Classical and Intuitionistic Models of Arithmetic

Abstract

Given a classical theory T, a Kripke structure ${\bf K} = (K, \leq, (A_{\alpha})_{\alpha \in K})$ is called T-normal (or locally T) if for each $ \alpha \in K $, $ A_{\alpha} $ 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, \leq) $ are locally ${\bf PA}$. They also proved that models of ${\bf HA}$ over the frame $ (\omega, \leq) $ contain infinitely many Peano nodes. We will show that such models are in fact ${\bf PA}$-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 ${\bf HA}$.