GÃ¶del's Dialectica Interpretation

Gödel’s **Dialectica** **Interpretation**Klaus Frovin JørgensenSection for Philosophy and Science Studies, RUCMay 6, 20101 / 33

Question to be AnsweredKurt Gödel devised (around 1941) an interpretation of intuitionisticarithmetic into a calculus of functionals.Together with Gödel’s ¬¬−translation (1933) this was also aconsistency proof classical arithmetic.What are the philosophical conclusions?2 / 33

The Context of the **Interpretation**The 1920s and 1930s:• The Grundlagestreit• Hilbert’s programme (1920s)• The development of intuitionistic logic (Heyting, Kolmogorov∼ 1930)• The incompleteness theorems (1931) and the Gödel/Gentzen¬¬-translation (1933)• Church’s and Turing’s negative answer to theEntscheidungsproblem (1935-6).• Gentzen’s proof of consistency of formal number theory (1936)3 / 33

Heyting’s Proof **Interpretation**p : ASome of the clauses of Heyting’s interpretation are:(→) p : A → B iff p is a construction taking any q such that q : Ainto p(q) such that p(q) : B.(∨) p : A ∨ B iff p is a pair (p 0 , p 1 ), p 0 ∈ {0, 1} and p 1 : A ifp 0 = 0 and p 1 : B if p 0 = 1. q : A into p(q) such thatp(q) : ⊥.(∀) p : ∀xA(x) iff p is a construction taking any t from theintended domain into p(t) such that p(t) : A(t).(∃) p : ∃xA(x) iff p is a pair (p 0 , p 1 ), where p 0 is an object of thedomain and p 1 : A(p 0 ).4 / 33

Intuitionistic Propositional LogicThe following is Spector’s (1962) formulation:A → A, ⊥ → A,A → A ∨ B, B → A ∨ B,A ∧ B → A, A ∧ B → B,A A → BBA ∧ B → CA → (B → C)A → B B → CA → CA → (B → C)A ∧ B → CA → B A → CA → B ∧ CA → C B → CA ∨ B → C5 / 33

Tertium Non DaturTertium non datur:A ∨ ¬Ais not sound under the proof interpretation.6 / 33

Intuitionistic Quantifier RulesB → A(b)B → ∀xA(x)A(t) → ∃xA(x),∀xA(x) → A(t),A(b) → B∃xA(x) → BHere b is eigenvariable meaning that b is not allowed to occur freein B.7 / 33

Disjunction Property and ExistenceProperty• Existence property: If S ⊢ ∃xA(x) then S ⊢ A(t) for a certainterm t.• Disjunction property: If S ⊢ A ∨ B, for A, B closed thenS ⊢ A or S ⊢ B.8 / 33

Impredicative Methods (1/2)A definition is impredicative if it refers to a collection whichcontains the object to be defined. If one sees definitions assomehow creating or constructing, then circularity is involved.For example the ‘least upper bound’ of a set is defined to be the‘smallest among the upper bounds’. This is seen, for instance, inthe completeness axiom:Any non-empty subset of real numbers bounded above has a leastupper bound.Example: There exists a real number x such that x 2 = 2.Another example is the Russell set R.9 / 33

Impredicative Methods (2/2)Yet an example is the intuitionistic meaning of implication: Theproof interpretation). We know A → B precisely when we knowwhat may count as a proof of A → B. A proof of A → Btransforms any proof of A into a proof of B.10 / 33

The System Σ (1/3)The ground type consits of the natural numbers. There aresymbols for zero and successor and variables of all types.If F is an operation of type σ → τ this is written as F σ→τOperations in T are defined from combinators (which introduceλ-abstraction)K(x, y) = xS(x, y, z) = x(z)(yz)The combinators give us λ-abstraction λx.t for terms t, with thefollowing equality:(λx.t[x])s = t[s].Moreover, we have primitive recursionR(x, y, 0) = xR(x, y, (z + 1)) = y(R(x, y, z), z)11 / 33

The System Σ (2/3)Quantifier free inductionA(0) A(x 0 ) → A(Sx 0 )A(x 0 )Substitution:A(x σ )A(t σ )12 / 33

The System Σ (3/3)We can do elementary arithmetic i Σ:x + y :≡ R 0 x(λw, u.Sw)yprd :≡ λx.R 0 0(λw, u.u)xx . − y :≡ R 0 x(λw, u.prd(w))y|x − y| :≡ (x . − y) + (y . − x)Cond :≡ λx σ , y σ , z 0 .R σ x(λv σ , w 0 .y)zmax :≡ λx 0 , y 0 .Cond(y, x, x . − y)min :≡ λx 0 , y 0 .Cond(x, y, x . − y)The prime formulas of Σ are decidable (proved by induction on thecomplexity of the terms). Therefore, Σ actually has classical logicwhich can, moreover, be represented in the system.13 / 33

Definition of D-translation (1/2)To each formula A of L(HA) is now associated its **Dialectica**translation A D which is a formula of Σ.A D ≡ ∃x∀yA D (x, y),where A D is quantifier free. Intuitively: If A is provable thenaccording to the translation A D there are x making A D ‘true’ forany y.The lengths and types of the fresh variables x and y depend onlyon the logical structure of A.14 / 33

Definition of D-translation (2/2)Let A D have the form ∃x∀yA D (x, y) and B D the form∃u∀vB D (u, v).Definition.A D :≡ A D :≡ A, if A is prime,(A ∧ B) D :≡ ∃x, u∀y, v(A D (x, y) ∧ B D (u, v)),(A ∨ B) D :≡ ∃z 0 , x, u∀y, v ( (z = 0 → A D (x, y)) ∧(z ≠ 0 → B D (u, v)) ) ,(∃zA(z)) D :≡ ∃z, x∀yA D (x, y, z),(∀zA(z)) D :≡ ∃X∀y, zA D (Xz, y, z),(A → B) D :≡ ∃U, Y ∀x, v ( A D (x, Y xv) → B D (Ux, v) ) .15 / 33

The Translation of ‘→’Assume ∃x∀yA D (x, y) → ∃u∀vB D (u, v). A reasonable reading is:∃U∀x ( ∀yA D (x, y) → ∀vB D (Ux, v) ) . (1)What could a possible interpretation of ∀xC(x) → ∀yD(y) be?Given any counter-example to D we can construct acounter-example to C, i.e. ∃X∀y ( ¬D(y) → ¬C(Xy) ) . Thisimplies:∃U∀x∃Y ′ ∀v ( ¬B D (Ux, v) → ¬A D (x, Y ′ v) ) . (2)The quantifier free formulas are stable. Therefore, ‘by’ the proofinterpretation we get:∃U, Y ∀x, v ( A D (x, Yxv) → B D (Ux, v) ) .16 / 33

On ‘Constructive Meaning’The following principles are not constructively valid according tothe proof interpretation by Heyting.• Different forms of independence-of-premise:( A → ∃yB(y)) → ∃y( A → B(y)) ,where y /∈ FV(A) and different restrictions on A.• Markov’s principle: ¬¬∃xA qf (x) → ∃xA qf (x).Gödel’s translation changes the intuitionistic meaning:HA ω + IP ω ∀ + MP ω + AC ⊢ A ↔ A D .This is not necessarily a bad thing.17 / 33

SoundnessTheorem [Gödel 1941].If HA ⊢ A, then Σ ⊢ A D (T , y),where T is a sequence of terms which can be extracted from aproof of A in HA.18 / 33

Examples from the Proof of Soundness(1/2)The proof is by induction on the length of the proof of A in HA.Case 1. Axiom A → A. This translates to∃X, Y ∀x, y(A D (x, Y xy) → A D (Xx, y)).From this we see that with T 1 :≡ λx.x and T 2 :≡ λx, y.y we haveΣ ⊢ A D (x, T 2 xy) → A D (T 1 x, y).19 / 33

Examples from the Proof of Soundness(2/2)Case 2. Modus Ponens. Assume as induction hypothesis(i)(ii)Σ ⊢ A D (T 1 , y),Σ ⊢ A D (x, T 2 xv) → B D (T 3 x, v),for given T 1 , T 2 , and T 3 . Find T 4 such that Σ ⊢ B D (T 4 , v). Setx in (ii) to T 1 and let y in (i) be T 2 T 1 v. Then use MP (in Σ) toobtainΣ ⊢ B D (T 3 T 1 , v).Let T 4 be equal to T 3 T 1 .Note the similarities to the proof interpretation.20 / 33

Gödel’s ResultsIn 1941 the following results are mentioned:1 For a certain quantifier free formula A(x) letC ≡ ¬∀x(A(x) ∨ ¬A(x)). Then HA + C is consistent.2 If HA proves ∃xA(x) then Σ proves the translated formulaA D (t), for a term t.3 ¬¬-translation (1933) together with the new interpretationproves consistency of classical arithmetic relative to T.The interpretation was ultimately published in **Dialectica** in 1958:“Über eine bisher noch nicht benützte Erweiterung des finitenStandpunktes”.21 / 33

Philosophy (1/3)22 / 33

Weak versus Strong Counterexamples(1/3)π is transcendental, but we can approximate π arbitrarily well:π = 4 1 − 4 3 + 4 5 − 4 7 + 4 9 − 411 + · · ·Let A be the statement: “There exists one hundred 9s in a row inthe decimal expansion of π.”23 / 33

Weak versus Strong Counterexamples(2/3)1 Start the computation of π.2 We start writing a real number a. The first digit is 0 followedby a point:0.3 Construction of the n-th digit of a:• If the decimal expansion of π up to digit number n − 1 has notverified A, then the n-th digit of a is 0,• Otherwise 1Is a = 0 or a ≠ 0?24 / 33

Weak versus Strong Counterexamples(3/3)Gödel produces with his interpretation a strong counterexample.¬∀x(A(x) ∨ ¬A(x)) is demonstrably incompatible with classicallogic.25 / 33

Constructivity is Understood Locally (1/3)The following principles are not constructively valid according tothe proof interpretation by Heyting.• Different forms of independence-of-premise:( A → ∃yB(y)) → ∃y( A → B(y)) ,where y /∈ FV(A) and different restrictions on A.• Markov’s principle: ¬¬∃xA qf (x) → ∃xA qf (x).26 / 33

Constructivity is Understood Locally (2/3)Different interpretations validate different principles:• Modified realisability validates full extensionality, IP ω ef and AC.• Functional interpretation validates MP ω , IP ω ∀ and AC.Can there be a better interpretation; one which is more optimalwith respect to these principles?27 / 33

Constructivity is Understood Locally (3/3)Two incompatible constructive theories:• The proof interpretation is a global interpretation (a rule ofthumb); locally one can accept more if the goal is computableexistence.• The combination of extensionality, Markov’s principle andrestricted forms of independence-of-premise is a subtle issue.WE-HA ω + IP ω ∀ + MP ω + AC + Γ,Γ is any set of universal true sentences; has existence property,disjunction property and is closed under various rules.E-HA ω + IP ω ef + AC + Γ,Γ is any set of true ∃-free sentences; has existence property,disjunction property and is closed under different rules, exceptMarkov’s rule.28 / 33

Philosophy (2/3)29 / 33

On the Existence and DisjunctionPropertiesGödel mentions the following result:• If HA proves ∃xA(x) then Σ proves the translated formulaA D (t), for a term t.But this does not suffice for proving the existence and disjunctionproperties for HA, as the original formula is not in generalintuitionistically provable from the translated. In 1945 Kleene (andNelson) proved that realisability by numbers can be used forshowing that.30 / 33

Philosophy (3/3):What should we think of the consistency proof?31 / 33

CoherenceGödel’s interpretation is paradigmatic with respect to coherence (Σis now called T):— PA is interpreted in T— T is consistent, we can prove strong normalisation, by:· Howard’s strong computability predicates(uses König’s lemma)· Tait’s method of ascribing ordinals < ε 0 to terms of T— Fits with Gentzen’s partial cut-elmination (which again fitswith Schütte’s full cut-elmination)— Tait’s proof of termination fits with Gentzen’s characterisationof PA as ε 0— The no-counter-example interpretation can be derived fromboth the **Dialectica** interpretation and Gentzen’scut-elimination32 / 33

Neurath’s ShipWelcome AboardWe cannot eliminate impredicativity. There is no secureArchimedian point—there is no cogito.We give up foundationalism: Instead we can account (I think) foractual mathematics—contemporary as well as historical.33 / 33