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Finite Type Arithmetic

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Finite Type Arithmetic Finite Type Arithmetic Computable Existence Analysed by Modified Realisability and Functional Interpretation H Klaus Frovin Jørgensen Master’s Thesis, March 19, 2001 Supervisors: Ulrich Kohlenbach and Stig Andur Pedersen Departments of Mathematics and Philosohy University of Roskilde Abstract in English and Danish Motivated by David Hilbert’s program and philosophy of mathematics we give in the context of natural deduction an introduction to the Dialectica interpretation and compare the interpre- tation with modified realisability. We show how the interpretations represent two structurally different methods for unwinding computable information from proofs which may use certain prima facie non-constructive (ideal) elements of mathematics. Consequently, the two inter- pretations also represent different views on what is to be regarded as constructive relative to arithmetic. The differences show up in the interpretations of extensionality, Markov’s prin- ciple and restricted forms of independence-of-premise. We show that it is computationally a subtle issue to combine these ideal elements and prove that Markov’s principle is computa- tionally incompatible with independence-of-premise for negated purely universal formulas. In the context of extracting computational content from proofs in typed classical arith- metic we also compare in an extensional context (i) the method provided by negative trans- lation + Dialectica interpretation with (ii) the method provided by negative translation + ω A-translation + modified realisability. None of these methods can be applied fully to E-PA , since E-HAω is not closed under Markov’s rule, whereas the method based on the Dialectica interpretation can be used if only weak extensionality is required. Finally, we present a new variant of the Dialectica interpretation in order to obtain (the well-known) existence property, disjunction property and other closure results for typed in- tuitionistic arithmetic and extensions hereof. Hence it is shown that functional interpretation can be used also for this purpose. ◭◮ Vi giver med udgangspunkt i Hilberts program og hans matematikfilosofi en introduktion til Dialectica fortolkningen inden for naturlig deduktion. Denne fortolkning sammenlignes med modificeret realiserbarhed, og vi viser, hvordan de to fortolkninger repræsenterer to strukturelt forskellige metoder til at uddrive beregnbar information fra ikke-konstruktive be- viser. S˚aledes repræsenterer de to fortolkninger forskellige syn p˚a, hvad der er konstruktivt. Forskellighederne viser sig især i forbindelse med ekstensionalitet, Markovs princip og prin- cippet om uafhængighed af præmis. Vi viser, at det er et subtilt emne at kombinere disse ideal-elementer,og at Markov’s princip og princippet om uafhængighed af negeret universelle præmisser i særdeleshed er uforlignelige med hensyn til beregnbarhed. Med hensyn til beregnbart indhold af klassiske beviser inden for typet aritmetik sammen- ligner vi i en ekstensionel kontekst (i) metoden givet ved negativ oversættelse + Dialectica fortolkning med (ii) metoden givet ved negativ oversættelse + A-oversættelse + modificeret realiserbarhed. Ingen af disse metoder kan benyttes i fuld udstrækning inden for E-PAω, da E-HAω ikke er lukket under Markovs regel. Metoden baseret p˚aDialectica fortolkningen kan derimod anvendes, hvis det kun kræves, at teorien har svag ekstensionalitet. Til sidst introduceres en ny variant af Dialectica fortolkningen. Denne benyttes til at vise (de velkendte) eksistensegenskab, disjunktionsegenskab og andre egenskaber med hensyn til lukkethed for typet intuitionistisk aritmetik inklusive en ekstension. S˚aledes vises det, at funktional fortolkning ogs˚akan benyttes til dette form˚al. iv Preface This master’s thesis is written for obtainment of a master’s degree in both mathematics and philosophy. Hence, the present text treats topics from both fields, as they are connected in logic. The specific purpose is to give a systematic analysis of certain prima facie non- constructive elements of mathematics by using different interpretations and translations pro- vided by proof theory. My hope is that the reader will acknowledge this to have significance both within mathematics and philosophy. My gratitude goes to my supervisors: Ulrich Kohlenbach in mathematics and Stig An- dur Pedersen in philosophy. Ulrich Kohlenbach has been most generous with his time and expertise, and all results presented here are obtained in collaboration with him. Stig Andur Pedersen introduced me several years ago to logic and proof theory and to a very fruitful, I think, view on mathematics. They have both in a very friendly and constructive way sup- ported and guided me. I also want to thank Ulrich Bergerfor giving me the idea, that one should give a Dialectica interpretation within natural deduction. I have gained a lot in pursuing this. With respect to the final proof reading I want to thank Vincent F. Hendricks for his thorough treatment; also Dan Temple adjusted my language. Finally, thanks also to Jesper H. Andersen and Ivar Rummelhoff for useful comments. v Contents 1 Setting and Object of Survey 1 1.1 Hilbert’sprogramandviewonmathematics . ...... 3 1.2 Failure of Hilbert’s program—failureof separation . ........... 5 1.3 TheproofinterpretationbyHeyting . ..... 7 1.4 G¨odel’sviewonaconstructivefoundation . ........ 9 1.5 Motivationforandfocusofthisthesis . ...... 11 2 Introduction to Godel’s¨ Dialectica Interpretation 15 2.1 A Hilbert system for weakly extensional Heyting arithmetic in all finite types 15 ω 2.2 Axioms and rules of WE-HAH ......................... 17 ω 2.3 WE-TH as the quantifier free subsystem of WE-HAH ............. 21 2.4 Definition andanalysis of Dialecticatranslation . ........... 22 ω 2.5 Interpretation theorem for WE-HAH ...................... 27 3 Interpretation Theorems within Natural Deduction 29 ω 3.1 Formulation of WE-HAND and WE-TND ..................... 29 3.2 Discussionofthedeductiontheorem . ..... 33 3.3 Translationofderivationsunderassumptions . ......... 36 ω 3.4 Interpretation theorem for WE-HAND ...................... 37 3.5 Relation between the Dialectica interpretation and the Diller-Nahm interpre- tation ...................................... 48 3.6 Intuitionistic linear logic and the Dialectica interpretation.. .... ... .. 49 ω ω ω 3.7 Interpretation theorem for WE-HA + MP + IP + AC ........... 51 ω ω ω 3.8 The non-constructive theory WE-HA + IP + MP∀ ............. 52 ¬∀ 4 Classical Arithmetic and Kuroda’s Negative Translation 55 4.1 Formulation of WE-PAω ............................ 55 4.2 From classical to intuitionistic logic: Kuroda’s negativetranslation . 56 ω 4.3 Interpretation theorem for WE-PA + QF-AC and consistencyof the theory . 59 4.4 Extractiontheoremandconservativeness . ........ 60 4.5 The philosophical significance of interpretation theorems........... 62 4.6 PA as a subsystem of WE-PAω ......................... 65 4.7 The no-counterexample interpretation of Peano arithmetic .......... 65 5 Modified Realisability, A-translation and Applications 69 5.1 Definition of E-HAω andmodifiedrealisability . 70 ω ω 5.2 Realisability interpretation of E-HA + AC+ IPef ............... 72 5.3 Contraction and negatively occurring universal quantifiers .......... 74 5.4 Markov’s principle intuitionistically unprovable . ............. 75 5.5 E-HAω IPω ACnotclosedunderMarkov’srule . 75 ± ef ± vii viii Contents 5.6 The Friedman-Dragalin A-translationforformulasofHA . 77 5.7 Realisability with truth: Closure properties . ........... 80 6 Closure under Rules by Functional Interpretations 84 6.1 There is no ‘Dialectica-with-truth’ interpretation of WE-HAω ......... 84 6.2 Definition and soundness of Q-translation ... .... .... ... .... 86 6.3 Closure properties of intuitionistic arithmetic showed by Q-translation . 98 6.4 Q-interpretationisnotclosedunderdeductions . ...... 99 ω ω ω 6.5 Closure properties of WE-HA + IP + MP + AC+ Γ byDialectica . 100 ω ω ω 6.6 Constructiveness of WE-HA + IP ∀+ MP + AC+ Γ .............101 ∀ 7 Conclusions 102 7.1 Interpreting mathematics by methods from proof theory . ...........103 7.2 Evaluation of modified realisability and Dialectica interpretation . 105 7.3 Twostrongconstructivetheories . .106 7.4 Dialectica as a tool in proof mining and reductive proof theory . 106 7.5 Different interpretations—different validations: Mathematics and language . 108 Bibliography 110 CHAPTER 1 Setting and Object of Survey This thesis concerns proof theory and the significance of certain proof theoretical investiga- tions. Proof theory is the part of mathematical logic which studies the concepts of mathemat- ical proofs and mathematical provability. Proofs are an indispensable part of mathematics and therefore proof theory is also a study of the foundations of mathematics. Let us elaborate on the connection between proof theory and philosophy of mathematics. First, proof theory studies proofs as formal objects whereas, generally, mathematical proofs are informal and to a certain degree imprecise. There is, however, in practice a close affinity between formal and informal proofs. Parts of proofs used in mathematical arguments can be more or less formal. But in case there is a disagreement on a proofonecan try to make the problematic part of the proof more formal in order to be precise on the matter. One can, in other words, formalise in different degrees and the complete formal proof can be viewed as some kind of idealisation of the mathematical proof. It is in this respect proof theory studies mathematical proofs.
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