Classical and Intuitionistic Models of Arithmetic

Given a classical theory T, a Kripke structure K = (K,≤, (Aα)α∈K ) is called T-normal (or locally T) if for each α ∈ K, Aα is a classical model of T. It has been known for some time now, thanks to van Dalen, Mulder, Krabbe, and Visser, that Kripke models of HA over finite frames (K,≤) are locally PA. They also proved that models of HA over the frame (ω,≤) contain infinitely many Peano nodes. We will show that such models are in fact PA-normal, that is, they consist entirely of Peano nodes. These results are then applied to a somewhat larger class of frames. We close with some general considerations on properties of non-Peano nodes in arbitrary models of HA. 1 Preliminaries A Kripke structure for a language L is a triple K=(K,≤, (Aα)α∈K ) such that (K,≤) is a (nonempty) partial order (called the frame of K) and for each α ∈ K, Aα is a classical L-structure Aα = (Aα,=α, (Rα)R∈L, ( fα) f ∈L) (not necessarily normal, that is, =α need not be true equality on Aα), with the proviso that the following monotonicity conditions be fulfilled. Whenever α ≤ β, then 1. Aα is a subset of Aβ ; 2. for every relation symbol R of L (including equality =), Rα ⊆ Rβ; 3. for every n-ary function symbol f of L, fα is fβ restricted to Aα. Throughout this paper, L will be some suitable version of the arithmetical language with or without symbols for all primitive recursive functions. Forcing, , is defined as usual. We are treating ⊥ as a basic connective (so that ⊥ counts as an atomic formula); negation is defined as ¬ψ :≡ ψ → ⊥. Since in HA atomic formulas are decidable, we assume without loss of generality for K|= HA that every Aα is a normal structure (i.e., =α is true equality on Aα) and that for α, β ∈ K, if α ≤ β, then Aα ⊆ Aβ (Aα is a substructure of Aβ) (cf. Markovic [3], Smorynski [4]). Lα is L(Aα), that is, L plus constant symbols a for each element a ∈ Aα. We often write ‘α |= φ’ instead of ‘Aα |= φ’, meaning that Aα classically satisfies φ, whereas ‘α φ’ means that φ is forced at α in K. Received November 8, 1995; revised June 25, 1996 CLASSICAL AND INTUITIONISTIC MODELS 453 Decidability of atomic formulas in HA also entails (cf. [3]) that for any K |= HA, every α ∈ K and every 0-sentence φ ∈ Lα, α φ ⇐⇒ α |= φ. A node α in some Kripke structure K is called classical if α forces every Lα-sentence of the form ∀x1 . . . xn(φ ∨ ¬φ). Note that in any Kripke structure, terminal nodes are classical. The following lemma gives some other characterizations of classical nodes. Lemma 1.1 Let K be any Kripke structure with the property that α ≤ β implies that Aα is a substructure of Aβ, and suppose α ∈ K. The following are equivalent. 1. α is a classical node. 2. α forces every L-sentence of the form ∀x1 . . . xn(φ ∨ ¬φ). 3. For every Lα-sentence φ, α φ ⇐⇒ α |= φ. 4. Whenever α ≤ β ≤ γ, Aβ ≺ Aγ . Proof: The equivalence of the first three conditions is well known and was already mentioned in [5]. It remains to show that condition 4 is equivalent to the others. First suppose that 1-3 hold. By 2 and the persistence of forcing, every β ≥ α is also classical. Let α ≤ β ≤ γ and let φ be an Lβ-sentence. First assume that Aβ |= φ. Since β is classical, β φ and hence γ φ. But γ is also classical and thus Aγ |= φ. Now assume that Aγ |= φ. Since γ is classical, γ φ. Now β is classical and β ¬φ (since β ≤ γ and γ φ), so β φ and Aβ |= φ. Now suppose that 4 obtains. We will prove by induction on φ that for each β ≥ α and each Lβ-sentence φ, β φ ⇐⇒ β |= φ. If φ is atomic, our claim follows by definition. The cases of conjunction, disjunction and existential quantification follow easily from the induction hypothesis. Let us consider the case of implication, say φ is of the form ψ → χ. First suppose β ψ → χ and β |= ψ. By the induction hypothesis, β ψ and hence β χ. By the induction hypothesis again, β |= χ. Now suppose that β ψ → χ. Then, for some γ ≥ β, γ ψ and γ χ. Then clearly β χ and thus, by the induction hypothesis, β |= χ. Now since γ ψ, by the i.h. γ |= ψ and, since Aβ ≺ Aγ , β |= ψ. The case of universal quantification can be treated analogously. Given L-formulas φ and ρ, the Friedman translation of φ by ρ, denoted φ, is obtained from φ by replacing each atomic subformula P in φ by P ∨ ρ (where it is understood that no variable occurring free in ρ is bound in φ). Some properties of the Friedman translation are the following. 1. ρ → φ is provable in intuitionistic logic. 2. Classically, φ ↔ φ ∨ ρ. 3. HA φ ⇒ HA φ. A formula φ is semipositive (cf. Buss [1]) if, whenever ψ → χ is a subformula of φ, ψ is atomic. In particular, only atomic formulas can occur negated (since negation is defined in terms of implication and falsum). Note that classically, every formula is equivalent to a semipositive one (simply eliminate subformulas ψ → χ in favor of ¬ψ ∨ χ and put the resulting formula in negation normal form). Semipositive sentences have the following property.