Classical and Intuitionistic Models of Arithmetic

Abstract

Given a classical theory T, a Kripke structure K = (K,≤,(Aα)α∈ K) is called T-normal (or locally T)if for eachα∈ 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 modelsof 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 modelsarein 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. © 1996 by the University of Notre Dame. All rights reserved.