Not all Kripke models of $\sf HA$ are locally $\sf PA$

Abstract

Let K be an arbitrary Kripke model of Heyting Arithmetic, HA. For every node k in K, we can view the classical structure of k, M k as a model of some classical theory of arithmetic. Let T be a classical theory in the language of arithmetic. We say K is locally T, iff for every k in K, M k |= T. One of the most important problems in the model theory of HA is the following question: Is every Kripke model of HA locally PA? We answer this question negatively. We provide two new Kripke model constructions for this matter. The first one is a direct construction of a Kripke model K HA + ECT0 (ECT0 stands for Extended Church Thesis) with the root r such that Mr |= I∆1 and hence K is not even locally I∆1. Not only the existence of this model completely solves the problem, but this result is also almost tight in terms of the power of induction axioms that can be failed in M k because every node of a Kripke model of HA classically satisfies induction for formulas that are provably ∆1 in PA. The second Kripke model construction is an implicit way of doing the first construction which works for any reasonable consistent intuitionistic arithmetical theory T with a recursively enumerable set of axioms that has Existence property. From the second construction we construct a Kripke model K HA + ¬θ + MP (θ is an instance of ECT0 and MP is Markov principle) with the root r such that Mr |= I∆1. Also, we will prove that every countable Kripke model of intuitionistic firstorder logic can be transformed into another Kripke model with the full infinite binary tree as the Kripke frame such that both Kripke models force the same sentences. So with the previous result, there is a binary Kripke model K of HA such that K is not locally PA.