Classical and Intuitionistic Models of Arithmetic

Abstract

Given a classical theory T, a Kripke structure K = (K, ≤, ( A α ) α∈K ) is called T-normal (or locally T) if for each α ∈ K, A α 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, ≤, ( A α ) α∈K ) such that (K, ≤) is a (nonempty) partial order (called the frame of K) and for eachnecessarily normal, that is, = α need not be true equality on A α ), with the proviso that the following monotonicity conditions be fulfilled. Whenever α ≤ β, thenThroughout 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 A α is a normal structure (i.e., = α is true equality on A α ) and that for α, that is, L plus constant symbols a for each element a ∈ A α . We often write 'α |= ϕ' instead of ' A α |= ϕ', meaning that A α classically satisfies ϕ, whereas 'α ϕ' means that ϕ is forced at α in K.

Weak Arithmetics and Kripke Models

“…So we get a non-I Π n -normal Kripke model of i Φ n . On the other hand, as it was observed in [1] (in the proof of 2.1(iv)), that any theory of the form H(T ) is closed under Friedman's translation and so by [10], each finite Kripke model of it is H(T ) c -normal. So, by [7, Lemma 1.2], it must be T -normal.…”

A Semantic Approach to Conservativity

“…This fact follows from the properties of axiomatization of the class of T c -normal Kripke models and was proven in [4]. For details, see also [13]. Now, since in HA all atomic formulae are decidable, P(HA) is exactly the set of Σ 1 formulae of the language of arithmetic.…”

Every Rooted Narrow Tree Kripke Model of HA is Locally PA

“…al in [1] give a positive partial answer for some special tree Kripke models of HA. They proved that every finite tree Kripke model of HA is locally PA and furthermore, every ω-chain Kripke model of HA has infinitely many nodes which are classical model of PA. K. Wehmeier in [3] improved these results to all nodes of an ω-chain Kripke models of HA and to Kripke model of HA of the following frames: …”