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.
