MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

74 Robert Goldblattwhich shows that ✷p → ✷✷p is PA-valid and hence a G-theorem. However theother S4-axiom ✷p → p is not PA-valid, and indeed not even the formula ✷⊥ → ⊥is a G-theorem, since (✷⊥ → ⊥) φ isBew(0 ≠ 0) → 0 ≠ 0,which is not PA-provable by Gödel’s reasoning above.Robert Solovay [1976] demonstrated that G is identical to Segerberg’s logicK4W, discussed in section 5.3, which is characterised by the class of finite strictlyordered (i.e. transitive and irreflexive) Kripke frames. The validity of the axiomW, i.e.✷(✷p → p) → ✷p,follows from an answer given in [Löb, 1955] to a question raised by Leon Henkinin 1952 about the status of sentences that assert their own provability. Any PAformulaF (v) has fixed points: sentences σ for whichPA ⊢ σ ↔ F (σ)(this is usually called the Diagonalisation Lemma). A fixed point of Bew(v) hasPA ⊢ σ ↔ Bew(σ)so is equivalent to the assertion of its own provability. Must it in fact be provable? 63Löb answered this in the affirmative by proving thatif PA ⊢ Bew(σ) → σ, then PA ⊢ σ.Equivalently, if Bew(Bew(σ) → σ) is true then so is Bew(σ), i.e. the sentenceBew(Bew(σ) → σ) → Bew(σ)is true. But more strongly it can be shown that this sentence is PA-provable forany σ, including σ = α φ , giving the PA-validity of W.Solovay’s completeness theorem for G is a remarkable application of the machineryof arithmetisation and recursive functions to show that any finite strictlyordered frame (K, R) can be “embedded into Peano Arithmetic”. A recursivefunction h : ω → K is defined that is in fact constant, but which cannot be provento be constant in PA. Each element x of K is represented by a sentence σ x expressing“lim n→∞ h(n) = x”. This sentence is consistent with PA, i.e. PA ¬σ x .The construction has a flavour of self-referential paradox similar to that of Gödel’sincompleteness proof, because the sentences σ x are used to define the function hitself. But that is resolved by some version of diagonalisation. 64 The structure ofthe ordering R is represented in PA by the fact that if xRy thenPA ⊢ σ x → ¬Bew(¬σ y ),63 This is a generalisation of Henkin’s question: see [Smoryński, 1991] for discussion.64 Solovay’s argument used Kleene’s Recursion Theorem on fixed points in the enumeration ofpartial recursive functions.