'(w':1 w ->(Sat,,(y,w') -+Sat,,(Sub(y,'v1",Sv1),w'))) -+Vw'(w' =1 w ->Sat,,(y,w')). This formula in turn is equivalent to: Sat,,(Sub(y,'v1", ),w) /\Va:(Sat,.,(y,s(1,:1:,w)) ->Sat,,(Sub(y,'v1", Sv1),s(1,:1:,w))) ->\7':z:Sat,,(y, s(1,:r:, 212)), where s(1, 1:,w) is as defined in Definition 3.3.5. This last formula is then, by a proof of length of the order of n2 of a "call by name = call by value" lemma analogous to the one proved in Lemma 3.3.13, equivalent to the following formula: 5atn(y.S(1,0,w)) AV1=(5atn(y.S(1.rI:,w))-+ 5atn(3/,3(1\ SI,w))) -+V:cSat,,(y,s(1,:r:,w)). This looks almost like an instance of induction. However, because Satn is not Ao,we replace it by its A0( Val,#, | |, )-equivalent Sat,,,A,as is allowed by Lemma 3.3.15 and the assumption Fmla,,,A(y), and we obtain the equivalent formula Sat,,'A(y,s(1,0,w)) /\V:z:(Sat,,,A(y,s(1,:1:,w)) ->Sat,,,A(y,s(1, S:r:,w))) ->\7';r:Sat,,,A(y,s(1, :12, As a true instance of A0( Val,#, | |, Lgxj)-induction, the formula above is at last prov- able from the assumptions. QED Now that we have the partial truth predicates in hand, we can proceed with the proof proper of the main theorem of this chapter. We suppose that the reader is familiar with IAo+Q1-cuts and IAo+Q1-initialsegments, and also with Solovay's method of shortening cuts (see definition 2.6.1, definition 2.6.2 and lemma 2.6.6). We have the following: Lemma 3.3.17 If K is an IAO+ Q1-initial segment, then IA0 + 01 l- \:/a:Pr0v('_K(E)1), where E stands for the "efiicient numeral" based on the binary expansion of 1:. 3.3. THE SMALL REFLECTION PRINCIPLE 57 Proof. See lemma 2.6.8. It is not difficult to see that the proofs of K(:f) are of length oftheorder However, in the formalized context in which we will use the result, the length of the formula K and the length of the proof p1(K) of Vy(K(y) ->K(Sy)) and the proof p2(K) of 'v'y(K(y)-+ K(SSO -y)) also play a part in the computation of the length of the total proof, thereby making the length of the total proof of the order |a:|2-|K|+|p1(K)|+|p2(K)|. In fact, if we analyze the proof we find that IA0+ 01 F VJ'v':1:(D(J"is an initial segment") -> QED Definition 3.3.18 We formally define the following: LPrf,,(u, "x") := "u codes a proof of xin IAO+ 91 involving only formulas of length 3 v". Lemma 3.3.19 The following is provable in IAO+ 01: V1=P7"0v(fVvS f(Prf(y, 7) H LP7'f..;(y.'?l))') Proof. Formalize the following observation: if a formula v occurs in a proof y where y 3 :c, then Len(v)3 |v| 3 |y| 3 QED Theorem 3.3.20 (Small reflection) For all sentences upthe following holds: IAO+ 01 lVa:Prov(r\7'y3 :f(Prf(y, ->cp)1) Proof. By Lemma 3.3.19, it suffices to prove IAO+ 91 l- \7':cProv('\'/y3 f1:'(LPrf|:|(y, ->cp)"). We reason inside IA0 + 91, and we take an :1:which we shall use to make a cut. The idea behind the proof is to find a Gédel number KI standing for a formalized "Prov-initial segment" such that we have PT0v(Kz(f)r -> Vv S f(LPrf.=.(v, 7) -> <p)")- (By abuse of notation we write K,(f) for the Godel number that results by the appropriate application of the substitution function to K3). In the construction of the Prov-initial segment K3, we will need the formalized versions of the lemmas which we proved above about the existence and the properties of partial satisfaction predicates for formulas of length smaller than some standard numeral n. In our formalized context, plays the role of "standard numeral", as will become clear when we define KI. Again by abuse of notation, we let Sat,,,|(v, w) stand for a Godel number instead of a formula; we will use the appropriate formalizations of lemmas we proved about the formulas Sat,,(v, w) to derive formalized facts about the Godel number Sat|,,|(v, w). Keeping these cautionary remarks in mind, we start the proof by defining the Godel number J, of a formalized "Prov-cut" (later to be shortened to the Provinitial segment K3 that we need) as follows: J_.,_.(s):= r\7'y,v 3 s(LPrf,,,l(y,v) ->Vw(Evalseq(v,w) ->"Sat|,,1(v,w)'))". 58 CHAPTER 3. A SMALL REFLECTION PRINCIPLE By the formalized version of Lemma 3.3.7, we may assume that this Godel number exists, because the length of Sat|,|(v,w) is linear in |:z:|. (Note that we are reasoning inside IAO+ Q1 all the time!) It is not difficult to prove directly from the definition of J, (and from the fact that J, is small enough) that the following holds: Prov(J,(6)r A 'v'y'v'z("'J,(z)"/\ y 3 z -+"J_.,(y)")"). To prove that J, is closed under successor, we remark that Prov("LPrf|,,(y,v)->Len(v)3 Therefore, we can formalize Lemmas 3.3.12, 3.3.13, 3.3.14 and 3.3.16 to conclude by a proof of length of the order |:r|2 that Sat|,|('u, w) is preserved by all logical and non-logical axioms and rules for formulas of length 3 |:1r:|,and thus indeed, P7'0v('Vy("J=(y)' -* "J=(5y)')"), proving J, to be a Proucut. By a formalization of the proof of Lemma 2.6.6, we can shorten the Proucut J; to a Prov-initialsegment K, of length linear in The proof that K, is a Prouinitial segment is of length polynomialin Carefully analyzing the proof of Lemma 3.3.17 (see the remark at the end of that proof), we find, by proofs of length polynomial in |2:|, Prov(K:(i)) APr0v(K3(?)). And thus, because we have Prov("\7'y("K_.,(y)"->"J,,(y)')"), we conclude that, by defini- tion of J,,, we have the following: Pr0v('-'v'y3 _r:(LPrf...(y. -> Vw(Evalseq(?,w) ->1Sat|,|(Fg7,w)'-))'). Because we have Prov('Vy 3 f(LPrf|¢|(y,'7p_fi) --+Fmla|,,(F?))'), we can apply the for- malized version of Lemma 3.3.11, taking note that cpis a sentence. Therefore, Prov(rVy 3 f(LP1"f|z|(Z/» ~> 'v'w(Evalseq(Fc?,w) -->cp))'). This in turn is equivalent to the desired P7'0v('Vy S f(LP7'f:=|(y, '7/7) -+ <p)")- Stepping out of IAO+ 91 again, we conclude that indeed IAO + Q1 l- \7':cPr0v(r'v'y 3 iE(LPr_fl,_.|(y, "TE ) ->90)"). QED Remark 3.3.21 Looking carefully at the proof of Theorem 3.3.20, we notice that it is also possible to derive the following result, which is a little bit stronger: IA0 + Q, l- 'v'v(Sent(v) ->'v':1:Prov(rVy 3 :'c'(LP7'f|z|(2/.F13') ->'v'')-'). Theorem 3.3.20 and its proof can also be adapted for the case that gois a formula instead of a sentence (or in the stronger result mentioned above: Fmla(v) instead of Sent(v)). 3.3. THE SMALL REFLECTION PRINCIPLE 59 Corollary 3.3.22 (Svejdar's principle is provable in IAO+ Q1) For all sentences (,0,ib, we have the following: IAO + Q1 l' Clcp -> Cl(Cl1,D S Dtp -->1p), i.e. IA0 + 91 l3$Prf(:1:, rap") ->Prov("E|y(Prf(y, F17) /\ V2:-5 y-wPrf(z, ->2,0'). Proof. We work inside IAo+Q1 and suppose Prf(:1:,rcp"). By provable 2','-completeness, this implies Pr0v("Prf(f, Hence, we have P7"0v(r33/(P7'f(y.F17) /\ V2 S 3/"1'?/(2.7/31)) -* 3y S f8P7"f(y.7)")- Theorem 3.3.20 gives P7'ov(r3y 3 EPr_f(y,Fip_") ->1/2.'); therefore, we have the following: P7'0v(r3y(P7'f(y. '77") /\ V2 S y-~P7"f(Z.3.7)) ->1/7)- Jumping outside IA0 + 91 again, we conclude that IAO + Q1 l3:rPrfl:z:,rcp") ->Pr0v(r3y(Prf(y,F1:7) /\ V2 3 yfiPrf(z, 333)) -+1/2"). QED Remark 3.3.23 Analogously to remark 3.3.21, we may strengthen §vejdar's principle to the following: IAO + Q1 |Sent(u) A Sent(v) /\ Prov(u) ->Prov(rProv(v) 3 Prov(u) -+1v). Svejdar introduced a modal system in order to study generalized Rosser sentences, and he derived the formalized version of Rosser's Theorem in it [Sv 83]. Because of Corollary 3.3.22, Svejdar's system is sound with respect to IAo+Q1, and Rosser's Theorem holds in IAo + 91. Below, we use an argument similar to §vejdar's to derive a more general theorem. For the case of PA, this theorem has been proved by Montagna and Bernardi (see [JM 87]). Theorem 3.3.24 (Montagna-Bernardi in IAO+ 01) For every function h which is Ell'-definablein IAo+Q1 and maps sentences to sentences, there is a sentence C such that IA0 + Q1 |Prov(rC") <-+Pro'u(h('-C")). 60 CHAPTER 3. A SMALL REFLECTION PRINCIPLE Proof. Define C by diagonalization such that IAO+ 01 lC <->Prov(h('C")) 3 Prov("C"). Reason inside IAO+ 01 and assume first that Prov("C"). Then by definition, Prov('Prov(h(rCfi)) 3 Prov("C")"). Meanwhile Corollary 3.3.22 gives Prov("C-') ->Prov('_Prov(h("C")) 3 Prov("C") ->h("C")'). Combined, these two yield Prov("C") -+Prov( ("C")). For the other side, we assume that Prov(h('C")). This implies Prov('Prov(h('C"))"), and thus Prov('Pr0v(h("C-')) 3 Pro'u("C") V Prov('"C") 3 Pr0v(h(rC"))"). By definition of C, we derive Prov("C V Prov('"Cfl) 3 Prov(h('PC"))"). Now we apply Corollary 3.3.22 to conclude that, because Prov(h("C")) -+Prov(rProv("Cj) 3 Prov(h('C")) -->C1), indeed Prov(h('C")) ->Prov('C"). QED Note that the formalized version of Rosser's Theorem follows immediately from this construction. If we take R such that IAO + 91 lR <-+Prov("-IR") 3 Prov(rR1), we derive IAO + 91 lPr0v('R") <->Prov("-wR"), and thus IAO + 91 lProv(rR") -> Prov('-.L")and IAO + 91 lProv('-wR") ->Prov('J_"). 3.4 Injection of small (but not too small) inconsis- tency proofs Using the small reflection principle, we can strengthen Hajek's, Solovay's and Krajfcek and Pudlak's results on the injection of inconsistencies into models of IAo+EXP [Ha 83, So 89, KP 89]. Instead of only injecting an inconsistency proof, we also take care to respect a fair number of consistency statements. Moreover, we do not need full exponentiation in our original model. We cannot immediately apply the lemmas of [KP 89], but the essential steps in our proof are the same as in their article. We first apply Pudlak's version of G6del's Second Incompleteness Theorem (see [Pu 86, Theorem 3.6]) to show that we can indeed inject an inconsistency proof; then we use the Omitting Types Theorem to prevent extra elements from creeping into the lower part of the new model that contains our injected inconsistency proof. 3.4. INJECTION OF SMALL INCONSISTENCY PROOFS 61 Theorem 3.4.1 Let T Q IAO + 91 be a 2'1'-asciomatizedtheory for which the small reflection principle (see Theorem 3.3.20) is provable in IAO + 01. Let Con»_r(2:)be a formalization of the consistency of T up to proofs of length 2:. Let M be a nonstandard countable model of IAo+Q1. Let a, c be nonstandard elements of M such that the following conditions hold: a exp(a°) E M, o M I= C'onT(a'°) for all k < co. Then there exists a countable model IC ofT such that a 6 IC and 1. M [ a = IC [ a, 2. M [exp(a") Q ICfor all k < co, 3. IC |= -vConT(a°), 4. IC }= ConT(a") for all k < co, 5. IC |= 2°C 1. Proof. Define N := {:5E Mla: < exp(a'°) for some k < co}. Then exp(a°) E M \/V', thus M is a proper end-extension of N. Therefore, by Theorem 1 of [WP 89], N I: B21. (Remember that B21 is IAO + the scheme Vt('v':z:< tElycp(a:,y) -»Ela'v':1:< t3y < a<p(:c,y)) for cp6 2?.) Also, it is easy to see that N |= 91. On the other hand, one of our assumptions is that M I: ConT(a'°) for all k < (.4). By A0-overspill we conclude that there is a nonstandard d < c in M such that M }= ConT(a"'). Thus, by Theorem 3.6 of [Pu 86], there is a k < (.4)such that M |= ConT+,Co,,T(aa)(ag), so certainly M l: ConT+-Co,,T(,,c)(alE). Indeed, because 3%is non- standard, we even have N l= Con(U), where U := T + -=ConT(a°). At this point we need some definitions analogous to the ones in [KP 89]. Let L(./V) be the language of arithmetic expanded with domain constants for the elements of N. We define a translation t from L(/\/) to N by t(A(a1, . . . , a,.)) := 'A(a;, . . . ,cT,;)",where a, is the efficient numeral of a,-. We need one more definition: U' ;= {A(a) e L(/\/)|/V 5: ProvU(t(A(a')))}. It is easy to show that U' is closed under the rules of predicate logic; that U Q U'; and that Diag(/V) Q U'. Also, because N l: Con(U), we can conclude that U' is consistent. Moreover, by the small reflection principle for IA0 + 01, we have N l: V:z:ProvU("C'onT(|f|)"), thus for all k < co, ConT(a") E U'. Finally, using Solovay's cuts, we can show that N l: \7':z:ProvU('2'1 "), thus 2°C16 U'. We construct the required model ICby the Omitting Types Theorem in order to take care that ICwill contain no new elements below a. Let r be the type in L(/V')defined by 7'(1:):={:c§a}U{a:-,£b|bEM[a}. Claim 1: U' locally omits r. 62 CHAPTER 3. A SMALL REFLECTION PRINCIPLE Proof. Take any A(:1:),and suppose that for all b 3 a in N we have U' l- -wA(b),and that U' lA(a:) ->:1:3 a. We want to show that U' l- -a3:z:A(:i:).By definition of U', it is sufficient to prove the following: N l: Vb 3 aProuU(r-A(b)") ->ProuU(r'v':z:3 anA(:f)"). So, suppose N I: Vb3 aProvU("-:A(b)"). By B21, there is a q E N such that N l: Vb3 ailp < qPrfU(p, F-aA(b)"). Now we can use Ao(w1)-induction to show that we can combine these proofs for all b 3 a into one proof p of Va:3 anA(:r), where |p| 3 a - (|q| + k - |a|) 3 a"' for some standard k, n, m, thus 19E N. We conclude that indeed N l: ProvU('V:c 3 anA(:c)"). QED At last we can construct a model }Cof U' omitting 7'. Using the facts that we proved about U', we conclude that ICsatisfies all the properties that we want. QED In Theorem 3.4.1, we require that T Q IAO + 01 is a Z3';-axiomatized theory for which the small reflection principle is provable in IAQ+ 01. Examples of such theories are finite extensions of IA0 + Q1 itself, IA0 + EXP and PA. We hope to give an exact characterization of theories amenable to methods analogous to those of section 3.3, [Pu 86] and [Pu 87] in a later paper. Theorem 3.4.1 is only a slight extension of [KP 89, Theorem 2.1]. We use the small reflection principle only to show that the length of injected inconsistency proofs can be bounded from below as well as from above. A variation on the proof of theorem 3.4.1 gives the following theorem. Its proof con- tains a more surprising use of the small reflection theorem than the proof of theorem 3.4.1: In theorem 3.4.3 we use it even in our application of the Omitting Types Theorem. Recently, some papers (see [WP 89, Ad 90, Ad 93]) appeared that partially answer the end extension problem, which was formulated by Kirby and Paris in 1977 as follows: does every model of IAO+ B21 have a proper end extension to a model of IA0? The theorem below gives a sufficient condition for a countable model of IAo + B21 to have a proper end extension to a model of IAO: if the model additionally satisfies 01 + Con(IA0) and provable completeness for H3-formulas, then it does have such an end extension. First we need a definition. Definition 3.4.2 Cl'I'2'(U)is the scheme A(a1, . . . ,a,.) ->ProuU("A(m, . . . , aTk)') for A(a1, . . . ,a;.) 6 H3. Theorem 3.4.3 Let U Q Q be a Elf-aziomatized theory, and suppose N is a countable model of B21 + 01+ CHg(U) + Con(U), then there exists a countable model IC ofU such that IC is an end-extension of N. Proof. Define U' from U,N exactly as in the proof of Theorem 3.4.1. Again, we construct the required model ICof U' using the Omitting Types Theorem. This time, we define for all a E N the type Ta in L(N) by: 'r.,(a:):= {a:3a}U{:1:;éb|bE.M ra}. 3.4. INJECTION OF SMALL INCONSISTENCY PROOFS 63 Claim 2: U' locally omits Ta for all a E N. Proof. Take any a E N and any formula As in the proof of Claim 1, it is sufficient to show the following: N l: Vb 3 aProvU('FA(l_7)") ->Pr0vU('\7':z:3 do/1(E)"). So, suppose N l: Vb3 aProvU("fiA(l3)"). By B21, there is a q E N such that N l: Vb 3 ailp < qPrfU(p, r-A(l_>)_'). Now by CHg(U), we derive N l= 3qProvu(rVb S @310< ?lP7'fU(P»""'A(5)")il- Therefore by the small reflection principle, N l: ProvU('_Vb 3 EL-vA(5)"). QED We can now construct a countable model ICof U' omitting all Tafor a E ./V'.As before, it is easy to see that U Q U' so [C l: U. By the way, note that by the small reflection principle for IA0 + 91, or simply by the isomorphism, we have ConU(|f|) E U' and thus IC |= ConU(|f|) for all 1:E N. QED

Chapter 4 Provable completeness for Z1-sentences implies something funny, even if it fails to smash the polynomial hierarchy But what's so blessed-fair that fears no blot? Thou mayst be false and yet I know it not. Shakespeare, Sonnets, no. 92 4.1 Introduction In chapter 3, we proved that, if NP 7éc0-NP,then Dcompleteness for witness comparison formulas is not provable in bounded arithmetic, i.e. IAO + 91 l7"v'b\7'c (Ela(Prf(a, c) /\ V2 _<_afiPrf(z, b)) v --+Prov("3a(P'r_f(a,E)/\ V2 3 awPrf(z, The above result does not give any information about Z31-sentences.If bounded arith- metic would prove completeness for Z31-sentences, then we could adapt Solovay's Com- pleteness Theorem and prove that L is the provability logic of bounded arithmetic. In this chapter we show that provable completeness for all Z31-sentencesis unlikely. Unfortunately we have to work under an assumption (namely P aé NPr'1co-NP) in which complexity theorists have less faith than in the assumption NP aéco-NP that we used in chapter 3. 65 66 CHAPTER4. PROVABLE COMPLETENESS FOR E1-SENTENCES 4.2 If S; proves completeness for all Z1-sentences, then NP (7co-NP=P Theorem 4.2.1 Let k 2 1. If Al: 75 Efi 0 Hi, then there is a sentence 0 of the form 3:z:<p(:z:),where up is a Hi-forrnula, such that S; l7'0 ->Prov5;("a'). Proof. We prove the theorem for k = 1. For k > 1 the proofs are analogous. Suppose that P 76 NP 0 co -NP. Let the 2'1'-formula A(:1:)represent a predicate in NP (1co -NP, but not in P. Thus there is a Z'1'-formulaB(:r) that represents the complement of A. Now define C'(:2:,y) := (y = 0 A V (y = 1/\ B(:z:)). It is easy to see that (.0l: V2333;g 1C(:r:,y). Now let 0 be the Z1 sentence defined by 0 := 3:z:'v'y3 1fiC(:z:, y), and suppose that S; l0 ->Pro'u("0"), which is equivalent to 51}lV1:(3y 3 1C(:z:,y) V 3yPrf(y, raj)). Next, by Buss' main theorem, we find a polynomial time function f such that w l= V1=(C(-'6,f(I)) V P7"f(f(1=),"0l))- But 0, being a false sentence, is not provable in S21,so we have actually an l= V:cC(:z:,f(:z:)). This means that f is the characteristic function of A, hence A is in P, contrary to our assumption. QED Chapter 5 On the provability logic of bounded arithmetic C noxaayeuocrn MLICOMr<panHnM- Happy furtherrnost cape of provability- (Mapnna Llneraena, Honoroimee, 1925) (Marina Tsvetaeva, New Year's Poem, 1925, translation Joseph Brodsky) Abstract. Let PLO be the provability logic of IA0 + $21. We prove some containments of the form L Q PLQ C Th(C) where L is the provability logic of PA and C is a suitable class of Kripke frames. 5.1 Introduction In this chapter we develop techniques to build various sets of highly undecidable sentences in IAo+Q1. Our results stem from an attempt to prove that the modal logic of provability in IAO+ Q1, here called PLQ, is the same as the modal logic L of provability in PA. It is already known that L Q PLQ. We prove here some strict containments of the form PLQ C Th(C) where C is a class of Kripke frames. Stated informally the problem is whether the provability predicates of IAQ+ 01 and PA share the same modal properties. It turns out that while IAO+ 91 certainly satisfies all the properties needed to carry out the proof of G6del's second incompleteness theorem (namely L Q PLQ), the question whether L = PLO might depend on difficult issues of computational complexity. In fact if PLQ 7,45L, it would follow that IAO + Q1 does not prove its completeness with respect to Z?-formulas, and a fortiori IA0 + 01 does not prove the Matijasevic-Robinson-Davis-Putnamtheorem (every r.e. set is diophantine, see [Ma 70], [DPR 61]). On the other hand if IA0 + 91 did prove its completeness with respect to Z?-formulas, it would follow not only that L = PLQ, but also that NP = co -NP. The possibility remains that L = PLO and that one could give a proof of this fact without making use of provable XI?-completeness in its full generality. Such a project is not without challenge due to the ubiquity of Z3?-completenessin the whole area of provability logic. 67 68 CHAPTER 5. ON THE PROVABILITY LOGIC OF BOUNDED ARITHMETIC We begin by giving the definitions of PLQ. For the definitions of L and T-interpre- tation, we refer the reader to section 2.2. Definition 5.1.1 Let PLQ be the provability logic of IA0 + Q1, i.e. PLQ is the set of all those modal formulas A such that for all IA0 + Q1-interpretations *, IAO+ 01 lA'. It is easy to see that PLQ is deductively closed (with respect to modus ponens and necessitation), so we can write PLQ lA for A E PLQ. Our results arise from an attempt to answer the following: Question 5.1.2 Is PLQ = L? {Where we have identified L with the set of its theorems.) The soundness side of the question, namely L Q PLQ, has already been answered positively. This depends on the fact that any reasonable 2'1'-axiomatized theory which is at least as strong as Buss' theory S21satisfies the derivability conditions needed to prove G6del's incompleteness theorems (provided one uses efficient coding techniques and employs binary numerals). For the completeness side of the question, namely PLO Q L, we will investigate whether we can adapt Solovay's proof that L is the provability logic of PA. We assume that the reader is familiar with the Kripke semantics for L (see defini- tion 2.2.2) and with the method of Solovay's proof as described in [So 76]. In particular we need the following: Theorem 5.1.3 L lA ifl A is forced at the root of every finite tree-like Kripke model. (It is easy to see that A will then be forced at every node of every finite tree-like Kripke model.) Solovay's method is the following: if L if A, then the countermodel (K, 4, Il-)provided by the above theorem is used to construct a PA-interpretation * for which PA l7'A'. The reason Solovay's proof is difficult to adapt to IAO+ 01 is that it is not known whether IAO+ Q1 satisfies provable Z3?-completeness (see definition 5.2.1) which is used in an essential way in Solovay's proof. 5.2 Arithmetical preliminaries Definition 5.2.1 Let I' be a set of formulas. We say that a (Elf-axiomatized) theory T satisfies provable I"-completeness, if for every formula o(:i:') E F, T lo(;r1, . . . ,2:,,) -+ Prov1~('o(:z:'1,...,:T,)-'). It is known that PA, as well as any reasonable theory extending IAO+ EXP, satisfies provable Z?-completeness. De Jongh, Jumelet and Montagna [JMM 91] showed that Solovay's result can be ex- tended to all reasonable Z3?-soundtheories T satisfying provable X3?-completeness. More precisely it is sufficient that the provability predicate of T provably satisfies the axioms of Guaspari and Solovay's modal witness comparison logic R'. So Solovay's result holds for ZF, 12,, and IAO+ EXP. On the other hand it is known that if NP 75co -NP, then IAO+ Q1 does not satisfy provable E?-completeness or even provable A0-completeness. In chapter 3 we proved that, if NP 75 co -NP, IAO+ 01 does not even satisfy provable completeness for the single 5.2. ARITHMETICAL PRELIMINARIES 69 )3?-formula a(u,v) E El:c(Prf;Ao+g,(x,u) A Vy < :1:-»Prf;,_«_0+n,(y,v)).(See also chapter 4 for a related result.) In view of the above difficulties, we try to do without E?-completeness. In the rest of this section we state some results about IAo+ D1 which in some cases allow us to dispense with the use of E?-completeness. The following proposition is proved by [WP 87] (see also theorem 2.3.24): Theorem 5.2.2 IAO+ 91 satisfies provable 2'1'-completeness. By abuse of notation we will denote by DA both the arithmetization of the provability predicate of IAO+ 91 and the corresponding modal operator. 0A is defined as -DnA and D+A as DA A A. If A(:z:) is an arithmetical formula, we will write 'v':z:D(A(:v))as an abbreviation for the arithmetical sentence which formalizes the fact that for all 1:there is a IAo+Q1-proof of A(.f), where E is the binary numeral for :13.If A and B are arithmetical sentences, DA 3 DB denotes the witness comparison sentence 3;z:(Prf;A0+Q,(:c, FA") /\ Vy < :2:-=Prf;A0+g,(y, "B")). Similarly DA < DB denotes 3r(PrfIAo+Q1($7 I-/1-')A S $-'PT.fIAo+Q1(yirB-l))- D,,A is a formalization of the fact that A has a proof in IAO+ 91 of Godel number 3 k. So DA < DB can be written as El:c(D,A/\ -wD,B). (Note that all the above definitions are only abbreviations for some arithmetical formulas and are not meant to correspond to an enrichment of the modal language.) Remark 5.2.3 Since the proof predicate can be formalized by a Ell'-formula,we have IAO + Q1l' GA ->DD/l and IAO + Q1 l' l'_-l_.,,A->DD:/l. We suppose that the reader is familiar with IAO+ Q1-initial segments (see defini- tion 2.6.2). Given an IA0 + Q1-initial segment I, IAO+ Q1 can formalize the fact that I defines a model of IAO+ 01. It follows that for any arithmetical sentence 6 we have: Proposition 5.2.4 IAo+Q1 l-D(6l) ->D(6'), where 6' is obtained from 0 by relativizing all the quantifiers to 1. Note that if a Z?-formula is witnessed in an initial segment, then it is witnessed in the universe. Thus we have: Remark 5.2.5 For every IA0+Q1-initial segment I, and every 2?-formula a(:c1, . . . ,:c,,), IAo+Q1l-1:16 I/\.../\:z:,.€ I/\a'(:z:1,...,:z:,,) ->a(.z1,...,:z:,,). The use of binary numerals is essential for the following proposition (see lemma 2.6.8 and [Pu 86]): Proposition 5.2.6 For any IAO+ Q1-initial segment I, IAO+ 01 |- \7':1:D(a:E 1). Making use of an efficient truth predicate, we proved the following result in section 3.3: Theorem 5.2.7 (Small reflection principle) IAO+ 01 |VkD(D,,A->A). 70 CHAPTER 5. ON THE PROVABILITY LOGIC OF BOUNDED ARITHMETIC An immediate corollary is the following principle (originally stated by Svejdar for PA; see corollary 3.3.22 for a proof): Corollary 5.2.8 (Svejdar's principle) IAO+ Q1 lDA ->Cl(ClB3 DA ->B). Using Solovay's technique of shortening of cuts, it is easy to prove the following: Proposition 5.2.9 There is an IAo+Q1-initial segment J, such that for each Z'1'-for7nula a(a:1, . . . ,a:,,) we have: IAo+Q1lJ(:z:1)/\ . . . /\J(:z:,,)/\aJ(:c1, . . . ,a:,,) ->CJa(:r1,. . .,1:,,). Proof. The proof is similar to the proof of provable El,'-completeness for IAO+ 91 (see theorem 2.3.24 and [WP 87]). Therefore we only give a sketch of the proof. By induction on the structure of the formula, one can prove that for each 130formula A with free variables :31,. . . ,;1:,,, there are k, l and m such that IA0 + Q1l- 'v'x'v'1:1,...,:c,,3 a:(y = e2:p(e:cp(|'A"|'°- + m) /\ A(x1, . . . ,:1:,,)-+Elz3 yPTf]A0+Q1(Z,rA(j1,...,j2)1)). Now let J be the initial segment, which can be obtained by Solovay's shortening methods (cf. lemma 2.6.6, lemma 2.6.9), such that o IAO + Q1lV:::(J(;z:) ->3z(z = 23)) and o IAO + Q1l- 'v':r,y(J(2:) A J(y) -»J(:I: + y) A J(1: y) A J(2l""'y')). For this initial segment, we have for all A0-formulas A, IA0+ Q1l- \7'a:1,...,:z:,,(J(:r1)/\ ... /\ J(:I:,,) /\ /l(.'I31,...,.'En) -+3ZPTf[A0+Q1(Z,rA(j1,...,.T2)1)). The result immediately follows. QED In the sequel of this chapter 'J' will always refer to the initial segment of proposi- tion 5.2.9. Corollary 5.2.10 If S, = 1,. . . ,k) are XI?-sentences,then IAO + 01 l' D(V --*D(V l-_-l+S,'). Proof. Let J be as in proposition 5.2.9. Work in IA0+Q1 and suppose E1(V,-S,-)holds. Since J (provably) defines a model of IAO+ 01, it follows Cl(V,By proposition 5.2.9 and remark 5.2.5 D(S,.J --+CJ+S,-) and the desired result follows. QED The above corollary was originally proved by Albert Visser [Vi 91b] as a consequence of the following more general result: Theorem 5.2.11 (Visser's principle) IfS and S,-(z'= 1,. . . ,k) are Z?-sentences, then IA0 + 01 l' Cl(/\t-(Si -->D5,) -9 ->US, 5.3. TREES OF UNDECIDABLE SENTENCES 71 5.3 Trees of undecidable sentences We will rephrase the problem of whether PLO = L as a problem concerning the existence of suitable trees of undecidable senctences. Let C be a class of finite tree-like strict partial orders. Without loss of generality we assume that for all (K, <) E C, K = {1,...,n} for some n E w, and 1 is the root (i.e. the least element of K By Th(C) we denote the set of all those modal formulas that are forced at the root of every Kripke model whose underlying tree belongs to C. Let j be the non-strict partial order associated to 4. Definition 5.3.1 Given a tree (K,<) with root 1 and underlying set K = {1, . . . ,n}, we say that (K , <) can be embedded (or simulated) in IAO+ 91 if there are arithmetical sentences L1, . . . , L,, (one for each node) such that, letting [3 denote formalized provability from IAO+ 01, the conjunction of the following sentences is consistent with IAO+ 91: 1. L1 2. EJ'' 01 D + P' 1 DJF 6' "1 x K». 3 X The following lemma is inspired by Solovay'sproof of the fact that L is the provability logic of PA. Lemma 5.3.2 In order for PLO Q Th(C) to be the case it sufiices that every tree (K, <) E C can be embedded in IAO+ 91. Proof. Suppose A ¢ Th(C). Then there is a Kripke model (K, 4, H»)such that (K, -<) E C, K = {1, . . . ,n}, 1 is the least element of K, and 1Il--»A. By our hypothesis there exists a model M of IAO+ 01 and sentences L1, . . . , L,, satisfying, inside the model M, the conditions 1 5 of definition 5.3.1. Define a IAO+ Qpinterpretation ' by setting, for every atomic propositional letter p, p" E V,-,,,,L,-. It is then easy to verify by induction on the complexity of the modal formula B, that for every 2'E K: I ill-B=>Ml=D+(L,--+B"'); . z'Il~a3 =;. M }= u:1+(L, -)-IB*)_ The induction step for Clis based on condition 4 and the following consequence of condi- tions 2 and 5 of definition 5.3.1: M e cm. ~ a<v L.>>. j>z' Since 1 II--aA, it follows that M l: -A', hence IAO + 01 l7'A' as desired. QED Corollary 5.3.3 If every finite tree (K, <) can be embeddedin IAO+ 01, then PLQ = L. 72 CHAPTER 5. ON THE PROVABILITY LOGIC OF BOUNDED ARITHMETIC W X Y Figure 5.1: The trees W, X, Y Proof. Let C be the class of all finite trees. If our hypothesis is satisfied, then L Q PLQ Q Th(C) = L. QED It can be easily verified that the sufficient condition of lemma 5.3.2 is also necessary. For, suppose that some tree (K, 4) E C with root 1 and underlying set K = {1, . . . ,n} cannot be embedded in IAO+ 91. Then the negation of the conjunction of 1-5 (see definition 5.3.1) is easily seen to be in PLQ \ Th(C). Thus PLQ Q Th(C) iff every (K, 4) E C can be embedded in IAO+ 01 . Hence a very natural question to ask is: Question 5.3.4 Which finite trees can be embedded in IA11+ 01? Note that a complete answer to the above question, although interesting by itself, may not suffice to characterize PLO. In fact if C is the set of all finite trees that can be embedded in IA0 + Q1, we can in general only conclude PLQ Q Th(C). In order to describe the results proved in this chapter and in previous papers, we need to define what it means for a tree to omit another tree. Definition 5.3.5 Let (T1,41) and (T2,42) be (strict) partial orders. An homomorphic embedding of (T1,41) into (T2,42) is an injective map f : T1 ->T2 such that for all :z:,y 6 T1, :1:41 y 4-»f(:1:) 42 f(y). If there is no homomorphic embedding of T1 into T2 we say that T2 omits T1. If we try to adapt Solovay's proof to IAO+ 91 in the most straightforward manner, the only trees that we can embed in IAO+ Q1 are the linear trees, namely trees omitting (K, 4) where K = {1,2,3}, 1 4 2, 1 4 3 and 2 is incomparable with 3. A first improvement can be achieved using Svejdar's principle: let C1 be the class of all trees that omit the tree W = (W, 4), the least strict partial order with underlying set W = {1,2,3,4} such that 1 4 2, 1 4 3 4 4 (see Figure 1). In [Ve 88] it was proved that for trees in C1 Solovay's proof can be adapted using Svejdar's principle. In other words, PLO Q Th(C1). Moreover it was proved that the inclusion is a strict one. In subsequent unpublished work I showed, using both Svejdar's and Visser's principles, that PLQ is included in the modal theory of C2, the class of all trees of height g 3. A new improvement [BV 91] was achieved by analogous techniques but using a different definition of the Solovay constants. In this way it was proved that PLQ Q Th(C3), where C3 is the class of all trees that omit the tree X = (X, 4), the least strict partial order with underlying set X = {1,2,3,4,5} such that 1 4 2 4 4 4 5, 1 4 2 4 3. Finally in Section 5.4 of the present chapter, we improve these earlier results, by proving: 5.4. UPPER BOUNDS ON PL!) 73 Theorem 5.3.6 PLQ Q Th(C4), where C4 is the class of trees that omit the tree Y = (Y,-<), the least strict partial order with underlying set Y = {1,2,3,4,5,6} such that 1<2<3<&1<2<4<& In particular, theorem 5.3.6 implies that we can embed X. Note that the trees in C4can have an arbitrarily large number of bifurcation points, but each bifurcation point except the root can have at most one immediate successor which is not a leaf. The root can have any number of immediate successors which are not leaves. On the other hand, we prove in Section 5.5 that for many classes C of trees (and espe- cially for the classes C1,. . . ,C4 defined above), we cannot have PLQ = Th(C). Therefore, all inclusions mentioned above are strict. More precisely we prove that if PLQ = Th(C), then every binary tree can be homomorphically embedded in some tree belonging to C. So it is unlikely that PLO is the theory of a class of trees, unless PLQ = L. 5.4 Upper bounds on PLQ Our task in this section will be to prove PLO Q Th(C4) using lemma 5.3.2. Definition 5.4.1 Given (K, <) 6 C4, we say that i E K is a special node, iffi is a leaf, and some brother ofi is not a leaf. For example, in the tree X of Figure 1, the only special node is 3. Definition 5.4.2 Let (K, <) 6 C4. Without loss of generality assume that K = {1, . . . ,n} and 1 is the root. Let J be the initial segment of proposition 5.2.9. By a self-referential construction based on the diagonal lemma, we can simultaneously define sentences L1, . . . , L,,, and auxiliary functions v,w, S, such that the following holds: 1. Ifi E K is not special, let = ,az:Cl,-=L,-(with the convention that = oo if <>L,);ifi E K is special = pa: 6 JD_.,wL,-(with the conventionthat = 00 if <>"L,-).We agree that oo is a specific element greater than any integer. Note that the definition of to can be formalized in IAO+ Q1. 2. Ifj is an immediate successorofi in (K, <), let v(i,j) = w(j); otherwise v(i,j) = oo. 3. S : K -->K is defined as follows: S(i) = i if for noj E K we have v(i,j) < oo; otherwise among all the j E K with v(i,j) < oo, pick one for which v(i,j) is minimal, and set S(i) = S(j). (Note that there exists at most one suchj because if w(j) = w(j') < oo, then there is one single proof of both fiLJand fiLJ-»,so j = j'.) 4. IAO+ Q] l' 144-?E-114/\7l= The important point to observe, is that the definition of S can be formalized in IAo+Q1 and that IAO+ Q1 proves that S(1) is always defined. This depends on the fact that, although S is defined in a recursive way, to compute S(1) one only needs a standard number of recursive calls, namely at most d where d is the height of the tree (K, <) (in fact at each recursive call we climb one step up in the tree). Note also that S depends self-referentially on L1, . . . , L,,. Finally note that, if a, b are distinct immediate successors of i, then the statement v(i, a) < v(i, b) is equivalent to a witness comparison sentence in which some quantifiers are relativised to J. In particular, if a and b are not special, then v(i, a) < v(i, b) is equivalent to the 2'1'-sentenceDaL,, < El-~Lb. 74 CHAPTER 5. ON THE PROVABILITY LOGIC OF BOUNDED ARITHMETIC Remark 5.4.3 The main differences with Solovay'sconstruction are the following: 1) we do not use an extra node 0 (but this is a minor point since we could define L0 as <>L1). 2) In our construction we can only jump one step at a time, namely at each recursive call 5 we can only move from one point to some immediate successor (but this is also an inessential difference). 3) While Solovay employs a primitive recursive function from w to K whose definition is not directly formalizable in IAO+ Q1, we use instead a function S : K ->K which is provably total in IA0 + 01. 4) We jump to a special node i E K only if we find a proof of nL, belonging to the initial segment J. Given (K, <) as above, we will show that L1, . . . , L,, constitute an embedding of (K, <) in IA0 + 01. We need the following lemma. Lemma 5.4.4 Let L1, . . . , L,, and (K, <) be as in definition 5.4.2. Then: 1. |*-El-1L1->L1V...VL,,. 2. lL, -->fiL,foriaéj in K; 3. lL, -->CJ-aL,fori E K. 4. L1 is consistent with IA0 + 91; 5. Ifj,j' E K are brothers, then lCl-»L,-<-+EHL,-I. 6. l-L,--><>L,-fori<j inK. 7. lL, ->D-aL,fori <j in K; 8. Ifi is above {i.e. :) a brother ofj, then lL, ->El-=L,-;if moreoverj is a leaf, then l" Lj -*D'|L,'. 9. Letj > 1 be an immediate successor of the root 1. Then lL, ->IZlCl(-L,~); 10. lL, -->E1+(L, -->Cl-:L,-)whenever i,j are incomparable nodes of K; where ' l- ' stands for 'IAo + 01 l- '. Proof. It will be clear from the context at which places we reason inside IAO+ 01. (1) and (2) are clear from the definition of the sentences L, and the fact that S : K -> K is a total function. (3). L, implies that Cl-L1 /\i = S(1). Ifi = 1, El-=L,follows immediately; otherwise we have w(i) < oo, and therefore Cl-L,. (4). If L1 is inconsistent with IAO+ 91, then El-L1 holds in the standard model, so by (1), one of the sentences L, must hold in the standard model. This is absurd since each of these sentences implies its own inconsistency. (5). First note that l' ClpL,- ->El(:cE J A ClpL,-). Thus, regardless of whetherj is special or not, lCHL, -+D(w(j) = ,u:r:D,~L,-). Sincej and j' are brothers, lL,-«-> 5.4. UPPER BOUNDS ON PLO 75 w(j') < w(j) (because j' = S(1) implies_w(j') < w(j)). Therefore lD-aL,--->Cl(L,-I -> Cl-1L,-2< El-»L,-). On the other hand by Svejdar's principle and we can conclude l-E]-uL,-2 Cl-L,-I. (6). L6b's logic L proves OA /\ C1(A->OB) ->OB. Hence by arithmetical soundness IOL,, A D(L,, -->OL,,) -><>L,,. It follows that in the proof of (6) we can assume without loss of generality that j is an immediate successor of 2. Working inside IAo+Q1, assume L,-. Then i = S(1). Hence w(j) = oo. Now ifj is not a special node, then w(j) = oo <-->0L,- and we are done. Ifj is a special node, from w(j) = oo we can only conclude OJLJ-, so we need an additional argument. This is provided by point In fact by definition of special node, 2 has certainly one immediate successor j' which is not special. Hence from L, we can derive <>LJ~reasoning as above. By point (5), <>LJ~+-><>L,~vand we are done. (7) can be derived through the chain of implications: L, ->CJwL,-->ElElfiL,~ ->EHL,-, wherethe last implicationuses point (8). Let 2 be above a brother ofj. Then by (5), (7) and (3) lL, ->D-aL,-as desired. To prove the second part, assume further that j is a leaf. We need to show 1-L, ->CHL,-. We can assume that 2 is strictly above a brother 3''ofj (for ifi itself is a brother ofj the desired result followsfrom (3) and But then j must be a special node, and therefore w(j) = pa: 6 JD,-wLJ-. So w(j) < w(j') is equivalent to a E?-formula relativized to J, namely w(j) < w(j') <--+3:1: 6 J(Prf1A0+g1,(:r:,'fiLJ7)/\'v'y 3 :21-Prf1110+g1,(y,r-1L,-21)). Thus by the properties of the initial segment J (and by theorem 5.2.6), *' WU) < WU') -* DWU) < WU')- Now the desired result follows by observing that lL, -+w(j) < w(j') (as 1-3'= 5(1) -+ WU) < WU')) and I' Li -+ WU') < WU)- (9). By (1) and (3), 1L1 ->D(V,H L,-). So to prove IL1 -+DD-«L,-, it suffices to show that for each i > 1 we have l-D(L,- -»Cl-»LJ-).This follows from (8) , (3) and (10). If the incomparable nodes i and j are in one of the situations covered by point (8), then 1L, -> D-1L,-, and a fortiori 1L1 -+ El+(L,- -> D-1L]-) as desired. Since (K, -<) omits Y, (8) can always be applied except when the largest node (with respect to j) below i and j is 1 (the root). So assume that this is the case. By (2), we have 1L1 ->(L, ->D-wLJ-). In order to show that also 1L1 ->Cl(L,- ->Cl-IL,-), we will take in account the properties of the initial segment J (see proposition 5.2.9). Let i',j' be the least nodes with 1 4 i' _<_2' and 1 < j' j 3'. So 2" and j' are brothers. It follows from (9) that 1L1 ->El(Cl-«L,-I). Therefore, by proposition 5.2.4, IL1 ->D(E1J-L,~r). In the presence of C1'-1L,-1,the sentence w(i') < w(j') is equivalent to a E?-sentence relativized to J. Therefore, by proposition 5.2.9, lL1 ->|Z1(w(i') < w(j') -»El(2u(2")< w(j'))). The desired result now follows from the fact that L, provably implies i = S(1) which entails w(i') < w(j'), while L, provably implies w(j') < w(2"). QED Corollary 5.4.5 If(K, <) and L1, . . . , L,, are as above, then the conjunction of the fol- lowing sentences is consistent with IAO+ 01: 76 CHAPTER 5. ON THE PROVABILITY LOGIC OF BOUNDED ARITHMETIC 1. L1 + + ( (--+-«L,-)f0ri;£j inK. +(L,-><>L,-)fori-<j in K. +(L,- ->Cl-aL,-)fori 74]' in K. Proof. By (1) and (3) of lemma 5.4.4, and noting that lD-=L1-»UD-»L1,we derive lL1 -+Cl+(L1 V . . . V L,,). Next, (3) implies that lL1 ->D+(L,- ->5L,-) for i aéj in K. By (6) we have L1 -->C1+(L,-->OL,-) for i <3' in K. Finally the corollary follows by (10)and Note that the derivation of corollary 5.4.5 from lemma 5.4.4 follows from a straight- forward argument which can even be formalized in the decidable theory L". (The axioms of L"'are all the theorems of L and all the instances of DA ->A. The only rule is modus ponens.) QED We have thus shown that every tree of C4can be embedded in IAO+ 01. Thus: Theorem 5.4.6 PLQ _C_Th(C4). .n L1v...vL,,). .::z L, .Cl - .Cl En-¢\C«ot\°: 5.5 Disjunction property In this section we prove the following: Theorem 5.5.1 If PLQ = Th(C), where C is a class of finite trees, then every binary tree can be homomorphically embedded in some tree belonging to C. In particular, since the binary tree Y cannot be embedded in any member of C4, it will follow that the inclusion PLQ Q Th(C4) is strict. We will use the fact that PLO has the 'disjunction property' as proved by Franco Montagna (personal communication). Definition 5.5.2 A modal theory P has the disjunction property if for every pair of modal sentences A and B, if P lDA V DB, then P lA or P lB. It is known that L has the disjunction property. Theorem 5.5.3 (Montagna) PLO has the disjunction property. Proof. Suppose that for some IAo+Q1-interpretations ° and ' we have lAo+Q1 l7'A(p'°) and IAo+ 01 i7'B(p'°), where p'contains all propositional variables occurring in the modal formulas A and B. We have to prove that there is an IA11+ 01interpretation * such that IA11+ Q1l/(CIA V DB)' By multiple diagonalization, define for all p,-6 {fan arithmetical formula p; such that me + 91* 193*-*(DA(I3"')S UB(1'>")/\ P?) V (DB(i>") < UAW) /\ PS)- We will show that IA11+ 91 I7'(DA V DB)'. So suppose, to derive a contradiction, that [A0 + Q1llIlA(p°') V DB(p"'). Then 1A0 + 91*' UA(15") S UB(1F')V 9305") < DA(I3"')- Thus, because IAO+ O1 is a true theory, either 5.5. DISJUNCTION PROPERTY 77 1. ClA(p'*) 3 DB(p"') and IA0 + Ql lp; 4-»pf for all i (by definition of 13"), or 2. DB(fi") < DA(13'*) and IAO + Ql lp; +->pf for all i. In case 1, we have IAo + Ql l--A(;')'*), so IAO+ Ql F A(;5'°), contradicting our assump- tion. Similarly, case 2 contradicts the assumption IA0 + Ql I7'B(;5"). QED In order to prove theorem 5.5.1 we need the following definition. Definition 5.5.4 We define D,, by induction. 0 D0 = T o D,-+l(;5',r) = <>(D,-(ff) A El+r) /\ <>(D,-(5) /\ D+-wr), where 15'is of length i, and all propositional variables in (5, r) are different. The main property of the formulas D,, is expressed by the following lemma. Lemma 5.5.5 IfK is a finite tree-like Kripke model with root k such that k Il-D,,, then we can homomorphically embed (see definition 5.3.5) the full binary tree T,, of height n {and 2"" -1 nodes) into K. Proof. By induction on n. Base case. Trivial: To contains only one point. Induction step. Suppose that k It D,~+l(p',r), i.e. k |l-<>(D,-(;5) A an) A <>(1),-(lr) A a+~7~). Then there are nodes kl,kg such that k j kl,k j kg,kl llD,-(5) /\ El+r and kg Il- D,-(ff) /\ Cl+<r. By the induction hypothesis, we can homomorphically embed a copy of the full binary tree T, of bifurcation depth i into the subtree of K that consists of all points : kl. Analogously, we can homomorphically embed a copy of T, into the subtree of K of points : kg. Because kl |lC1+r and kg |lD+-«r,we may conclude that kl and kg are incomparable and that the two images of T, are disjoint. Therefore, we can combine both homomorphic embeddings into one and subsequently map the root of T,-+l to k. Thus an homomorphic embedding of T,-+l into K is produced. QED Theorem 5.5.1 is now an immediate consequence of the following: Theorem 5.5.6 Let C be a class offinite trees such that Th(C) has the disjunction prop- erty. Then for every n, Th(C) U {D,._}is consistent. Therefore every binary tree (thus every tree) can be homomorphically embedded in some member of C. Proof. Let P = Th(C). Note that P Q L. We prove by induction on n that P U {D,,} is consistent. Base case. Trivial. Induction step. Suppose as induction hypothesis that for 13'consisting ofi different propositional variables, P U {D,-(13 is consistent. In order to derive a contradiction, suppose that P l- -.D,-+l(p',r), that is P l-t:1(D+r -§~D,-(5)) v E](D+-Ir -+~D,-(5)). Then by the disjunction property, either 78 CHAPTER 5. ON THE PROVABILITY LOGIC OF BOUNDED ARITHMETIC 1. P l' D+r -->-1D,-([7) or 2. P |Cl+-1r ->-1D,-([7). We show that 1 cannot hold. By the induction hypothesis, P I7'wD,~(f)').Since r does not appear in D,-(13),we can take r = T. But then P lCl+r, so P l7'D+r ->wD,-(5). By an analogous proof, we can show that 2 cannot hold, which gives the desired contradiction. QED Note that in the proof of the fact that Th(C)U{D,,} is consistent we have only used the fact that Th(C) is a consistent modal theory extending L and satisfying the disjunction property. The same proof can therefore be applied to PLQ, yielding: Proposition 5.5.7 PLO U {D,,} is consistent. Remark 5.5.8 For a strengthening of proposition 5.5.7 due to Berarducci, we refer the reader to [BV 93]. We are now able to strengthen theorem 5.5.1 as follows: Theorem 5.5.9 If there exists a binary tree H which cannot be homomorphically embed- ded in any member of C, then Th(C) Q PLQ. Proof. Under our assumption there is some n such that the full binary tree of height n cannot be embedded in any member of C. Hence Th(C) U {D,,} is inconsistent. On the other hand PL!) U {D,,} is consistent. QED Part III Metamathematics for Peano Arithmetic

Chapter 6 Feasible interpretability Sometimes we see a cloud that's dragonish: A vapour sometime like a bear or lion, A tower'd citadel, a pendent rock, A forked mountain, or blue promontory With trees upon't, that nod unto the world And mock our eyes with air... Shakespeare, Anthony and Cleopatra Abstract. In PA, or even in IA0+EXP, wecan define the concept of feasible interpretability. Informally stated, U feasibly interprets V (notation U l>f V) iff: for some interpretation, U proves the interpretations of all axioms of V by proofs of length polynomial in the length of those axioms. Here both U and V are E';-axiomatized theories. Many interpretations encountered in everyday mathematics (e.g. the in- terpretation of ZF + V = L into ZF) are feasible. However, by fixed point constructions we can find theories that are interpretable in PA in the usual sense but not by a feasible interpretation. By making polynomial analogs of the usual proofs, we show that the bimodal interpretability logic ILM is sound for feasible interpretability over the base theory PA. Here, A D B is translated as PA + A' D; PA + B', where * is the translation. Moreover, we can prove in PA a polynomial version of Orey's theorem for feasible inter- pretability. This paves the way for a polynomial adaptation of Berarducci's proof of arithmetical completeness of ILM with respect to PA. Thus, we show that ILM is arithmetically sound and complete with respect to feasible interpretability over PA. 6.1 Introduction In this chapter, we investigate a novel concept of interpretability -we call it feasible inter- pretability -in which the complexity of proofs associated to the interpretation is bounded. 81 82 CHAPTER 6. FEASIBLE INTERPRETABILI TY The concept was invented by Albert Visser, who called it effective interpretability in his paper [Vi b]. In order to define this concept, we slightly change the usual definition of interpretability (see section 2.4). First we give a half-formal definition of U l>f V (pronounced as "U feasibly interprets V": U t>f V <-+3K3P("Kis an interpretation and Pis a polynomial" /\ V0«(av(0«)-* 3P("|P| S P(|a|)" /\ Prfu(p, aK)))) (6-1) If we want to formalize this concept, we need an evaluation function for coded poly- nomials and we need to be able to prove that the exp of this function is total. We remind the reader that e:i:p(thevalues of polynomials in |:c|) corresponds to the values of #-terms in 2:, where :c#y = e1:p(|a:|- |y|) as in definition 2.3.2. Thus, since there is an evaluation function for formalized terms containing # that is provably total in IAO+ EXP, we see that the formalization of feasible interpretability can be carried out in IA0 + EXP. We will not carry out the details, and for ease of reading we will keep using the half-formal definition (6.1). However, it is clear that the formula U >; V is 22. As we know that, for reasonable theories U extending PA, {A | U l> U + A} is a H3-complete predicate, it would be interesting to find out whether {A | U I>f U + A} is Z3-complete. Chapter 7 provides a positive answer to this question. In [Vi b], Visser gave proof sketches to show that ILM is arithmetically sound with respect to feasible interpretability over PA. Moreover, he gave an Orey-Hajek like char- acterization for feasible interpretability over PA', where PA' is defined as follows: C is an axiom of PA' iff C is the conjunction of the first n axioms of PA for some n. He then surmised that, using this characterization, Berarducci's arguments from [Ber 90] could be adapted to show that ILM is the modal interpretability logic for feasible inter- pretability over PA'. In this chapter, we show that ILM is indeed arithmetically sound and complete with respect to feasible interpretability over PA itself. The rest of the chapter is organized as follows. In section 6.2, we show that some well- known interpretations from the contexts of set theory and bounded arithmetic are feasible. For the subsequent sections, the horizon is narrowed down to Peano Arithmetic. Thus we prove in section 6.3 and section 6.5 that ILM is exactly the modal interpretability logic for feasible interpretability over PA. Section 6.4, meanwhile, gives two counterexamples to show that, for reasonable theories U extending PA, feasible interpretability over U is a definitely stricter concept than normal interpretability. 6.2 Feasible interpretations in various settings For an intuitive introduction to feasible interpretability, it is useful to define feasible interpretability also for settings other than arithmetic. The informal definition is as follows. U l>fV if and only if there is an interpretation K of V into U which is feasible, i.e. for which there is a polynomial P such that for all axioms cpof V, there is a proof of length 3 P(|cp|) in U of cpK 6.2. FEASIBLE INTERPRETATIONS IN VARIOUSSETTINGS 83 Here |cp| denotes the length of cp. In this section, we look at some well-known inter- pretations from different settings and show that they are feasible. As a first remark, it is clear that every interpretation of a finitely axiomatized theory into some other theory is feasible: a constant polynomial, namely the maximum of the lengths of the proofs of the interpreted axioms, suffices. We will prove an easy lemma which can be used to show that many well-known interpretations are feasible. First we state some conditions on the length of formulas and proofs. Remark 6.2.1 Of course the definitions of |cp| and of the lengths of proofs depend on the setting. For example, it is not always convenient to define |cp| as "the length of the binary expression of the Godel number of p". In all examples in the rest of the chapter, the length measure is polynomially related to the length of the binary expression of the Godel number. In general, we have to keep in mind that a few conditions on the definition of the lengths of formulas and proofs are necessary to make lemma 6.2.2 applicable. The length of formulas should be defined in such a way that the following conditions hold: 1lnwl Z|10|+1, 3- |1b°X|Z|w|+|X|+11'0I' °€ {/\,V»-h*-*}. 3. |Q:cw| 2 [112]+ 1 for Q E {\7',El}, and 4. for all formulas go, |cp| 2 2. The last of these conditions is not necessary, but it just simplifies the computations by allowing us to work with polynomials P(n) of the form n'''only. Moreover, we suppose that the proof system and the corresponding length of a proof is defined in such a way that applications of /\-rulesand Modus Ponens do not make the proofs explode to gargantuan proportions; e.g. we suppose that we do not use a tableau system or a sequent calculus. A sufficient condition is the following. There is a constant c such that the following conditions hold: 1. if |A| 3 |B|, and we have a proof of length lA of the formula A, and a proof of length lA_.Bof A ->B, then there is a proof of length 31,, + lA_.3 + c- |B| of the formula B; and 2. if |A|,|B| 3 |C|, and we have a proof of length lA of A, a proof of length lg of B and a proof of length lAA3_.C of A /\ B ->C', then we have a proof of length 3 lA + l3 + lA,\3_.c + c - |C| of the formula C. Lemma 6.2.2 Let L be a language and U a theory satisfying the conditions in Re- mark 6.2.1. Let F be a function from L into LU such that there is a polynomialP such that for all cpE L, |F(cp)| g (This is always the case when L is finite.) Moreover, suppose that the following four conditions hold: 1. U lfflffi F(cp) for all atomic gpE L; 2. U lfli' Fw) -»Fwy) for all w e L; 84 CHAPTER 6. FEASIBLE INTERPRETABILITY 3. U l'°-"'MFe») A F(x) -»Fw o x) for an 1.0.xe L and o e {A. v. a, H}; 4. U ljflw) ->F(Q:m,b)for allzbE L andQ E {V,3}. Then there is a polynomial R such that for all go6 L, U lF (cp) by a proof of length 5 R(|<p|)- Proof. We do not need to find the smallest possible polynomial bound R, which makes the proof quite simple. Take a constant d 2 2 such that 1. for all n 2 2, P(n) 3 nd and 2. for all cp E L, c- |F(<p)| 3 |<,o|",where c is as in Remark 6.2.1 in the condition on the length of proofs. Define the polynomial R(n) := n'''". We will prove by induction on the construction of gothat for all cpE L, U |F(gp)by a proof of length 3 Basic step By the assumption we know that for atomic formulas cp,U lF(<p)by a proof of length 3 But by definitionof d, P(|cp|) 3 |<p|d3 |cp|"+2. w-step Suppose as induction hypothesis that U lF(w) by a proof of length 3 |zD|"'+'.By assumption, U lF(¢) ->F(-aw) by a proof of length 3 P(|-=1/2|)3 |fi1b|°'(where the last inequality holds because of clause 1 of the definition of d). Therefore by the first clause in the condition on the length of proofs in Remark 6.2.1, we have U lF(~zp) by a proof of length 3 |1,D|"+2+ I-it/Jld+ c- |F(-=w)| 3 |¢|d+2 + I-vzbld+ I-wzb|d(where the last inequality holds by clause 2 of the definition of d). Since we assume that |fi2/2|2 |w| + 1, we have |¢|"'+2 + |-nwld+ |-z,b|d 3 |<zp|"+' by an easy computation using the binomial theorem and the fact that d 2 2. Connective step Let 0 E {/\,V,-+,<->}.Suppose as induction hypothesis that U l- F(1,b)by a proof of length 3 |1p|"+', and U lF(x) by a proof of length 3 |x|"+2. By assumption, U lF(1,D)/\F(x) ->F(zl)ox) by a proof of length 3 P(|wox|) 3 |1,Dox|d. The second clause in the condition on the length of proofs in Remark 6.2.1 now implies that U lF(1,l2o X) by a proof of length 3 |1p|"'+2+ IxI"+2 + Ill)0 xld + c- |F(2p o X)| g |¢»|°'+2+ |X|°'+"'+ |zpo X|d + |1j;o X|" (where the last inequality holds by clause 2 in the definition of d). Since we assume that |z,[2o X| 2 lwl+ Ixl + 1, we can again use the binomial theorem to show that I'!/J|d+2+ Ixld" + |2,bo X|" + |w o )(|'' 3 |w o X|"+2, as desired. Quantifier step The quantifier steps are analogous to the n-step, so we leave them to the reader. QED Remark 6.2.3 For applications of lemma 6.2.2, we usually take F(¢) to be a schema involving '(Dand W', where K is an interpretation. When we want to prove that some interpretation K of V into U is feasible, we can often use lemma 6.2.2 in the following way. Suppose all but a finite number of axioms of V have the form <I>(w),where <I>is a formula scheme. The feature we need in order to apply lemma 6.2.2 is the fact that both |<I>(z,b)|and |1,l2K|are polynomial in 6.2. FEASIBLE INTERPRETATIONS IN VARIOUSSETTINGS 85 An adapted version of lemma 6.2.2 works in case attention is restricted to A0-formulas. An example of a function F for which the A0-version of lemma 6.2.2 could be applied is the scheme F (w) := 1,04-»W' for 1,06 A0, where K is a fixed interpretation such that |1,bK| is polynomial in |z,b|. As a first example, in which we do not yet need lemma 6.2.2, we will show that the usual interpretation of IAO+ 01 into IAO by a cut is feasible. Theorem 6.2.4 IAO I>f IAO+ 01 by a cut. Proof. Let J be a cut constructed by Solovay'smethods such that IA0 proves that J is a cut closed under +,- , and wl (see lemma 2.6.10). Define cpJ to be the formula cpwith all quantifiers restricted to J. It is well-known that J is an interpretation of IAo + 01 into IA0; so to show that it is a feasible interpretation, it suffices to find a polynomial P such that for all A0-formulas cp,the following holds: by proofs of length 3 P(|"cp'|): mo Mo) A V~'v(<p($)-»«p<s:c>>~» V:c<p(2=)l"- First, it is easy to see that there is a polynomial P1 such that for all A0-formulas up, IA0 ll) J(a) ->(<p(a)4-»cp(a)") and IA0 [Ii Vamp-->(\7':z:cp)".Second, there is a polynomial P2 such that for all A0-formulas go,the following holds: IA. va me) Avzsa (<p($)-» </>(51=))~ vxsa ¢<z>1. In fact one uses only the induction axiom for V1:g a cp(:1:),the fact that VaV:z:(S:c3 a. -> :1:3 a), and some predicate logic. Combining P1 with P2, we then find a polynomial P3 such that for all A0-formulas go,the following holds: IA. (Va[<.0(0)AV-'vSa(<P($)~ <P(5$))-»vxsa <p(1=)l)'~ Now it is easy to find a polynomial P from P3 such that for all A0-formulas cp, the following holds: IA. mo) AV$(<P($)~ <P(5-Tl)~ vw<z>1'. We use only the fact that Va(a _<_a.) and some predicate logic. Thus, J is a feasible interpretation of IAO+ 01 into IA0. QED Next, we will prove that the usual interpretation of ZF + V = L into ZF is feasi- ble. Because Z F consists of a finite list of axioms plus the schemata of separation and replacement, we can restrict our attention to feasibly proving these schemata relativized to the universe L of constructible sets. We will first prove that the schema of separation relativized to L follows feasibly from the reflection theorem for L, and then give a feasible proof of the reflection theorem itself. Finally we give a proof of the schema of replacement relativized to L. For the reflection schema, we will try to follow the elegant proof in terms of closed unbounded collections, which unfortunately becomes much less elegant when forced into the strait-jacket of the calculation of lengths. We will not stray far from the straightforward presentation given in [Ku 80], where all details about the constructible universe that we omit here can be found. The length |c,0|of a formula upof ZF is defined as the number of appearances of symbols in cp;without loss of generality, we can take the length of all variables to be 1. Likewise, we define the length of a proof in ZF to be the total number of symbols appearing in the proof. In the following lemmas, quantifiers in greek letters range over the ordinals, while those in roman letters range over all sets. The next lemma corresponds to lemma IV.2.5 of [Ku 80]. 86 CHAPTER 6. FEASIBLE INTERPRETABILITY Lemma 6.2.5 ZF proves the followingbyproofs of lengthpolynomial in V2,i'2'€L {$62 I cpL(a:,2,i'2')}€L --> V2,i7EL 3y€L [:0 E y 4-»:1:E 2/\ cpL(:z:,2,'D')] Proof. Immediate; note that (:1:E y)L := 2: E y, so the succedent is feasibly equivalent to the comprehension schema for cp,relativized to L. QED The following lemma corresponds to a part of lemma VI.2.1 of [Ku 80]. Lemma 6.2.6 ZF proves the followingbyproofs of lengthpolynomial in (Va 3[3>aV2,.7:,i')'ELg [<,oL(:c,2,27) +->c,oLf'(:2:,2,i7)]) -> V2,17EL{:I:E2 | cpL(a:,2,i7)} E L Proof. It is easy to see that the usual proof in ZF is feasible: suppose 1. Va 3fi>a V2,:z:,z7E Lg [goL(a:, 2,17) 4-» 901'/-"(:I:,2,z'2')]and 2. 2,17 6 L From 2 it follows that there is an a such that 2,276 L0,. Now let H > a be such that V1:6 L5 [<,oL(:I:,2,i7)+->cpL5($,2,i7)]. Then, using the fact that L is transitive and that :1:E 2 is absolute for L3, L, we find that {$62 | cpL(:z:,2,17)}= {:r€Lg | (:1:E 2 A <p(:z:,2,17))L"}E D6f(Lg) = L544, so {$62 | goL(:c,2,27)}E L. QED From lemma 6.2.5 and lemma 6.2.6, we conclude that in order to feasibly prove the comprehension schema, we only need polynomial length proofs of Va 3B>a V2, :3,176L3 [<,oL(:I:,2, 17)4-»4,oL"(1:,2, For a proof of this reflection theorem, we need a few more definitions. Definition 6.2.7 A collection C of ordinals is o unbounded iff Va 3B>a (C E C); 0 closediffVa(a7E@/\a§C->sup aEC); 0 closed unbounded (c.u.b.) iff C is both closed and unbounded. Lemma 6.2.8 ZF l- "IfC and D are c.u.b., then C D D is c.u.b. as well" Proof. An easy application of lemma 11.6.8 of [Ku 80]. QED Definition 6.2.9 A collection C of ordinals is a closed unbounded ap-mirror iff 1. C is closed unbounded, and 2. La reflects cpfor all ordinals a E C, i.e. Va (a E C ->V176 La [c,oL(i7) <->cpL°('U)]) 6.2. FEASIBLE INTERPRETATIONS IN VARIOUS SETTINGS 87 Suppose cp is a formula and D is a first-order definable collection of ordinals. Using definition 6.2.7, we are able to construct new first-order formulas CU B1), CUBp,,,, and REFS, with the following intended meanings: 1. CUBD := "'Dis closed unbounded" 2. CUBp_,,, :="D is a closed unbounded cp-mirror" 3. REF,,,:= " there is some collection of ordinals that is a closed unbounded cp-mirror" The next lemma roughly corresponds to theorem IV.7.5 of [Ku 80]. Lemma 6.2.10 (Reflection theorem) ZF proves thefollowingbyproofsof lengthpoly- nomialin Va 3B>a Vz,:z:,17€Lg [<pL(:r, 2,27) +->cpL"(:r, z, 27)] Proof. First we note that ZF proves (a < B ->LO,Q L5), "if 7 is a limit ordinal, then L», = Uo.<-,La" and L = UQEORLa. We will prove the reflection theorem by induction on the construction of cp. A straight- forward application of lemma 6.2.2 implies that for the reflection theorem to have a proof of length polynomial in |<,o|,it is sufficient to find a polynomial bounding the lengths of the induction steps. Thus, we need to find a polynomial P such that by proofs of length S P(|<p|)» resp. S P(|'=d»|)a reSpS P(|I/2 0 Xi)» reSpS P(|Q:m/2|), ZF proves the following: 1. for atomic cp: Va Elfi>a Vz,1:€L5 [c,0L(:c,2) 4-DtpL5(:c, 2)] /\ C'UB0R,,, 2. the -w-step: Va 35 > a we Lg [wL(z7)H zbL5(z'2')]A REF, H Va 35 > a we Lg [~wL(z7) H -w15Ll'(17)]A REF_,) 3. the connective step, where o E {A, V, ->,+->}: Va 35 > a we L5 [wL(27)H 51-23(27)]A REF),/\ Va 35 > a we L5 [XL(1Zi)H XL5(u'i)]A REF, H Va 35> a We I45 ['(/JL0 XL('[)',215)H we o xLv(z7,13)] A REFW 4. the quantifier step, where Q E {El,V} : Va 35>a Vz,17ELg[1bL(Z,'l7)H zpLa(z,27)]A REF, H Va Elfi>a V276Lg [Q2 6 L tpL(z,27) <-+Q2 6 L3 wLB(z,17)] /\REFQ,,), Finding polynomials bounding the lengths of the proofs of 1, 2 and 3 is very easy: we can use the feasibly provable fact that atomic formulas are absolute for any La, L, some propositional reasoning independent on the specific 1/1,x, and an application of lemma 6.2.8 for step 3. We will show how the proofs of the 3-case in step 4 can be bounded by a polynomial; we can then find a bound for the V-stepby rewriting V as -13-: and using the bounds for the -v-stepand the 3-step. Define D := {5 IV17€Ln [32 e L wL(z.«7) 3 azem wL<z,z7>1}. 88 CHAPTER 6. FEASIBLE INTERPRETABILITY It is easy to see that Z F proves the following by proofs of length polynomial in |Elz1b|: Va 30> a Vz,176 L1. [1/2L(z,17)H 1bLB(z,17)]ACUBCW /\ CUB1; -1 Va ElB>a V176 L1; [Elz6L 1,bL(z, 17) 4-» 326 L131bL(z, 17)]A C'UBc¢r1-D,32¢, In fact, we only use lemma 6.2.8 and the fact that VB(L13C_ZL). Thus we need to find a polynomial P such that Z F lCU B1; by a proof of length 3 P(|Elz1b|). Immediately from the definition, it is clear that ZF l- "D is closed'' by a proof of length polynomial in |3z1,b|. Thus, it suffices to show by a proof of length polynomial in |3z1,b|that Z F proves that 'Dis unbounded, that is: Va 3B>a V176L11[3z6L 1l2L(z,17)-> 3z6L11 1pL(z,17)], Va 3B>oz V176 L13Elz6L13 [Elz6L 1,bL(z,17) ->1,bL(z,'l7)]. We will reason in ZF, taking care that all steps are applications of general ZF -theorem schemas that do not depend on the specific formula 1b. Take any ordinal a. We know using only predicate logic that V17eL,, 3z€L [3z€L 1bL(z,17)-1 1bL(z,17)]; therefore, VUELC,am, (013= fl{B>a|Elz6L13[3z e L 1bL(z,17)-.1bL(z,17)]}). by the unrelativized replacement and union axioms, there is a B1such that B1= sup{ag|176 La}. Continuing in this way, we can define by recursion a sequence B,,,for p 6 cu, where for all p 6 w, VUeLfip 3z6Lgp+, [ElzeL 1,bL(z,17)-1 1bL(z,17)] (6.2) Define B := sup{B, | p 6 cu}. Because a = Bo < B1 < B2 < ..., we infer that B is a limit ordinal > a. Now using (6.2) and the fact that L1;= LJK1,L.,, we find that v17eL,, 3-zeL,, [ElzeL 1bL(z,17)-1 1bL(z,17)], as desired. QED Lemma 6.2.11 For all «,0,ZF feasibly proves the comprehension schema for 1,0,relativized to L; i.e. by proofs of length polynomial in |Lp|,ZF proves the following: Vz,176L 3y6LV:2:6L [:136 y 4->:1:6 z /\ cpL(:z:,z,17)] Proof. Combine lemmas 6.2.5, 6.2.6 and 6.2.10. QED Lemma 6.2.12 For all go,ZF feasibly proves the replacement schema for 1,0,relativized to L; i.e. by proofs of length polynomial in |cp|, ZF proves the following: Va,176L [V:1:6a3!y6L<,oL(:z:,y,17) -+ 3c€LVz/EL (y 6 c H 3-1:€as0"(r,y,17))l 6.2. FEASIBLE INTERPRETATIONS IN VARIOUS SETTINGS 89 Proof. We already have feasible proofs of the relativized comprehension schema for the formula 3/ E b A 32:E a cp(:c,y, 27). So we can (feasibly) prove that it suffices to show the following by proofs of length polynomial in |go|: ZF lVa.,'z7EL [Vscea 3!yeL <pL(:z:,y,17)_. ElbeL (V:1:Ea 3yEb<,0L(:c,y,17))] The last proof works, as in lemma 6.2.10, by general theorem schemas of ZF that do not depend on the specific <,0.Work in ZF and suppose a.,17E L and V:1:Ea3!y€ L<,oL(:r:,y, 27). Now Vzea 3% (B: = flla I 3.1./EL...<pL(=v,y.I7)}); then by replacement and the union axiom we find 6 such that fi = U{fi, | :1:E a}, and we let b be I43. Then \7'a:Ea 33/Eb cpL(:z:,y,1'2'). QED Theorem 6.2.13 ZF [>fZF+V = L Proof. We take the usual interpretation L of ZF + V = L into ZF. Because ZF is axiomatized by a finite list of axioms plus the schemata of comprehension and replacement, the lemmas 6.2.11 and 6.2.12 immediately imply that L is a feasible interpretation. QED Remark 6.2.14 Looking carefully at the proofs of the lemmas leading up to theo- rem 6.2.13 and using an analog of lemma 6.2.2 for polynomial time instead of polynomial length, one can observe that theorem 6.2.13 can be strengthened: the proofs in ZF of the interpreted ZF + V = L-axiomscpl'are not only of length polynomial in |cp|. There is even a deterministic polynomial time Turing machine M such that if the input of M is the code of an axiom (,0of ZF + V = L, then M outputs the code of a ZF-proof of cpl'. It is a known result that PRA lCon(ZF) ->Con(ZF+V = L) (see [Sm 77, Corollary 5.2.4]). One of the referees suggested that by theorem 6.2.13 this could perhaps be strengthened to IAO+ exp |C0n(ZF) ->Con(ZF + V = L). The observation about the polynomial time computability of the proofs of the interpreted axioms, however, even leads to the conjecture that IA0 + Q1 l- \7'a(azp+v=L(a) ->ElpPrfzp(p, aL)), where (1zF+\/_-_L is a Al,'-formula axiomatizing ZF + V = L (cf. [Bu 86, Theorem 5.6]). Then, by a standard argument (involving Parikh's Theorem), we would have IAO+ Q1 lC'on(ZF) --+ Con(ZF + V = L). Contrary to our expectations, the usual interpretation of ZF + V 95L into ZF(M) (by forcing with generic extensions), although much more complex, is still feasible. We checked this following the lines of the proof in [Ku 80]. Our proof relies so heavily on the many details of Kunen's proof, that it would be incomprehensible to readers not conversant with that book. Therefore, we do not give it here. In the literature there are also proofsof ZF l>ZF+V 74L and ZF+AC I>ZF+AC+ -CH which entirely avoid the use of the transitive countable collection M. A sketch of such a proof can be found in [Co 66, Section IV.11], and a completely different full proof 90 CHAPTER 6. FEASIBLE INTERPRETABILITY in [VH 72, Ch. V, VI]. It appears that these proofs can also be analyzed to show that the interpretations in question are feasible. Other well-known interpretations, such as the one of PA into ZF, and those of IAO+ 0,, into Q [HP 93, Section V.5] are also feasible, as the reader may check for her/himself. All in all it seems that the only examples of theories U and V such that U D V but not U Df V are contrived theories obtained by fixed-point constructions like the ones in section 6.4. It would be nice to find a more natural counterexample. It would also be interesting to investigate severely restricted kinds of interpretability which do distinguish between interpretations used in everyday mathematics. For exam- ple, one could restrict the complexity of formulas allowed to occur in the proofs of the interpreted axioms. Sam Buss suggested the following restricted definition of feasible interpretability to us: U Dfm V <->3K3M("Kis an interpretation and Mis a deterministic polynomial time Turing Machine" /\ Va(av(a) -->PrfU(M(a),aK))). (6.3) This definition is more in line with the conventional use of the word "feasible" in the context of polynomial time computability. The clause PrfU(M(a),aK) in (6.3) is a P-like formula, while the clause 3p("|p| 3 P(|a|)" /\ PrfU(p,aK)) in the definition of feasible interpretability used in this chapter is an NP-like formula. However, all interpre- tations considered in this section can also be shown to be feasible in Buss' sense: as in remark 6.2.14, we only need an easy analog of lemma 6.2.2. 6.3 Soundness of ILM for feasible interpretability over PA In this section, we restrict our attention to feasible interpretability over PA. We show that the modal interpretability logic ILM is PA-sound even if the intended meaning of A D B is "PA + A feasibly interprets PA + B". The definition of ILM can be found in section 2.5. Definition 6.3.1 A feasibility interpretation is a map * which assigns to every proposi- tional variable p a sentence p' of the language of PA, and which is extended to all modal formulas as follows: 1. (A r>B)*=PA+A* 1>,PA+B* 2. (CIA)* = ProvpA(A") 3. * distributes over the boolean connectives. Here D; abbreviates the formalization of feasible interpretability. We will prove that ILM is arithmetically sound for feasible interpretability, i.e. that for all modal formulas A, if ILM lA, then for all feasibility interpretions *, PA lA'. Thus, we have to check that the axioms J1 to J5 are valid in PA when A D B is read as PA + A D; PA + B. Whenever possible, we will prove generalizations of these axioms to theories U, V 2 PA. Also we prove a generalization of the property M, where an infinite set of E'1'-sentencescan be added on both sides instead of one Cl-sentence only. 6.3. SOUNDNESS OF ILM 91 Lemma 6.3.2 PA proves all feasibility translations of J1 to J5. Proof. The proofs for J1 through J4 can be found almost verbatim in [Vi b]. We reason in PA. J1 Suppose for some theory V and some p that Prf1/(p, 'A"). Then by the identity interpretation and the polynomial bound P(n) = n + Ip|, V D; V + A. So in particular, if EJp,1(A->B), then PA + A D; PA + A + B, and surely PA + A D; PA + B. J2 Suppose c U D; V by interpretation K1 and polynomial P1, and a V D; W by the interpretation K11and polynomial P2. As in the usual case, U D W by the interpretation K1 0 K2. We need to show that there is a polynomial bound for the proofs of the translated axioms. So let b code an axiom of W, and p a proof in V of b"? with |p| 3 P2(|b|). If we take the K1-translations of all formulas appearing in the proof coded by p, and add some intermediate steps, we can construct a U-proof of (bK')K1from K1-translations of axioms of V as assumptions; the number of steps in this proof will be 3 k - |p|, where k is a constant depending on the translation K1. Now we only have to add proofs of the translated V-axioms; the axioms themselves have codes of length 3 |p|, so their K1-translations have proofs with codes of length _<_P1(lPl) S P1(P2(lbl))- All in all, even in the worst case where the U-proof of (bK')K1 consists wholly of assumptions, there is a q with |q| 3 k-P2(|b|)-P1(P2(|b|)) such that PrfU(q, (bK')K'). In particular,if PA+A D;PA+B and PA+B D;PA+C, then PA+A D;PA+C. J3 Suppose a U + A D; V by interpretation K1 and polynomial P1, and 0 U + B D; V by interpretation K2 and polynomial P2. As in the usual case, we have U+AVB DV by the disjunctive interpretation M which equals K1 in case A holds and equals K2 in case -A holds. To find a polynomial bound, we observe that for all C, lA ->(CM 4-»CK') and l- -:A ->(CM 4-»CK') by proofs of length 3 P(|C|), where the polynomial P depends on K1 and K2. Now suppose that c codes an axiom of V, that p1 codes a U + A-proof of ck' with |p1| 3 P1(|c|), and that P2 codes a U + B-proofof CK'with |p2| 3 P2(|c|). But then there is a constant k such that 0 we can find p'1such that PrfU(p'1,"A ->"cM)with 3 k-(|c|+P(|c|)+P1(|c|)); and 0 we can find p'2such that PrfU(p;,,'fiA/\B ->"cM)with 3 k-(|c| +P(|c|)+ P2(|C|))- Combining p', and pi',and their respective polynomial bounds, we find p and P' such that Prfu(p,"A V B ->"cM) with |p| 3 P'(|c|). Thus U + A V B D; V. In particular, we have: if PA + A D; PA + C and PA + B D; PA + C, then PA+AvB D;PA+C. 92 CHAPTER 6. FEASIBLE INTERPRETABILITY J4 Because (PA + A D; PA + B) ->(PA + A D PA + B), we have by the soundness of J4 for normal interpretability immediately (PA + A D; PA + B) ->(<>A-+OB). J5 In an easier variation of lemma 6.5.11, we use a claim proved in [Vi 91b], which is stated in our chapter as lemma 6.5.10. Suppose B is a 2'1'-formulaaxiomatizing a subset U of a 2'1'-language L. We will prove that IAO + 91 + <>3T I>f U i.e. IAo+Q1+<>UTDf By lemma 6.5.10, there is a polynomial P1 such that PA l' DIAo+Q1+Con(fi)0071(5) -* P ElKVaE Sent(L)[IAo + 01+ Con(fi) lg ("Ugo ->"aK)]. Of course we also know that PA l- |3IA0+n1+C,,,,(5)Con(B),so PA + 3KVa e Sent(L)[IAo + 91+ C'on(fi) y"'-""fl("ow -."a'')]. On the other hand, by provable 2'1'-completenessthere exists a polynomial P2 such that PA I- \7'a.(B(a.)_. [mo + 91 + Con(fi) ii) (ra,,a,')]). Combining the last two results, we have a polynomial P3 such that PA lElK\7'a(fi(a.)-.[mo + 01 + C'on(B) ii) (aK)]), so PA I- (IAo + 01 + <>3T) >, U. In particular, we have for any sentence A: PA l- (IAO + 01+ <>pAA) I>f PA + A, thus PA l- (PA + <>pAA) l>f PA + A. QED We want to prove that Montagna's principle M holds for feasible interpretability over PA in its general version, where we can add an infinite set of Z?-sentences on both sides. In order to ensure that the usual arguments can indeed be polynomialized, we do not formulate the proof in the usual model-theoretic way, and we give many details that are not given in most proofs of Montagna's property for normal interpretability over PA. The example we give in theorem 6.4.1 of a set S of formulas such that PA lPA l> PA + S but w bé PA l>f PA + 8 also relies heavily on these details. Suppose U Z_>PA, V 2 PA. Now suppose U t>fV by the interpretation K (preserving =) with domain 6, and polynomial P. We want to find a polynomial Q such that for every )3?-sentence 0 there is a U + 0-proof p of UK with |"p"| 3 Q(|"a'|). First, we need some definitions and lemmas. Fix U, V,K, P as given above. 6.3. SOUNDNESS OF ILM 93 Definition 6.3.3 Define pism(s) for "s is a partial isomorphisrn" and the function G(j, y) as follows: pism(s) :seq(s) A (3)0 = OK/\ Vi < lh(s) -1((S)i+1 = SK(s),-) G'(j,y) := 3s(pism(s) /\ lh(s) = j + 1 /\ (s), = y) Lemma 6.3.4 U lVj3!s(pism(s) /\lh(s) = j + 1) and thus U l- \7'jEl!yG(j,y). Therefore, there is a function g corresponding to G'. Proof. By induction. QED Lemma 6.3.5 U proves that g is injective, and U lVjVy(G(j,y) -+6(y)). Proof. By induction. QED Lemma 6.3.6 U proves that g preserves O,S, +, -, and g. Proof. We will give some of the preservation proofs. It follows immediately from the definition of pism(s) that U lg(O) = OK and U l- \7':r(g(S:1:)= SK(g(:c))). We now prove by induction that g preserves +. (The proof for is analogous, and in the case of 3 we can use the fact that 3 is expressible via +.) We have U lg(:c + O) = g(:r:) = g($) +K OK = g(a:) +K g(O) and U lg(:1:+ y) = flfl+"g@)~9&H4w%=d5@+yD=5"@@+yD=5"@@O+"flw%=dfl+" = g(:1:)+K g(Sy), So by induction on y, U P V:z:Vy(g(a:+ y) = g(1:)+K g(y)). Lemma 6.3.7 The range ofg is 'closed downwards', i.e. U l- 'v'3:\'/u(6(v.)Au <K g(:c) -> 3y<flu=Mw» Proof. Before we start the proof proper, we note a useful fact. V includes PA and K is an interpretation of V into U. Thus, as 1. PAl-\7':1:Vu(u<:I:+1->u<:z:Vu=:z:) and 2. U lV:c(g(:z:)+K1K = g(:c +1)), we also have 3. U lV:z:\7'u(6(u)/\ u <K g(:z:+1) ->v. <K g(:c) V n = Now we can start with the proof by induction on :1:that U l- 'v'a:Vu(6(u)A u <K g(;z:) -> 3y<fiu=MwO x = 0 We have U l- -wElu(6(u) A u <K g(O)), so U l- 'v'u(6(u) /\ u <K g(O) ->Ely < O(v. =no» Induction step Work in U and suppose 'v'u(6(u)/\ u <K g(:c) ->3y < :z:(u = g(y))) (induction hypothesis). Moreover, suppose 6(a) /\ u <K g(:1:+ 1). Then, by 3, u <K g(:z:)Vu = So by the induction hypothesis 3y < 1:(u= g(y)) Vu = g(a:), i.e. Ely< :c+1(u = g(y)). QED 94 CHAPTER 6. FEASIBLE INTERPRETABILITY Remark 6.3.8 Let I(:::) be the formula 33/(1:= g(y)). Note that if U I7'Va:6(a:), then U does not prove that I defines a cut. For, suppose that U l- [(0) A \7'a:(I(:z:)->[(13 + Then induction gives U lVz:3y(;r = g(y)), thus, by lemma 6.3.5, U lV:z:6(:r), contradicting our assumption. On the other hand, by the previous lemma we do have U lVa:'v'u(6(u)/\ I(:1:)/\ u <K :1: -> I(u.)). Lemma 6.3.9 For all formulas go6 A0, U proves the following by proofs of length poly- nomial z'n |"cp(:z:1,. . . ,:2:,,)"|: Lp($17'--11:11)'-'((pK)(g(x1)1"' Proof. By induction on the construction of (,0.We will see below that the proofs for the atomic formulae 1/}are obviously of length linear in |'"z/Fl, and that all induction steps follow a given proof scheme in which the particular formulas at hand can be plugged in. So, because every (,0has at most |"cp"| subformulas, there is a polynomial R such that for all go, the U-proof of cp(:z:1,. . . ,a:,,) 4-» (cpK)(g(:c1), . . . ,g(:z:,,)) is of length 3 R(|'cp"|). We will do the atomic step and the V1:3 t-step of the proof, and leave the others to the reader. Atomic step By lemma 6.3.6, we have for all terms t by proofs of length polynomial in lf'tW': U |V1131,. . . ,:c,,(g(t(:c1, . . . ,a:,,)) = (tK)(g(:I:1), . . . ,g(:c,,))). So suppose p is the formula t1(:r1, . . . ,:c,,) = t2(a:1, . . . ,:z:,._)where 1:], . . . ,:z:,, include all variables appearing in t1 and t2. Then, because U proves that g is an injective function, Ult1(:z:1,...,:z:,,) = t2(:z:1,...,:1:,,) +->g(t1(:z:1,...,:c,,)) = g(t2(:r:1,...,a:,,))*6 = vg($n)) *4 ((t1 _ t2)K)(g($1)v vg($n)) Vx 3 t-step Suppose that <,0(x1,. . . ,:z:,,) = Va: 3 t(1:1,. . . ,:1:,,)w(:r,:r1,. . . ,:1:,,), and that U lw(:1:,:1:1,...,a:,,) 4-» (¢K)(g(2:),g(:r1),...,g(:z:,,)) (induction hypothesis). We will use the fact that, because of lemmas 6.3.5, 6.3.6 and 6.3.7, by proofs of length polynomial in |"t"|: Ul-'v'u,:z:1,...,:z:,, (3:z:[:z:3 t(.'C1,...,.'13,,)/\'u H 5('u) A u S" t"(9(-r1). . - . ,9(rn)))- II cc /5 H 1'. Thus, we have the following equivalences by proofs of length polynomial in |'cp(:t:1, . . . ,:c,,)"|: U l- <,o(:r1,...,:c,,) H V1: 3 t(2:1, . . . ,:z:,,)w(:c,:1:1, . . . ,:1:,,) H V17S t($1, - - - ,rn)(11»")(9(-'E),9(r1), - - - ,9(:rn)) (by indhyp-) H A11*SKtK(g($1)>' ' ' ag($n))_"¢K(Uag(I1)v' ' ' H (V1:3 t1[2)K(g(a:1), . . . ,g(:c,,)) (by def. of K) H ('p)K(g(1"1)v°"vg(xn))' 6.4. INTERPRETABILI TY AND FEASIBLE INTERPRETABILI TY 95 Now we can finish the proof of the uniform version of Montagna's property M for feasible interpretability. Theorem 6.3.10 Suppose 0 U satisfies full induction, 0 V extends PA in its language and 0 U D; V by interpretation K (preserving =) and polynomial P. Then there is a polynomial Q such that for every EC,'-sentence0 there is a U + 0-proofp of 0" with |'P'l S Q(|'0'|)- Thus, U + S D; V + 8 where S is a finite or infinite 2','-set of Z?-sentences. Proof. Suppose a E 8 is the E?-sentence Elx<p(x),where cp6 A0. By lemma 6.3.9, there is a polynomial R such that we can prove the following by a proof of length 3 R(|ro'l): U l' 3:c<p(-'6) -* 3:w"(g(=r)) -+ 33/(5(y) /\ <p"(y)) -> (3w(:r))"- Now we have U + S D; V + S by the interpretation K and polynomial Q := P + R. QED All results of this section also hold if we add the function symbol exp to the language of U and V, which we need in theorem 6.4.1. Let g be as defined in lemma 6.3.4. We will only give the result which needs some adaptation. The following preservation lemma corresponds to lemma 6.3.6: Lemma 6.3.11 Suppose exp 6 LU. Then U proves that g preserves 0, S',+, -, 3, and exp. Proof. We already have a preservation proof for by lemma 6.3.6. Preservation of exp then follows in the same way as preservation of + was proved from preservation of S in lemma 6.3.6. QED 6.4 Interpretability does not imply feasible inter- pretability Theorem 6.4.1 There is a set S of Ao(exp)-sentences such that PA lPA D PA + S, butw béPA D; PA+S. Proof. Define by Godel's diagonalization theorem (or rather by the free variable version as formulated by Montague) a Ao(exp)-formula ap(y)such that PA F <p(y)H Vzsy -Prflrc. "<p(z7)")- It is easy to see that if we diagonalize directly, there is a polynomial 0 such that for each n, |n| < |'cp(n)"| 3 0(|n|). Moreover, if cp(1'i)were false, then by definition we would have a proof of the Ao(exp)-sentence cp(n); so t,o(7'i)must be true. But then, since L,0(7-'L)is Ao(exp), we have the following: 96 CHAPTER 6. FEASIBLE INTERPRETABILITY 1. PA proves cp(fi), though 2. because <p(r't)is true, PA does not prove cp(n) by any proof whose Godel number is of length 3 n. Define S := {go(n)| n E w}. Then, by the identity interpretation, w l: PA D PA + 8. Actually, as in [JM 88, section 6], we even have PA I- 'v'yProv('cp(37)"),so PA lPA D PA + 8. Now suppose, in order to derive a contradiction, that anl= PA Df PA + S by inter- pretation K and polynomial P. Thus, for all n, PA lcp(r't)Kbya proof of length 3 P(|"<p(n)"|). We also know by lemma 6.3.9 (with U = V = PA) that there is a polynomial R such that for every n, PA Icp(7'z)H cp(n)Kby a proof of length 3 R(|"cp(n)"|). Now we can construct from R and P a polynomial Q such that for all n, PA lcp(n)by a proof of length 3 Q(|rcp('r'z.)"|). However,there willbe n suchthat n > 2 Q(|'cp(n)"|),and wehavea contra- diction with 2. QED A salient feature of the counterexample above is the trivial identity interpretation by which PA interprets PA + 8. To prove that interpretability does not imply feasible interpretability, it is not essential that the set of formulas added to PA be infinite like 8 above. In theorem 6.4.2 we show a counterexample where one sentence can be normally but not feasibly interpreted over PA. Of course in this case the normal interpretation cannot be the identity. The counterexample also shows that in general we cannot feasibly merge two compatible cases of feasible interpretability; i.e. it is not true that if U D; V, U DfBandU DV+B,thenU DfV+B(takeU=V=PA, B=A(n)orB=E* as below in the proof of theorem 6.4.2). Theorem 6.4.2 There is a sentence A such that w l: PA D PA + A, but cubé PA D; PA + A. In order to prove theorem 6.4.2, we need a well-known result and its proof, given in the- orem 6.4.4 below. Solovay proved that the set {A | PA D PA+A} is 1'13-complete[So 76b]. This result inspired Hajek to prove that, for every n, the set {A | A is H9,+,-conservative over PA} is also H3-complete [Ha 79b]. We have adapted the proof of theorem 6.4.4 from Visser's unpublished rendition of an alternative proof by Lindstrom of Hajek's general result (see [Vi 90b]). First we need a definition. Definition 6.4.3 Define DWB for "there is a proof of the formula B which only uses those axioms of U with Godel number 3 :13."Then define D58 := El:cEJU,,B. Theorem 6.4.4 Suppose U is a theory extending PA in the language of PA, such that for all B, PA lV:z:ElU(ElU,,B~> B) (reflection for U). Then for every H2-predicate P(:1:), there is a formula A such that PAl-OUT->Vx((UDU+ H 6.4. INTERPRETABILITY AND FEASIBLE INTERPRETABILITY 97 Proof. The proof is taken almost verbatim from [Vi 90b]. Let P(:1:) be any H2-predicate, say P(.7c)= V:z:S(z:,y), with S E 2?. Pick R by diago- nalization such that PA IR(:I:,y) 4-»S(a:,y) 5 EJUR(:z:,y). Let Q(:c,y) := DUR(:z:,y) j S(a:, 3;). Now we can prove the following: PA F VrvVy(DuR(x,y) H 3(:r, y) V Dull (6-4) In order to prove (6.4), work inside PA and suppose ClUR(:c,y). Then either R(a:, y) or Q(:::,y) holds. In case that R(2:, 3/)holds we have S(:::,y) by definition. In case that Q(a:, y) holds we have ElUQ(.7:,y) by 2?-completeness, and hence by definition both ClUR(:z:,y) and I:1U-»R(:c,y), thus DU_L. For the other direction, suppose S'(:c,y). Again we have either R(:c,y) or Q(:z:,y). From R(:c, y) we find ElUR(:c,y) by Z3?-completeness. From Q(:z:,y) we immediately derive DUR(:c,y). Finally ElUJ_gives ClUR(a:,y) as well. This finishes the proof of (6.4). Define A by diagonalization such that PA IA(:r:) +->C1,',-1A(2:)j 3yfiR(a:,y). Note that by (6.4) we have PA IOUT -->V:z:['v'yDUR(:r:,y) -->P(:::)] and PA I- 'v':c[P(:c) -+ \7'yDUR(:z:,y). For this A, we can prove PA l' OUT ->V.'II((UD U + +--> (6.5) First note that refiection for U allows us to apply the Orey-Hajektheorem in order to conclude that PA 1-V:t:(VyCJU<>U,yA(a:)+->(U l> U + We start the proof proper of (6.5). Work in PA and suppose OUT. ->-side Pick any :1:and suppose U [> U + A(:c). Then by the Orey-Hajek theorem VylIlU<>U,yA(a:). We will prove 'v'yEJUR(:1:,y). Pick any y. We have DU[Q(.'z:,y) -+ -=R(:c,y)]; therefore by definition of A, '3u[Q(-73.3/) -* "A($) V DU,y-'A($)] and hence by reflection DU[Q($»3/)" 'A($)]- But then there is a 1)such that '3U,vlQ(Il«',1/) -*"A($)]a so by 29completeness C'UDU.u[Q($,3/) -*"A($)l- Also by 2?-completeness, there is a w such that '3u[Q(1=.y) -' DU,wQ($ay)]' Combining the previous two facts, we find a u such that C'u[Q(w,y) -> Du,..-'A(=B)l, 98 CHAPTER 6. FEASIBLE INTERPRETABILITY and thus, by the assumption, CJUaQ(:z:,y). It follows that I--lU[DUR($vy) _' R($v y)l: hence by L6b's theorem DUR(:z:,y). We may conclude VyClUR(:z:,y),thus, because we have OUT, we conclude P(;z:). 4--side Pick an :1:and suppose Then VyClUR(:c,y) and thus VyClU(\'/z< yR(:z:, It follows by definition of A that V3/l3U(DU,y"A(17)-* A(I))- On the other hand, we have VyUu(Uu.y"'A(r) -> n/1(3)) by reflection, hence \'/yElU(<>U,yA(x)).But then by the Orey-Hajek theorem U D U + A(:::). This finishes the proof of (6.5), and thus of the theorem. QED Proof of theorem 6.4.2. Let P(1:) be some I13-complete formula, say P(:z:) = VyS(:z:,y), with S E 2?. Define the formulas R and A by diagonalization such that PA F R(r,y) H 5(r,y) : DpAR(-my) and PA lA(:c) +->Cl};,pA(:r) j Ely-wR(a:,y), where E1'is as defined in definition 6.4.3. Carrying out the proof of theorem 6.4.4 in True Arithmetic, and taking the theory U mentioned there to be PA, we find the following result: if PA is consistent (as we believe it to be), then w l: 'v':c((PA D PA + A(:c)) 4-»P(:z:)). Now suppose, to derive a contradiction, that wl: V:c[(PADPA+ <-+(PA D;PA+ Then w l: V:c[(PA D; PA + A(:r:))+-> However, it is easy to see that PA Df PA + A(:1:)is a Z33-predicate, contradicting the 1'13-completenessof P. Therefore, there is an n E w such that 0 wl=PA DPA+A(fi) but a wbéPA DfPA+A(fi). 6.5. ILM IS THE LOGIC OF FEASIBLE INTERPRETABILITY OVER PA 99 By this method we do not immediately find the value of a particular n that works, however. A. Visser pointed out that we can make a specific counterexample in a more direct way using the Lindstrom method. Because {e | e is the Godel number of a sentence E such that -u(PA l>f PA + E 113,we can construct a formula A as in theorem 6.4.4 for which the following holds: for all sentences E, w l: -w(PA [>f PA + E) 4-»PA > PA + A("E"). Now let E' be the sentence constructed by the fixed point theorem such that PA F E' +-+A("E'"). Then w|=-~(PAi>,PA+E*)<->PA r>PA+E*. Therefore, wl=PA l>PA+E* andwbéPA l>fPA+E'. Incidentally, such a specific counterexample can also be constructed by a straightfor- ward adaptation to 113of a theorem of Myhill (see [Od 89, Proposition III.6.2]). 6.5 ILM is the logic of feasible interpretability over PA In this section, we will show that Berarducci's proof of the arithmetic completeness of ILM with respect to interpretability over PA can be adapted to prove that ILM is also arithmetically complete with respect to feasible interpretability over PA. We have already proved in section 6.3 that for all modal formulas in the language of I LM we have: if ILM l- «,0,then for all feasibility interpretations ', PA lcp". Therefore, we will only need to show the converse: if ILM l7'(,0,then there is a feasibility interpretation ' such that PA I7'cp'. We suppose that the reader has a copy of [Ber 90] at hand in order to follow the original proofs. For the lemmas 6.5.5 up to 6.5.7, knowledge of [Pu 86], [Pu 87] or chapter 3 will be helpful to the reader. As in [Pu 87], we take the logical complexity of a formula to be its quantifier depth. We can then adapt the results obtained in [Pu 87] to find for every standard n a formula Satn, a satisfaction predicate for formulas of logical complexity 3 n, such that Sat,, is of length linear in n. Subsequently, we can find proofs of length quadratic in n of the Tarski conditions and of the truth lemma for these satisfaction predicates Satn. Moreover, all these results can be formalized in PA. In the formalized case, we read Sat,, and True" as Godel numbers found as function value in n. We will not go into the details here but refer the reader to the papers by Pudlak and to chapter 3. We apologize to the reader that in this chapter Fmlan, Sat" and True,, have different meanings than in chapter 3: here the complexity measure is logical complexity, not length. First, we (re)define some of the concepts that we use in the subsequent lemmas. 100 CHAPTER 6. FEASIBLE INTERPRETABILITY Definition 6.5.1 Formally, we define the followingconcepts: 0 S'ent(a) for "a is the Godel number of a sentence"; 0 Fmla(a) for "a is the Godel number of a formula"; Fmla,,(a) for "a is the Godel number of a formula of logical complexity 3 n"; C'l(a) for "the Godel number of the universal closure of the formula with Godel number a"; note that Cl denotes a function; Indaz,,(b) for "b is the Godel number of an induction axiom of logical complexity 3 n", i.e. Inda:z:,,(b)<=> Fmla.,,(b) /\ 3y[Fmla(y)A b = Sub(y, '"v1-', "0")" /\ \7'v1("y" ->"S'ub(y, "121","Sv1")") ->Vvfy. We need to discriminate between a few different kinds of restricted provability, as defined below. In this section, provability means provability in PA, unless we explicitly state otherwise. Definition 6.5.2 We formally define the following: o BPrf,,(a:, y) for ":1:codes a proof of the formula coded by y, where only formulas of logical complexity 3 71.appear in the proof"; 0 W |P-('il:1:for ":1:codes a formula that is provable in W by a proof of length 3 P(n)" where P is a polynomial; 0 W l-l'_'-l for "there is a polynomial P such that" 'v'nElp(|p|3 P(|n|) /\ Prfw(p,a:)); o Prov,,(:z:) for ":1:codes a formula that is provable by a proof which only uses those F '1 axioms of PA with Godel number 3 n"; abbreviation Uncpfor Prov,,( cp ); o Provw,,,(:c) for ":1:codes a formula that is provable by a proof which only uses those axioms of W with Godel number 3 n"; abbreviation Clwmcpfor Provw,,,("cp"). In the context of satisfaction predicates Sat,,(;z:,212),we need a few more concepts. Definition 6.5.3 We formally define the following: o Evalseq(w,:c) for "w encodes an evaluation sequence for the formula or term with Gédel number 2:; i.e. the length of the sequence w exceeds any 2'for which a variable 12,-occurs in the formula or term coded by 2:"; o s'(z',a:,w) for "the sequence which is identical to w, except that :1:appears in the 2'-th place"; note that s* denotes a function; 0 True,,(:z:) for Vw(Evalseq(w,:z:) -+Sat,,(:r,w)), where Satn is as in [Pu 87]; Remark 6.5.4 When we prove formalized results, we read Truen as a Godel number just as Sat". So in that case the appropriate definition is as follows: True,,(a:) for r\7'w(Evalseq(w,:c) ->"Sat,,(:c,w)')"'. 6.5. ILM IS THE LOGIC OF FEASIBLE INTERPRETABILITY OVER PA 101 Lemma 6.5.5 (feasible subformula property) There are polynomialsP and Q such that PA I- \7'k'v'a(Fmla(a)-»PA PER [Pmu,,(a) -+3q(BPrfQ(|k|+|a|)(q, a))]) Proof. In [Ta 75] Takeuti gives a proof of the free-cut elimination theorem for PA, where PA is formulated as a Gentzen system. In order to be able to use Takeuti's method, we first transform Hilbert proofs that use only axioms of Godel number 3 k into associated Gentzen proofs of length linear in the length of the original proofs, in which all non-logical axioms have Godel number 3 k and the induction rule is only applied to formulas of Godel number 3 Is. Free cut-elimination then works in such a way that all principal formulas of induction inferences in the new free cut-free proof are substitution instances of principal formulas of induction inferences in the old proof. From this result one derives a proof of the corre- sponding subformula property: all formulas in the free cut-free proof of a are substitution instances of subformulas of either a principal formula of the induction rule for a formula of length 3 Us],or an axiom of Q, or a itself. At this point we can transform the Gentzen proof back into a Hilbert proof, again with only a linear increase in the length of the proofs and of the axioms occuring. We can then formalize the proof of the subformula property in PA: we find a polynomial Q such that PA l- 'v'kVa(Pr0vk(a) ->3q(BPrfQ('k|+|a|)(q, a))). But then it is easy to see that there is a polynomial P such that PA l- \7'kVa(Fmla(a) ->PA |fl-"Ll-°l)-[Prov;.(a.) ->3q(BPrfQ(|k|+|a|)(q, a))]), as desired. QED Lemma 6.5.6 There is a polynomial P such that PA l- \7'kVa(Fmla(a) ->PA lillal) [3qBPrf|,,,+lal(q, a) ->True|k|+,a,(a)]) Proof. First, we work informally by induction on the construction of q. We work in PA, and we take any /4:and an a such that a is the Godel number of a formula. We have to prove by polynomial length proofs (where the polynomial is fixed in advance) that True|;,|+,a| preserves the axioms and rules as applied to formulas of logical complexity S lkl + |a|- As an example, we show how this works for the induction schema. We take 221as the induction variable in all our instances of the induction axioms. So suppose b codes an induction axiom of logical complexity 3 |k| + |a|, e.g. b = (Sub(y, P221','0")' /\'v'v1(-'yr-> -'S'ub(y,"v1","Sv1")") ->Vvfy). We have to prove the following by a proof of length polynomial in n := Us]+ |a|: True,,(Sub(y, F1217,"07)r /\ 'v'v1(_'y'"->"Sub(y, F211",rSv1")r) -+Vvlly). (6.6) By a proof of length quadratic in n of the Tarski properties for Sat" and a proof of length quadratic in n of a call by name / call by value lemma for Sat,, (cf. the proofs of lemmas 3.3.12 and 3.3.16, but remember that a different complexity measure was used 102 CHAPTER 6. FEASIBLE INTERPRETABILI TY there), we can find a proof of length polynomial in n that (6.6) is equivalent to the following: \7'w[Sat,,(y, s*(1, O,w)) /\V:c(Sat,,(y, s"'(1,.'c,u2)) ->S'at,,(y, s*(1, Sa:,w))) -> V:r:(Sat,,(y, s'(1,:1:,u2))]. (6.7) The formulas (6.7) are themselves instances of induction of length linear in n, so they are provable by proofs of length linear in n. A polynomial of the form P(n) = c-n3 should now suffice to carry out the proofs of (6.6). Again, we can formalize the argument to derive the following: PA IVk\:/a(Fmla(a) -»PA flfl ['v'b(Inda:r|,,,+|,,,(b)-+True,,,,,.,,(b))]). Similarly, we can show by polynomially short proofs that the other axioms of logical complexity 3 |k| + |a| are true, and that the rules preserve truth. We leave these proofs and their formalizations to the reader. QED Lemma 6.5.7 There is a polynomial P such that PA lVkVa(Fmla(a) ->PA |-E(-lflflfl(True|;,|+|,,|(a)r ->7C'l(a)) Proof. By a formalized Tarski's snowing lemma; cf. lemma 3.3.11. QED The following theorem corresponds to the reflection theorem 1.6 in [Ber 90]. Theorem 6.5.8 (feasible reflection theorem) There is a polynomial P such that PA IVlc'v'a(Sent(a)-»PA 51"") (rProv;,(a) -»"a)) Proof. Combine lemmas 6.5.5, 6.5.6 and 6.5.7. QED In the following lemmas and theorems, 3K abbreviates ElK("K codes an interpretation" /\ . . The next lemma was proved by Albert Visser [Vi 91a, section 6, Claim 3] in the course of a formalized Henkin construction in IAO+ 01. Lemma 6.5.9 Suppose U Q IAO+ 01 and B aziomatizes some subset of a Elf-language L. Then there is an r such that IAO + 91 lE1UCon(fi) -+3K\7'a E Sent(L)Elp < wj(a)Prfy(p,"Clga ->"aK). Proof. See [Vi 91b]. QED In remark 2.3.3, we pointed out that the values of an-terms in a correspond to ezp(the values of polynomials in |a|). Therefore, lemma 6.5.9 implies the following lemma: Lemma 6.5.10 Suppose U Z_>IA0 + 91 and B aziomatizes some subset of a El,'-language L. Then there is a polynomial P such that mo + Q1lDUC0n(fl) -»3Kva e Sent(L)U |-"-"°-"("ofia -»M"). 6.5. ILM IS THE LOGIC OF FEASIBLE INTERPRETABILITY OVER PA 103 The following theorem corresponds to Orey's theorem; see for example [Ber 90, The- orem 2.9] Theorem 6.5.11 (feasible Orey's theorem) Suppose that U 2 PA and W is given by a set of axioms defined by the Ell'-forrnula a. Then PA l- \/:c[U # (r<>.,,,,,T")]-» U >, W. Proof. Work in PA and suppose Va:[U Mi ('<>,,,,T")]. In U, we will do a Henkin construction for the Feferman proof predicate for W. First define: fi(:c) := a(:z:) A <>a,,+1T. As in Feferman's original proof, we can prove thatDU (For, reason in U and suppose PT'fg(.'I,J_), then for the axiom of B coded by the biggest Godel number y to appear in :1:we have a(y) /\ -u<>,,,_y+1T,thus wfi(y): a contradiction.) On the other hand, by provable 2'1'-completeness for a(a) and by the assumption Va:[U}$ ("<>a,,T")], we have a polynomial P1 given in advance such that: va<a<a>~ [U ("am A<>......T">>1. So, by definition of B, we have the following for a polynomial P2 fixed in advance: va<a<a>e [U <"a<a>">1. (6.8) But, using DUC'on(fi) we can apply lemma 6.5.10 to first derive, for a polynomial P3 fixed in advance: 3mm 6 Sent(L)[U lfllfl ("mafia-+"a")], and thus for a polynomial P4 fixed in advance: 3KVae Sent(L)[Uii) (ma) -»"am. (6.9) Finally we can combine 6.8 and 6.9 to get the desired conclusion that there is a polynomial P given in advance such that 3Kva<a<a>-+ {UM (a"))l. i.e. U :>, W. QED Now we can start the proof of the arithmetical completeness of ILM with respect to feasible interpretations (cf. definition 6.3.1) over PA. Theorem 6.5.12 If ILM l7'B, then there is a feasibility interpretation * such that PA |-/ B'. 104 CHAPTER 6. FEASIBLE INTERPRETABILITY The proof will in most places be identical to the one in [Ber 90]. First we will sketch the outline of the proof, then we will prove the propositions that we need in the feasible case but differ essentially from those used in [Ber 90]. Proof sketch. Suppose ILM I7'B, and take, by modal completeness of ILM with respect to simplified models (see definition 2.5.6), a provably primitive recursive I LM - Kripke model V =< V, R, S, b, II->, with b = 1 and 1 IVB. Extend V with a new root 0 with 0R2: for all as6 V, as in definition 5.1 of [Ber 90]. Adapting definition 5.2 of [Ber 90], we define a feasibility interpretation * such that for all propositional variables p, p":= "3:1:EVU{0}:L=a:A:z:II-p", where L is defined as the limit of the Solovay function F, which is in turn defined in definition 5.7 of [Ber 90]. We want to prove the following: whenever 1 IVA, then PA if A', (6.10) Then we will be done, as we have chosen V such that 1 IVB. To prove (6.10), we need to prove in PA a few properties of F and its limit L. Subsequently we need to prove by induction on the construction of the formula that for all formulas A, the feasibility interpretation ' respects A, i.e. PAl-V:I:EV(:z:II-A/\L=$->A") and PAl-V:cEV(:r:II--IA/\L=a:-+fiA"). It is clear from the definition of F that * is faithful on atomic formulas. Moreover, the induction steps for the propositional connectives and D immediately followfrom the proofs in [Ber 90]. Even the "negative" induction step for D has a straightforward proof: Work in PA and suppose :1:E V, :1:II--=(A D B), and L = 2:; then by part 2 in the proof of lemma 5.6 of [Ber 90] and by the induction hypothesis, -u(A* D B'). But then surely fi(A" D, B'), thus, as ' is a feasibility interpretation, a(A D B)'. For the "positive" direction, we need two extra lemmas. First we will prove in PA that F satisfies a feasible adaptation of Berarducci's property 5 , which we then use to finish the induction step for D. For 1: E V, let rank(:1:,n), the rank of :1:at stage n, be defined as in definition 5.7 of [Ber 90]. The following proposition is an analog of proposition 5.14 in [Ber 90]. Proposition 6.5.13 (F has feasible property S) PA proves the following: PAl- 'v':z:EVU{0}[L=:c-> PA ll? ("v'y,z E VU{O}(L =y/\:I:Rz/\ySz -+<>kL= z)")] Proof. We will prove the proposition by combining a few facts that are easy to check. For brevity's sake, we will leave out "E V U {0}" after quantifiers V:z:,Vy,Vz. Fact 1 PA IPA Hi,' ("vy(L = y -»<>;L = y)") Proof. Immediately from the refiection theorem 6.5.8. The formula L = y has a fixed length, so the polynomial found in the proof of the reflection theorem in this case depends only on |k|. QED 6.5. ILM IS THE LOGIC OF FEASIBLE INTERPRETABILITY OVER PA 105 Fact 2 PA lPA ||-',f'("v'y(L = y ->'v'n(I5< rank(y,n)))") Proof. Immediately from fact 1 and the definition of rank. The appearance of k as an efficient numeral keeps the length of the proof polynomial in |k|. (This is also the case in the other facts below) QED Fact 3 PA lPA [LEI("Vz(Cl;L 75z ->ElmVn Z m(rank(z,n) S h))") Proof. Immediately from the definition of rank. QED Fact 4 PA lPA l-"_°-'("'v'y\7'z(L= y/\Cl;L 79z ->3n(F(n) = yAn codes y/\rank(z,n) 3 E /\ rank(z, n) < rank(y,n)))")) Proof. From the definition of limit and fact 3: just take n big enough. We can take care that n codes y because we have an infinitely repetitive primitive recursive coding of the elements of V U Finally, to prove rank(z,n) < rank(y,n), we use fact 2. QED Fact 5 We have the following: PAlVa:(L=:z:->PA [lid ("Vy'v'z(L=y/\El;L7E z/\:r:Rz/\ySz-> 3n(n codes y A rank(z, n) < rank(y, n) /\ rank(z, n) 3 h /\ F(rank(z, n))S:1:RzA F(rank(z, n))Rz))")) Proof. For the part up to rank(z, n) 3 h, we use fact 4. For the last two conjuncts, we use the S-monotonicity of F and the property corresponding to M of Veltman ILM-frames. QED Fact 6 We have the following: PAI-'v'a:(L=a:--+PA ll] ("\7'yVz(L=y/\El,;L# z/\:cRz/\ySz-> 3n(F(n) = y /\ F(n +1) = z))") Proof. Immediate from fact 5 and the definition of the function F, clause 2. QED Now we can wrap up the proof: we see that 3n(F(n) = y/\F(n+ 1) = 2) is inconsistent with L = y, so in fact we have what we were looking for: PA l- \7':r[L= 2; -»PA # ('vyvz(L = y /\ $122 A ySz -»<>,;L= z)")] QED The following proposition corresponds to part 1 of Lemma 5.6 of [Ber 90]. Proposition 6.5.14 (positive induction step for 1>) Let " bethefeasibilityinterpre- tation defined in the proof sketch of theorem 6.5.12. Suppose as induction hypothesis that PAl-'v'y(L=y->(yIl-A<->A")) and PAl-Vz(L=z->(zIl-B4->B')). Then PAl-'v':c(L=:::/\a:Il-A l>B->(A l>B)'). 106 CHAPTER 6. FEASIBLE INTERPRETABILITY Proof. Let b be such that PA l-Vy(L= y -+(y IFA 4-»A")) and PA I-Vz(L= z-+ (zlk B +->B")), both by proofs that use axioms of Godel number up to b. Moreover suppose c is such that PA lVz(z II-B -> Elc(z H-B)); for this, any c 2 the Godel number of the biggest axiom of Robinson's arithmetic Q will do. Define d := ma:z:(b,c). By theorem 6.5.11, the feasible version of Orey's theorem, it is sufficient to prove the following: PAP-\7':c(L=2:Aa:H-A 1>B-+Vk2dPAl'-',f'('A*-+<>,;B*'). Again, we will state a list of easily provable facts from which the result immediately follows. Fact 1 PA}-V:n(L=1:A;r:II-A DB->Cl[A*->3y(L=yA:cRyAyI|-AA:I:H-A DB)]) Proof. L = :1:--»EJEly(L = y A 22123;)by property (-R), Cl(A* A L = y ->3/ It A) by the induction hypothesis, and D(:c H-A D B) by provable EC,'-completeness. QED Fact2 PA l-'v':c(L=1:A:z:HA DB ->D[A* -->3y3z(L =yA:cRyAyH- AAa:Il- A DBA:z:RzAySzAzIl-B)]) Proof. From fact 1 and the definition of 2: IFA D B. QED Fact3 PAI-vzvk3dPAp",f-'(z:+B_.a,;zu-B) Proof. From the definition of d, and the fact that k appears only as efficient numeral. QED Fact4 PA}-\/:c(L=a:Aa:H~A 1>B->\7'k2dPA|'-',f'('A'-+3y3z(L=yA:1:RyA :z:RzAySzAO,;L=zAD,;zII- B)")) Proof. From fact 2 for A' ->3yElz(L = y A 1:Ry A :1:Rz A 3/52 A z |I~B); fact 3 for a proof of length polynomial in k of .2 H-B ->El,~czII-B), and proposition 6.5.13 for a proof of length polynomial in |k| of L = y A :cRy A :r:Rz A 3132 -><>;L = z. QED Fact5 PA}-'v'2:(L=:z:A2:I+A>3--»vk2dPAlfl:i('A--»3z<>;(L=z/\z:+B)")) Proof. If k is large enough (and k 2 d will do), then by an easily formalized property of modus ponens, we have the following by proofs of length polynomial in |k| : PA l- 'v'z([D,,(z HB -> L 79 2) A Dkz It B] -> Cl,,L 75 z), and thus PA lVz(<>;.L = z A Dkz II-B -+<>,,(L = z A 2 llB)). This argument can be formalized and combined with fact 4 to derive fact 5. QED 6.5. ILM IS THE LOGIC OF FEASIBLE INTERPRETABILITY OVER PA 107 Fact6 PAl-Va:(L=a:/\:rIl-A t>B-->Vk2dPA|-"f-'('A*-><>,;B"') Proof. From fact 5 and the induction hypothesis; the fact that k 2 d is used at this place. We also use that PA + Wk 2 d[PA piifl (3.-;<>;(L = z /\ 2 II-B) -»<>,;3z(L = zA2IlQED From fact 6 and the feasible version of Orey's theorem, we may indeed derive PAI-V:v(:cIl-A l>B/\L=a:->(A I>B)*), as desired. QED Proof sketch of theorem 6.5.12, continued. Concluding by induction that * respects all formulas A, we have proved that ILM l7'B'. Therefore, ILM is arithmetically complete with respect to feasible interpretability over PA. QED.

Chapter 7 The complexity of feasible interpretability How is it that life is orderly and you are content, a little cynical perhaps but on the whole just so, and then without warning you find the solid floor is a trapdoor and you are now in another place whose geography is uncertain and whose customs are strange? (Jeanette Winterson, The Passion) Abstract. We prove that there is a Eyformula f such that {e | PA feasibly interprets PA + €(e)} is Z2-complete. The method of proof that we use combines a recursion- theoretical reduction and an adaptation of some lemmas from Lindstrom's paper [Li 84]. 7.1 Introduction In this chapter, we continue our investigation of feasible interpretability begun in chap- ter 6. We remind the reader of the half-formal definition of feasible interpretability: U D; V H ElK3P("Kis an interpretation and P is a polynomial" A Va(av(a) -* 3P("|P| S P(|a|)" /\ P7'fu(P,aK))))- (7-1) Similarly, we define a concept of feasible H1-conservativity, given as U Dn,f V <->3P("P is a polynomial" A 'v':I:'v'y(Fmlan,(:1:)A Prfv(y,:z:) -*3P("|P| S P(|3/I)" /\ PTfu(P,I))))- In section 6.2 we show that many interpretations encountered in everyday mathematics are feasible. For example, we have both ZFC D; ZFC+ CH and ZFC Df ZFC + fiCH. All in all it seems that the only examples of theories U and V such that U D V but not U DfV are contrived sets of sentences obtained by fixed-point constructions. Moreover, D and Df turn out to behave rather similarly with respect to their modal-logical properties. 109 110 CHAPTER 7. THE COMPLEXITY OF FEASIBLE INTERPRETABILITY However, when we study the definitional complexity of feasible interpretability, the difference with normal interpretability is striking. It is clear from (7.1) that the formula U D; V is 22. On the other hand Solovay in [So 76b] proved that {A | PA l>PA + A} is H2-complete. This result in turn inspired Hajek to prove that, for every n, the set {A | A is H9,+1-conservative over PA} is Hg-complete [Ha 79b]. Bearing in mind Rogers' observation in [Rog 67] that "almost all arithmetical sets with intuitively simple definitions that have been studied . have proved to be Z9,-complete or H9,-complete (for some n)", we would like to know whether feasible interpretability is complete for some level of the arithmetical hierarchy. Indeed, it turns out that there is a Z3?-formula£ such that {e | PA I>f PA + £(é)} is 'E3-complete, as we prove in section 7.6. From our methods we immediately derive that {e | PA I>PA + §(E) but not PA l>; PA + §(é)} is not only inhabited, but even rather wildly so -to be explicit, it is Hg-complete. Thus the two completeness results provide some precise evidence for the observation that normal interpretability and feasible interpretability over PA have substantially different extensions. The formula 5 that we use for the Z33-completeness results is as simple as possible. More precisely, it is easy to show that for any H?-formula {(113),{e | PA l>f PA + §(é)} is equal to {e | PA l- {(6)}, which is recursively enumerable. The rest of the chapter is organized as follows. In section 7.2 we give some preliminaries on partial truth definitions and some notational conventions. In section 7.3 we characterize feasible interpretability in terms of feasible H1-conservativity. Section 7.4 contains the main novelty of this chapter, namely a recursion-theoretical reduction by which we show that the set of (possibly infinite) theories feasibly interpretable over PA is E3-complete. Lindstrom provided in [Li 84] a general method by which one can replace every re- cursively enumerable set Y of E9,-sentences by a single 2?,-sentence 0 such that PA + 0 has the same H9,-consequences as PA + Y. In section 7.5, we prove a feasible version of Lindstr6m's lemmas. The proofs in this section are fairly straightforward. Finally, in section 7.6, we apply the methods of section 7.5 to the possibly infinite, but still suitably simple, sets of formulas constructed in section 7.4. Thus, using the characterization of feasible interpretability over PA as feasible H1-conservativity, we prove that there is a Z3?-formula{(2) such that {e | PA l>f PA + {(5)} is Z3-complete. The chapter is almost self-contained. However, for some details of proofs we refer the reader to chapter 6, and we suppose that the reader is at least slightly familiar with the terminology used in [Bu 86]. 7.2 Preliminaries and notation We use Pudlak's notation T )1 4pand T # '.p(n) as discussed in notation 2.6.7. We will also sloppily leave out some Godel brackets and numeral dots, in particular deeper nested ones. We use efficient numerals 17.of length linear in We suppose that all theories mentioned in the sequel are Elf-axiomatized. Definition 7.2.1 Prov,.,q~("A")stands for "there is a proof of A from T in which only axioms with Godel number 3 I: are used". Con,.(T) := -Prov;,,T(r_l_"). In this chapter we use two kinds of partial truth predicates. Pudlak in [Pu 87] intro- duced the first kind that we need. His True" are truth predicates of length linear in n 7.2. PRELIMINARIES AND NOTATION 111 which work for quantifier depth 3 n. Pudlak works with a relational language, whereas in chapter 3, the standard language of arithmetic including function symbols was used. In the present chapter, as in chapter 6, we use Pudlak's complexity measure, namely the logical complexity and not the length of formulas. We have the following Tarski lemma: Lemma 7.2.2 There is a polynomial P such that for all sentences A of length 3 |k|, PA |i"' True,,.,(A)H A. Proof. See [Pu 87]. (cf. chapter 3. Pudlak works with a relational language, whereas in chapter 3, the standard language of arithmetic including function symbols was used.) QED Lemma 7.2.2 can be formalized to get: Lemma 7.2.3 PA I3P'v'kVa(Sent(a) -»PA |"-""-"L"(True|;.,+|a|(a) «-»a)). Definition 7.2.4 A theory T in the language of arithmetic is feasibly essentially reflexive if there is a polynomial P such that for all sentences A and for all k, T pflfl (Prou.,T("A')-+A). Lemma 7.2.5 PA is feasibly essentially reflexive, even provably so, i.e. we have: PA 1-3P\7'kVa(Sent(a)-»PA E (Prov;,,pA(a)-»True,,.,,.,.,(a))), thus PA 1ElP'v'kVa(S'ent(a) -->PA lflfilill (Prov;.,pA(a) ->a)). Proof. See theorem 6.5.8 and lemma 7.2.3. QED We will need a similar result in section 7.5. First we need two definitions. Definition 7.2.6 We say that a set A is sparse if there exists a polynomial P such that for every n the number of elementsof A having length 3 n is bounded by Definition 7.2.7 A Z?-formula a defines a provably sparse relation if there is a polyno- mial P such that for all q, PAlfi)-'v'z(a(z)/\z§g-> z=x). 1:3?)/\a(:|:) Lemma 7.2.8 Let A be an extension of PA by a provably sparse set of axioms in the language of PA. Then we have A |iT-lVu, v 3 1_n(PrfA(v, u) ->True|,,,|(u)). 112 CHAPTER 7. THE COMPLEXITY OF FEASIBLE INTERPRETABILITY Proof. Suppose the set of new axioms over PA is given by the 2'1'-formulaa. The proof uses lemma 6.5.5 and lemma 6.5.6. In order to apply lemma 6.5.5, we work in PA and first transform proofs with Godel number 3 m of u from A into PA,,,-proofs of (par abus de langage) ( A 2:)!' _' flu; :r:$fi/\a(:l:) this can be done polynomially in |m|. Next we apply lemma 6.5.5 and lemma 6.5.6. Finally we note that, due to the provable sparsity of the set of new axioms defined by a, we have A # Truel,,,|( /\ 2:), :I:SrT./\a(:t:) Thus we derive the lemma. We leave the details, e.g. of formalization of /\,,,S,7,,,c,(¢)1:, to the reader. QED The second kind of partial truth predicates that we use are the standard ones related to the levels of the arithmetical hierarchy. Lemma 7.2.9 For all i Z 1 there are partial truth definitions Trueg, and Truem such that the Tarski lemmas have short proofs. More precisely, there is a polynomial P such that: 0 for all 1'1,--sentences7r, PA [1 7r<-+Truem('7r"), and 0 for all Z3,--sentences0, PA lffl 0 4-»Trueg'.("o"). Proof. Visser in [Vi 92] gives a Ao(e:rp) definition of satisfaction for A0-formulas. It is easy to construct from this partial truth definitions Trueg, and Truen, of length linear in i. We leave the reader the easy but tedious task of showing that the Tarski lemmas indeed have short proofs. QED We also need a result relating the two kinds of partial truth definitions. Lemma 7.2.10 Let F, E {X},-,lI,-}.Then PA |@ \7'u3 rTi[Fmlap'.(u) A True|,,,|(u) ->Truep,(u)]. Proof. We leave the proof to the reader. QED 7.3 Characterizations of feasible interpretability In section 7.6, we will use a characterization which says that, over suitable theories, feasible interpretability is equivalent to feasible H1-conservativity. We prove a formalized version of this equivalence in corollary 7.3.3. Non-feasible versions of all three characterizations below are well-known. We proved a feasible version of Orey's Theorem in chapter 6. We have the following Orey-Hé.jek-stylecharacterization of feasible interpretability. (Provably sparse relations are defined in definition 7.2.7.) 7.3. CHARACTERIZATIONS OF FEASIBLE INTERPRETABILITY 113 Lemma 7.3.1 (Feasible Orey-Héjek characterization) SupposeU 2 I 21 and V is an extension of PA by a provably sparse set of axioms in the language of PA. Then PA + [U ('4' Con,(U)] _. (U :>, v ._. [U ('4' Con,(V)]). Proof. -->We first state a useful fact. We axiomatize each PA;, using only one induction axiom. To be precise, there is a function 2'r-+ "6,-", with |'"6,-"|polynomial in 2',definable in IE1, and a 2','-formulaa;'v (standing for the axiom set that contains the induction axiom for 6w plus the sparse set of remaining axioms of V) such that 121% 'v'a:,z(a;,V(z) ->z 3 ac) and 1231 l- \7':z:(Prov,,v(_L) ->Pr0v,,;'v(_L)). Therefore we have, by instantiation, PA l- [U llél Prov"/(_L) ->Provo,;'v(_L)]. (7.2) We now start our proof proper by reasoning in PA and assuming that U }'i_'eon,(U). (7.3) If U [>f V by interpretation K, then there is a polynomial R such that V:z:Vz3 :z:(a;,v(z) -+U Ii 2K). Because a;,,, singles out a provably sparse set, we have provable completeness, thus we derive U Hi,' V2 g x(a;,,(z) -»U #*'-"fl2"). (7.4) Next we want to find a polynomial Q such that U |$ PT'0'Ua;'V(_l_)-->"'C0n2Q(|:;|)( (7.5) So reason inside U and suppose Prfo,;'v(q,J_). If we take K-translations of all formulas in q and add some intermediate steps, we find a quasi-U-proof p of .1.that still depends on some assumptions 2" where a;'V(z). But, since 2 3 2:,we know by (7.4) that these zK have U-proofsof length 3 We add these proofs to p in order to find a U-proof p' of _Lthat uses only U-axioms of length 3 Q(|:::|) for some polynomial Q given in advance. Stepping out of U again, we find that indeed (7.5) holds. However, by (7.3) we have U # Con2Q(.:.)(U), so (7.5) gives U <Provo,;_V(_J_). By (7.2) we finally conclude U @ Con,( V). 114 CHAPTER 7. THE COMPLEXITY OF FEASIBLE INTERPRETABILITY +-For this direction, which is a feasible version of Orey's Theorem, we do not need the assumption U # Con,(U), nor do we need the provable sparsity of V over PA. We make the desired interpretation by a feasible Henkin construction for a Feferman proof predicate of V. For details of the proof we refer the reader to theorem 6.5.11. QED Lemma 7.3.2 PA + [v i'f-'Con,(V)] -.(U >,,,, v H [U (E4 Con,(V)]). Proof. ->Reason in PA and suppose that U [>n,f V and V llél C'on,( V). Because C'on,( V) is a H1-sentence, this immediately gives us U [iii Con,( V). 4This direction of the proof does not depend on the assumption V |-lf-'Con,( V). So, reason in PA and suppose that U # Con,(V), Fmlan,(:1:)A PrfV(y,:z:). (We will use without mention the fact that a: _<_y.) First we analyze the proof of 2'1'- completeness, and we note that there is a fixed m - to be explicit, m is the Godel number of the largest axiom of Robinson's Arithmetic Q - for which we have the following: U |l-fl-Con,,,( V + 1:) ->2:. (7.6) Because the axioms of V are recognized in a 2'1'-way,we can again invoke provable Z?-completeness to show that Prfv(y, sc) implies U Hi,'-«con,( v + fix). (7.7) By our assumption U ll? Con,( V) we have U l|i_l Con,,,,,,(,,,,,,)(V), which we may combine with (7.7) to derive U ll-f_'lC'on,,,( V + 2:), thus by (7.6) U l'i_l 2:, as desired. QED Corollary 7.3.3 If U 2 I21 and V is an extension of PA by a provably sparse set of axioms, then PA l- [U 9? Con,(U)] /\[v +'f-'Con,(V)] -.(U :>, v H U :>,7,, V). Proof. --+By the ->-direction of lemma 7.3.1 and the 4--direction of lemma 7.3.2; we do not need the assumption V [lid Con,( V). <-By the ->-direction of lemma 7.3.2 and the 4--direction of lemma 7.3.1; we do not need the assumption U )5,' Con,( U). QED Because PA is provably feasibly essentially reflexive, we have the following useful characterization for feasible interpretability over PA: Corollary 7.3.4 For all formulas A, B in the language of PA, PA}-PA+A l>;PA+B+->PA+Al>n,,PA+B. Proof. Immediately from lemma 7.2.5 and corollary 7.3.3 QED 7.4. THE SET Of THEORIES FEASIBLY INTERPRETABLE OVER PA 115 7.4 The set of Elf-axiomatized theories feasibly in- terpretable over PA is Z32-complete In this section, we will prove that the set of theories that can be feasibly interpreted in PA is E2-complete. We assume that each 2'1'-axiomatized theory is given by a code of a non-deterministic polynomial time Turing machine that accepts exactly the Godel numbers of axioms of the theory in question, and we suppose that the coding of Turing machines is standard, e.g. as in [BDG 87]. By the way, the reduction that we use to prove theorem 7.4.1 can be easily adapted to yield a slightly alternative proof of Hajek's theorem that {e | the deterministic Turing machine coded by e works in polynomial time} is E2-complete (cf. [Ha 79a]). Let us turn to the technicalities. Take some Godel numbering which codes formulas in the language of arithmetic (which for our purpose includes a function symbol exp) as binary numbers. L, stands for the language accepted by the Turing machine with code e. By writing out the definitions we see that E := {e | e codes a deterministic Turing machine such that PA feasibly interprets PA+ the set of formulas whose codes are in L,} = {e | 31K EIP Va:Vy(2:is an axiom of PA or y is an accepting computation of e on :1:->3z(|z| 3 P(|;2:|) A Prj(z,:z:K)))} is in 22. It is well-known that {e | We finite} is Z32-complete (cf [So 87]). So in order to prove that E is in fact Z32-complete,the following theorem suffices. Theorem 7.4.1 There is a total recursive function F such that for all e: We is finite <=>F (e) codes a non-deterministic polynomial time Turing machine such that PA feasibly interprets PA + the set of formulas whose codes are in Lp(e). In order to prove this theorem, we first introduce a definition and prove a lemma. Definition 7.4.2 Define by G6del's diagonalization theorem (or rather by the free vari- able version as formulated by Montague) a Ao(e:cp)-formula go(y)such that PA l" <P(y) H V|$| S €IP(lyl)"P7'f(117,rS0(37)1)- This fixed point is a bounded analog to the fixed point that Godel used to prove his First Incompleteness Theorem. Informally, every <,o(n)says "I am not provable by any short proof". Part of the proof of the following lemma is reminiscent of G6del's argument. It is almost identical to the proof of theorem 6.4.1. Lemma 7.4.3 For any NP-subset X of the natural numbers defined by a Z3'1'-formulaa, we have X is finite <=>PA [>f PA + {<p(fi)|a(n)}. Proof. 0 It is easy to see that there is a polynomial 0 such that for each n, < |"<p(n')"|3 0(|n|)- o If cp(n) were false, then by definition we would have a proof of the A0(ea:p)-sentence cp('r'i);so cp('7"i)must be true. But then, since cp(7'i)is A0(e:rp), we have the following: 116 CHAPTER 7. THE COMPLEXITY OF FEASIBLE INTERPRETABILITY 1. PA proves cp(1'z),though 2. because cp(7"z)is true, PA does not prove <p(1'z)by any proof whose Géidel number is of length 3 2"". o If X is finite, then it is obvious that PA [>f PA + {cp(7'z)|a(n)}. 0 Suppose X is infinite and PA l>f PA + {<p(7'z)|a(n)}by interpretation K and poly- nomial P. Thus, for all n E X, PA IP(lrS"(fi)-'l) Since cp is Ao(ea:p), we also know by lemma 3.10 of [Ve 93] (with U = V = PA) that there is a polynomial R such that for every n E X, PA <p(7'7«)«-»<p(T7)K- Now can construct from R and P a polynomial Q such that for all n E X, 13,4| M,-,,)_ However, there will be n such that 2l"' > Q(O(|n|)) Z Q(|"<p(r'z)"|),and we have a contradiction with 2. QED Now we can prove the theorem by giving an appropriate reduction F. Proof. For e,t given, we describe the behavior of the deterministic Turing machine coded by F(e) on input t. In the rest of the proof we will sloppily mention the codes instead of the machines or functions that they code. As usual, |s| stands for the number of symbols that 3 consists of. IF t is not of the form "cp(§)" for any 3, THEN we halt and reject t; ELSE we find the 3 such that t is of the form "cp(§)'; first we simulate the behavior of e on inputs 1,. . . , |s|, for at most |s| steps each. IF |s| > Is - 1| AND e halts on |s| within |s| steps, OR there is an 2'3 |s| -1 such that e halts on 2'within |s| steps AND e does not halt on 2'within |s| -1 steps, THEN we halt and accept t; ELSE we halt and reject t. We want to show that F is the required function. To this end, first define .9e(i) := the smallest .3such that 0Island 0 e halts on 2'within |s| steps. 7.4. THE SET Of THEORIES FEASIBLY INTERPRETABLE OVER PA 117 Now we prove the theorem. (2) Suppose We is finite, say We = {i1, . . . ,i,,}. Then Lp(e) = {"<,o(se(i1))",...,'cp(se(i,,))"}. By lemma 7.4.3 we know that PA I>f PA + {'P(3e(7;1))a - - 290(3e(in))}° (<=) Suppose We is infinite. We know that for all i 6 We, i 3 |se(i)|. So the set Lp(e) = {"<p(W)" | i 6 We} is an infinite set, and by lemma 7.4.3 we do not have PA [>; PA + {<p(se(i)) | i 6 We}. QED Corollary 7.4.4 There is a total recursive function F such that for all e: We is infinite <=>F (e) codes a non-deterministic polynomial time Turing machine such that PA interprets PA + the set of formulas whose codes are in Lp(e), but does not feasibly interpret this set. Proof. We can take the reduction F as in the proof of theorem 7.4.1. The «--direction follows immediately from the old proof. For the ->-direction,we only have to remember the additional fact that PA 1-cp(n')for every n. QED In section 7.6, we will replace the possibly infinite sets Lp(e) by a single sentence {(6) which is just as strong as far as feasible interpretability over PA is concerned. In order to do this we make use of two properties of Lp(e) which make it suitable for replacement: 1. it is easy to compute whether y E Lp(e), thus, because we know already that PA is Alf-axiomatized, PA + Lp(e) is All'-axiomatized as well; and 2. Lp(e) is only sparsely populated (see definition 7.2.6). We now proceed to make these properties more precise and to prove them for our Lp(e). Remark 7.4.5 Note that, if we take some standard coding of Turing machines (see e.g. [BDG 87]), then to compute whether t E Lp(e) takes only time polynomial in |t| + |e|, for: o to compute whether t is of the form "cp(s)" and, if it is, to find this 3, takes time polynomial in |t|; 0 to simulate the behavior of e on inputs 1, . . . , |s|, for at most |s| steps each takes time polynomial in |s| + |e| 3 |t| + |e|; 0 to see whether |s| > |s -1| takes time linear in |s| 3 |t|; 0 to see whether e halts on some i 5 |s| within |s| but not within |s| -1 steps takes time polynomial in |s| + |e| 5 |t| + |e|. Remark 7.4.6 Note that for every e, the set Lp(e) is sparse. More precisely, all members of Lp(e) are of the form "<p(§)" for some 3 such that |s| > |s -1| (i.e. there is a k such that s = 2"). Generalizations of theorem 7.4.1 By some slight adaptations, the proof of theo- rem 7.4.1 immediately gives rise to other results. We mention two directions of general- ization. 118 CHAPTER 7. THE COMPLEXITY OF FEASIBLE INTERPRETABILITY 0 Sam Buss suggested the following restricted definition of feasible interpretability to LISZ U l>f,,,V 4-»3K3M("Kis an interpretation and Mis a determ. pol. time Turing Machine" AVa(av(a) ->PrfU(M(a),aK))). This definition is more in line with the conventional use of the word "feasible" in the context of polynomial time Computability. The clause PrfU(M(a), a") is a P- like formula, while the clause 3p("|p| g P(|a|)" A PrfU(p,aK)) in the definition of feasible interpretability used in this dissertation is an N P-like formula. It is easily seen that under the new definition, E' := {e | e codes a deterministic Turing machine such that PA "feasibly" interprets PA + the set of formulas whose codes are in L3} = {e | HK 3P-time polynomial Turing machine M V:z:Vy(:ris an axiom of PA or y is an accepting computation of e on :1:-+Prf(M(:z:),:cK))} is in 22. Moreover, by inspection of the proofs of lemma 7.4.3 and theorem 7.4.1, we see that E' is in fact Z2-complete. We could also define feasible interpretability by bounding the length of proofs used in terms of other standard function classes than the polynomials; e.g. the linear functions or the exponential functions would be a good choice. In the latter case we have to adapt the fixed point of lemma 7.4.3 in order to diagonalize out of the function class, but we still get Z2-completeness of the set of theories "feasibly" interpretable over PA. 7.5 Lindstr6m's general lemmas polynomialized Let A be an extension of PA by a provably sparse set of axioms in the language of PA, where the set of axioms of A is given by the Z'1'-formu1aa. We give a definition from [Li 84], and we adapt some of the lemmas from that paper. Definition 7.5.1 Let P be either H or E. For every i 2 1, we define the following: [I",-],,,(:I:,y) := 'v'u,v 3 y(Fmla1~'(u) /\ Prfc,(,)V,:,(v,u) ->True1~[(u)) The following lemma corresponds to [Li 84, Lemma 1]. Lemma 7.5.2 Let F be either H or 2. Then [1",~]o,(:1:,y)is a F,~-formulasuch that 1. PA l- [1",-]a(:c,y) A z 3 y ->[I",-]c,,(:1:,z); 2. For every e and every cp, A + cp(é) |-l",:'-l[I",-]o,("cp(e)',rn) 3. For every e and every cp, there is a polynomial P such that if?/J E I', and A+<,0(e) l212 via a proof coded by q, then A + [1",-].,('«,p(e)',q)[13 1/2. Proof. 1. See [Li 84, Lemma 1]. 7.5. LINDSTROM'S GENERAL LEMMAS POLYNOMIALIZED 119 2. By lemma 7.2.8, we have A + me) 919-'vu. v s fi(Prfa<z>v==w»<a>~(v,u> -»T7'"elm|("ll- Moreover, we have by lemma 7.2.10: A ['4' 'v'u(Fmlap,.('u) A True|m|(u) ->Truep_.(u)), so indeed A + we) l'.1' [r.1.<w>",-m>. 3. Suppose 1,06 I'; and A+g0('é) l212,via a proof coded by q. Then there is a polynomial P1 given in advance such that A 1% Fmlar.-("ab") A Prfa(z)Vz='<p(E)'(qarw1)a thus by definition of [I",-]o,,there is a polynomial P2 given in advance such that A + [11-la("<p(é)".a) ifl Tmer.<"w">. So by lemma 7.2.9, there is a polynomial P given in advance such that A + [11-]..("so(é)',<7)rm 2». QED The fixed points that we define below in definition 7.5.3 were introduced by Lindstrom in his paper [Li 84]. Our lemmas 7.5.4 and 7.5.5 are analogous to [Li 84, lemma 2]. The difference is that we keep track of the lengths of the proofs. Definition 7.5.3 Let i 2 1 and x E E,»be given. Define 5 by diagonalization such that PA |" {(5) H 3y('*lHila(r€(E)".y) /\ V2 S y x(és 2))- Dually, let i 2 1 and X 6 11,-be given. Define 0 by diagonalization such that PA F 6<e>H vy<[>:.-1..<r6<e>".in ~ x(é. in). Lemma 7.5.4 If )((:::,y) is a ZI,~-formula,then {(23) is also 23,-and the following holds: 1. For all e, A + {(6) ill,' 'dz 3 *rfix(e,z). 2. For every e, there is a polynomial P such that if 7r 6 IL and A + E(é) l7r via a proof coded by q, then A + V2 3 6 )((E, .2) PW) 7r. Proof. 120 CHAPTER 7. THE COMPLEXITY OF FEASIBLE INTERPRETABILITY 1. Take e fixed. By lemma 7.5.2(2) we have A + ((7) # [H.-1..(':(e)'.772), so by lemma 7.5.2(1), we have A + ((7) ('4' vz s 7'n[1T.-lo.("€(é)".z). (7.8) Now by definition of {, A + 5(7) )@ ay(~(H.1.(:(e). y) Avz s y x(e. 2)). (7.9) From (7.8) and (7.9), we finally conclude that A + {(6) V23 'Tn'x(é,z). 2. Suppose 7r 6 1'I,-and A + §(e) l77via a proof coded by q, (7.10) then by lemma 7.5.2(3), we have a polynomial P1 given in advance such that A+ [n.~1..(e(e):) vr. (7.11) By definition of 5, we have a polynomial P2 fixed in advance such that A + vz s 7)x(e. z) aw) v [IL-]a(€(é).q). (7.12) From (7.10), (7.11) and (7.12) we conclude that there is a polynomial P given in advance such that A + V2 3 Qx(e, 2) PW) 7r. QED We state the next lemma for reasons of symmetry only: it will not be used in the sequel Lemma 7.5.5 If x(:r,y) is a ll,--formula, then 0(:c) is also 11,-and the following holds: 1. For all e, A + 9(é) ll?' x(é,'r-n). 2. For every e, there is a polynomial P such that ifa E E. and A + 9(6) lfl 0, then A + vz 5 <7x(é. 2) P""" 0. Proof. We leave the proof, which is similar to the proof of lemma 7.5.4, to the dis- trustful reader. QED 7.6. FEASIBLE INTERPRETABILITY IS Z32-COMPLETE 121 7.6 Feasible interpretability is Z32-complete The result of section 7.4 is not yet entirely satisfactory: we seek an elegant result that corresponds more neatly to the idea that feasible interpretability is E2-complete. In order to do this, we would like to be able to replace each possibly infinite set Lp(,) of codes of axioms accepted by a Turing machine with code e by an instance £(é) of a single formula 5. Moreover, we would like this replacement to be such that LF(,) and §(é) have the same status with respect to feasible interpretability over PA. Luckily, one of the fixed points of section 7.5 can do our job. Definition 7.6.1 Let a be a 2'1'-formuladefining the set of axioms of PA. By remark 7.4.5, the relation y E Lp(e) is polynomial-time computable, so by [Bu 86, Theorem 3.2] it is All'-definable in IAO + 01. This means that there are a 2'1'-formula n(e,y) and a I'll,'-formula1/(e,y) that both correspond to the relation y 6 LF(,) and such that IA0 + Q1 ln(e,y) +->z/(e,y). Now we define x(e,y) := n(e,y) ->True~,;,(y). Clearly x is 2?. We define 5 as given by definition 7.5.3 for z'= 1 and this X. To be explicit, 5 is defined by diagonalization such that PA l' 6(5) H 33/("lH1la(r€(é)",y) /\ V2 S 3/X(77(5a2) -' T7"U€2:1(Z)))- Definition 7.6.2 For every e, let Xe be the set of Ao(e:::p)-formulaswhose Godel numbers are contained in LNG). Lemma 7.6.3 For all e, the following holds: 1. PA + {(6) >, PA + X,; 2. PA + X. t>n,, PA + {(5). Proof. 1. Suppose that cp E X... Because T)6 2'1',we have PA |'+" n(e, Hp"). (7.13) Next, by lemma 7.5.4.1, we have PA + so) #131n<e.*.o"> -»Tmez.<"«.o">. (7.14) Combining (7.13) and (7.14), we derive PA + {(6) )l-",flTrue;;,('cp"). Thus by lemma 7.2.9, PA + £(é) )'-'f_°'Lp.So we certainly have PA + §(é) l>f PA + X... 2. Suppose 7r 6 H1 and PA + {(6) l7r by a proof with Godel number q. Then by lemma 7.5.4(2), we have a polynomial P1 fixed in advance such that PA + 'v'z _<_(7x(é, 2) Plug') 7r. (7.15) Also we have a polynomial P2 given in advance such that PA + X...IE V2g q )((E,z) (7.16) 122 CHAPTER 7. THE COMPLEXITY OF FEASIBLE INTERPRETABILITY The reason for this is as follows. By a formalized version of remark 7.4.6, Xe provably contains only formulas 90(3) where 3 is a power of 2, i.e. PA ln(é, 2) -+ V 2 = '-<p(2'°)". l=SlZ| Moreover by All'-completeness (see [Bu 86]) we have a polynomial P3 such that for all n (n ¢ Xe => PA + Xe lflu fin(é, 17)),so there is a polynomial P4 given in advance such that PA}-fflQ'v'z(n(é,z)/\z§§-+ V z=.f). :I:$q/\:I:EXe A130, PA VA,'Lp(fi) -. Tme,e,(',o(n)"), thus PA + Xe Hi,' V2 3 q(n(é,z) -» Trueg,(z)). From (7.15) and (7.16), we conclude that there is a polynomial P given in advance such that PA + Xe lfli 7r,as desired. QED Corollary 7.6.4 For all e, the following holds: 1. PA + {(6) D PA + Xe; 2. PA + Xe l> PA + £(é). Proof. Directly from lemma 7.6.3. Alternatively, see Lindstr6m's original nonfeasible argument in [Li 93, lemma 5]. QED Theorem 7.6.5 {e | PA l>f PA + £(é)} is Z32-complete. Proof. By theorem 7.4.1, we have for all e: We is finite <=> PA I>f PA + Xe. By inspection of lemma 7.2.8 we see that for all e, PA + Xe is feasibly essentially reflexive. Therefore corollary 7.3.3 gives for all e: PA Df PA + Xe <=> PA Dnlf PA + Xe. Combining this with lemma 7.6.3 finally gives us for all e: WelSfinite <=> Dnlf + {(3), thereby giving a reduction from the E2-complete set {e | We is finite} to the set {e | PA Df PA + {(83)}. QED Theorem 7.6.6 {e | PA > PA + 5(6) A -=(PA [>f PA + £(é))} is H2-complete. 7.6. FEASIBLE INTERPRETABILITY IS E2-COMPLETE 123 Proof. It is easy to see that PA I>PA+E(é)/\fi(PA l>fPA+€(é)), being a conjunction of two H2-formulas, is again a H2-formula. From corollary 7.4.4, we conclude that for all e, We is infinite if and only if PA l>PA+X,/\w(PA l>,PA+X,). But by lemma 7.6.3 and corollary 7.6.4, we find that Weis infinite if and only if PA l>PA+ {(5) A-1(PA I>fPA + §(é)). Thus we have reduced the H2-complete set {e I Weis infinite} to {e | PA > PA + €(é) A w(PA >, PA + £(e))}. QED Generalizations of theorem 7.6.5 We can generalize theorem 7.6.5 on lines suggested at the end of section 7.4. However in this case it would not be a good choice to define feasible interpretability by bounding the length of proofs by linear functions.

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We introduceren ook de belangrijkste begrippen die in het proefschrift aan de orde komen: complexiteitstheorie, begrensde rekenkunde, bewijsbaarheidslogica en interpreteerbaarheidslogica. Hoofdstuk 2 bevat de technische beschrijving van de in hoofdstuk 1 geintroduceerde begrippen. Daarnaast geven we een beknopte opsomming van de stellingen uit de lite- ratuur die we bij het bewijzen van onze resultaten gebruikt hebben. Zo geven we enkele stellingen over definieerbare sneden en hun toepassingen in de rekenkunde. Het hoofd- stuk eindigt met een paragraaf waarin resultaten uit de literatuur besproken worden die de verschillen en overeenkomsten tussen enkele zwakke rekenkundige theorieén belichten. Interpreteerbaarheid en conservativiteit voor bepaalde klassen formules worden hier ge- bruikt om de kracht van de theorieén te vergelijken. Deel II is gewijd aan de begrensde rekenkunde. In hoofdstuk 3 bewijzen we eerst onder de cornplexiteitstheoretische aanname NP 75 co-N P dat de begrensde rekenkunde geen volledigheid bewijst voor alle formules voor het vergelijken van getuigen. In de Peano Rekenkunde speelt bewijsbare volledigheid voor zulke formules een belangrijke rol bij het bewijzen van de geformaliseerde versie van Rossers Stelling en So1ovay'sVolledigheidsstelling. Om toch 00k in de begrensde rekenkunde de geformaliseerde versie van Rossers Stelling te kunnen afleiden, bewijzen we een reflectieprincipe voor "kleine" bewijzen. Het bewijs daarvan maakt gebruik van partiéle waarheidspredicaten en definieerbare sneden. Als toepassing van dit principe geven we een bewijs van een stelling van Bernardi en Montagna voor de begrensde rekenkunde. Bovendien gebruiken we het "kleine" reflec- tieprincipe voor een simpele versterking van een bekende stelling over het injecteren van kleine bewijzen van inconsistentie. Tenslotte gebruiken we het principe, op een meer ver- rassende manier, in het bewijs van een stelling over het bestaan van echte eindextensies van modellen van de begrensde rekenkunde die aan een zware extra eis voldoen. In hoofdstuk 4 keren we terug naar het probleem van bewijsbare volledigheid. We bewijzen dat de cornplexiteitstheoretische aanname P 96 NP fl co-NP impliceert dat Buss' begrensde rekenkunde S; niet voor alle Z3?-zinnenvolledigheid bewijst. In hoofdstuk 5 presenteren we partiéle antwoorden op de vraag: wat is de bewijs- baarheidslogica van de begrensde rekenkunde? Omdat bewijsbare volledigheid voor zin- nen voor het vergelijken van getuigen op grond van resultaten uit hoofdstuk 3 en 4 dubieus 131 132 is, kunnen we niet rechtstreeks So1ovay'smethode gebruiken. Met behulp van het kleine reflectieprincipe uit hoofdstuk 3 en definieerbare sneden passen we voor een geschikte klasse van Kripkeframes de methode van Solovay aan. We geven een inbedding van modellen op zulke eenvoudige frames in de begrensde rekenkunde. Ook bewijzen we dat de bewijsbaarheidslogica van de begrensde rekenkunde in ieder geval niet de modale theorie van een klasse Kripkebomen kan zijn. De vraag wat de bewijsbaarheidslogica van de begrensde rekenkunde dan wel is, is op het moment van schrijven voor zover bekend nog open. Deel III behandelt een alternatieve definitie van interpreteerbaarheid. In hoofdstuk 6 definiéren we "uitvoerbare interpreteerbaarheid," waarbij de lengte van bewijzen van vertaalde axioma's begrensd is door een polynoom in de lengte van die axioma's zelf. We laten zien dat een aantal bekende interpretaties, zoals die van ZF + V = L in ZF, uitvoerbaar zijn. Aan de andere kant zijn niet alle interpretaties te vervangen door uitvoerbare interpretaties. Met behulp van diagonalisatie construeren we een theorie die weliswaar in de Peano Rekenkunde interpreteerbaar is, maar er niet op uitvoerbare wijze in geinterpreteerd kan worden. Verder laten we zien dat de interpreteerbaarheidslogica I LM arithmetisch correct en volledig is voor uitvoerbare interpreteerbaarheid over de Peano Rekenkunde. Hoofdstuk 7 behandelt de definitionele complexiteit van uitvoerbare interpreteerbaar- heid over de Peano Rekenkunde. We bewijzen, door een recursie-theoretische reductie te combineren met een aangepaste versie van een methode van Lindstrom waarin partiéle waarheidsdefinities een belangrijke rol spelen, dat uitvoerbare interpreteerbaarheid over de Peano Rekenkunde Z32-volledigis. En passant geven we een karakterisering van uit- voerbare interpreteerbaarheid in de stijl van Orey en Hajek. De 23-volledigheid van uitvoerbare interpreteerbaarheid over de Peano Rekenkunde staat in contrast met de H2-volledigheidvan standaard interpreteerbaarheid over de Peano Rekenkunde. Het blijkt dat standaard interpreteerbaarheid en uitvoerbare interpreteer- baarheid substantieel verschillende extensies hebben. We bewijzen dat de verzameling zin- nen die wel gewoon maar niet uitvoerbaar interpreteerbaar is over de Peano Rekenkunde, zelfs H3-volledig is.