U.U.D.M. Project Report 2016:47 Double Negation Interpretations for Typed Logical Systems Jonne Mickelin Sätherblom Examensarbete i matematik, 15 hp Handledare: Erik Palmgren, Stockholms universitet Ämnesgranskare: Vera Koponen Examinator: Jörgen Östensson December 2016 Department of Mathematics Uppsala University Double Negation Interpretations for Typed Logical Systems Jonne Mickelin S¨atherblom November 26, 2016 Abstract The G¨odel-Gentzen negative translation provides a method of proving conservativity of classical logic over intuitionistic logic for a wide class of formulas. Together with Dragalin's and Friedman's A-translation this can be extended to simple existence statements. We discuss how to apply this to translate theorems from classical Peano arithmetic as well as a stronger arithmetical theory based on G¨odel'ssystem T . Finally, we mention how the translations can be used to extract algorithms from classical proofs, and give a negative result in form of a theory where the negative translations cannot be used to translate theorems into constructive logic. Sammanfattning G¨odel-Gentzens negativa ¨overs¨attning ger en metod f¨or att visa att klassisk logik ¨ar konservativ ¨over intuitionistisk logik f¨or vissa klasser av formler. Tillsammans med Dragalin-Friedmans A-¨overs¨attning kan man visa att detta g¨aller ¨aven f¨or enkla existensp˚ast˚aenden.Vi diskuterar hur dessa kan till¨ampas p˚asatser ur klassisk Peanoaritmetik samt en starkare aritmetisk teori baserad p˚aG¨odels system T . Slutligen n¨amner vi hur man kan anv¨anda ¨overs¨attningarna f¨or att extrahera algoritmer ur klassiska bevis, samt ger ett negativt resultat i form av en teori d¨ar negativa ¨overs¨attningar inte kan anv¨andas f¨or att ¨overs¨atta satser till konstruktiv logik. Contents 1. Introduction 4 1.1. Many-sorted logic . .6 1.2. Free variables and substitution . .7 1.3. Formula schemas and theories . .8 1.4. Classical, intuitionistic and minimal logic . .9 2. The G¨odel-Gentzen negative translation 12 3. Consequences of the negative translation 16 3.1. Properties of schemas . 16 3.2. Conservativity results . 20 3.3. Identifying wiping, spreading and isolating formulas . 21 4. Heyting and Peano arithmetic 22 4.1. The arithmetical hierarchy . 22 4.2. Towards translations of arithmetic . 24 5. Provably recursive functions 25 5.1. Markov's rule . 25 5.2. The Dragalin-Friedman A-translation . 27 5.3. Conservativity for Π2-formulas in HA ............................ 29 6. Extracting algorithms 31 6.1. G¨odel'ssystem T ........................................ 31 6.2. Interpreting atomic formulas and optimizing extracted programs . 33 6.3. Extraction of program terms . 33 6.4. Formulas as specifications . 37 6.5. Applications to classical proofs . 40 7. Finite type arithmetic and the axiom of choice 42 7.1. A model of E-HA! ....................................... 42 7.2. The axiom of choice and constructive mathematics . 43 7.3. Translations of HA! ...................................... 45 7.4. Translations of choice . 46 A. Proof of Lemma 1.1 49 B. Proof of the soundness theorem for program extraction 52 3 1. Introduction Many proofs in mathematics make use of the so-called law of excluded middle (LEM), which says that for any statement ', either ' or its negation holds. In symbols: ' _:': (LEM) Proofs that rely on this principle often appear somewhat toothless, as illustrated by the following example: Theorem. There are two irrational numbers a; b such that ab is rational. p p 2 Proof. Consider the number 2 . It is either rational or irrational, by the law of excluded middle.p In the p p p 2 p first case, we can pick a = b = 2, since 2 is irrational. In the second case, we set a = 2 ; b = 2, since both are irrational and p p 2 p p p 2 p 2· 2 p 2 ab = 2 = 2 = 2 = 2: p p p 2 p 2 p 2 While the proof tells us that either 2 or 2 is irrational, it gives us no way of knowing which one it is. By extension, merely having a proof of a theorem that claims the existence of an object with a given property generally gives us no way to actually find such an object. The law of excluded middle was one of the protagonists of a dispute in the mathematical community during the beginning of the last century. The discovery of paradoxes in Cantor's naive set theory had caused some distress in the mathematical community, with regards to the foundations on which they built their practice. Possibly fuelled by these doubts, a topologist by the name of L.E.J. Brouwer presented a novel philosophy of the foundations of mathematics. Mathematics, he argued, should have a basis in mental constructions rather than mechanical manipulations of strings of symbols on a paper. Moreover, mathematics is a purely human endeavour, and not an exploration of objects living in some ideal world. This put him at odds with both the traditional Platonist philosophy and Hilbert's new ideas of Formalism. Among other, perhaps more controversial ideas, Brouwer argued that proofs themselves should be given as constructions. In his intuitionistic mathematics, we regard a statement '(x) as \true" if we can construct a proof of '(x), and \false" if we can construct a proof that the assumption '(x) leads to a contradiction. With this in mind, we cannot claim that the law of excluded middle holds without simultaneously giving either a proof or a refutation for every mathematical statement. Although Brouwer himself only implicitly formulated his idea of what constitutes a construction, some of his students gave a more formal description of how to construct proofs. This is the so-called Brouwer{Heyting{Kolmogorov (BHK) interpretation: (i) A proof of a conjunction ' ^ is a pair containing a proof of ' together with a proof of . (ii) A proof of a disjunction ' _ is either a proof of ' or a proof of , and some information that indicates which of the two it is. (iii) A proof of an implication ' ! is a method that transforms a proof of ' into a proof of . (iv) A proof of a universal quantification 8x:'(x) is a method that, for any element d, gives a proof of '(d). 4 (v) A proof of an existential quantification 9x:'(x) is an element d and a proof of '(d). (vi) There is no proof for absurdity ?. A proof of a negation :' := ' !? is then a method that given some hypothetical proof of ' produces a proof of absurdity. A proof of the general statement ' _:' under the BHK interpretation consists of either a proof of ' or a proof of :', together with a label that tells us which of the two it is. This means that we cannot prove the disjunction until we can claim to have a proof of one of the two disjuncts. Besides LEM, several other commonly used principles fail to have a construction. For example, the least number principle that says that every predicate on the natural numbers is well-founded: 9x 2 N:'(x) ! 9x 2 N:('(x) ^ 8y < x::'(y)): (LNP) Consider what would happen if '(x) was x = 2 _ (x = 1 ^ ) _ (x = 0 ^ : ): for some . We know that '(2) is true, so by LNP, there is a smallest number x satisfying '. This number cannot be 2, since then we would have : and :: (otherwise x = 1 or x = 0 would satisfy '(x)). But this is a contradiction. Thus, x = 0 or x = 1, but these are equivalent to and : respectively, so _: . So the least number principle implies LEM. Note, however, that we can prove ::LEM and ::LNP, which means we cannot possibly hope to refute these principles. All we have showed is that the principles are not compatible with Brouwer's idea of what should constitute a proof. Such principles are called weak counterexamples, to distinguish them from ordinary counterexamples, which immediately refute a formula. Philosophical objections aside, one has found connections between constructivism and computer science: While the phrase \method" in the BHK-interpretation is left undefined, we can interpret it to mean \algorithm". As such, the witnesses for constructive formulas are computer programs! This is, among other things, the basis for constructive type theory, which has seen some rise in popularity in recent years. This text will be concerned with translations of classical formulas into constructive logic. As it turns out, a considerable subset of classical logic, and indeed classical mathematics, can be made constructive. We can do this by trying to manually adapt non-constructive proofs, but for sufficiently well-behaved theories and theorems, the process can be automated. In general, for theories T1 and T2 in languages L1 ⊆ L2 and with possibly different underlying logical systems, we say that T2 is a conservative extension of T1 if T1 ⊆ T2 and for any formula ' 2 L1 we have T2 ` ' ) T1 ` ': If the implication holds for a class Γ of formulas, we say that T2 is a conservative extension of T1 with respect to Γ-formulas. In the following, we will prove for several classes Γ that certain classical theories are conservative over their constructive counterparts. While it is assumed that the reader has some basic familiarity with formal logic, we dedicate Section 1.1 to defining the logical systems we will use and summarizing some basic notions regarding formulas and derivations. In Section 2 we define the G¨odel-Gentzen negative translation and show how classical logic can be embedded within intuitionistic logic. Section 3 investigates sufficient conditions for proving conservativity of the classical version of a theory over its intuitionistic counterpart, with respect to certain formulas. 5 This will be applied to Peano arithmetic beginning in Section 4, and extended in Section 5 using the so-called Dragalin-Friedman A-translation.