Interpretations over Heyting's Arithmetic

Abstract

In this paper we experiment with a rather general notion of “interpretation in constructive arithmetical theories”. We prove a number of elementary properties of the notion introduced. We prove a number of negative results for interpretations that commute with disjunction. These negative results diverge markedly from what is known in the classical case. We briefly consider interpetations in formula classes of bounded complexity. In an appendix we show how to do (the interpretation version of) the Henkin construction for Intuitionistic Predicate Logic inside Peano Arithmetic. This construction cannot be given in Heyting’s Arithmetic.