Cuts, consistency statements and interpretations

Abstract

Interpretability in reflexive theories, especially in PA, has been studied in many papers; see e.g. [3], [6], [7], [10], [11], [15], [26]. It has been shown that reflexive theories exhibit many nice properties, e.g. (1) if T, S are recursively enumerable reflexive, then T is interpretable in S iff every Π1 sentence provable in T is provable in S; and (2) if S is reflexive, T is recursively enumerable and locally interpretable in S (i.e. every finite part of T is interpretable in S), then T is globally interpretable in S (Orey's theorem, cf. [3]).In this paper we want to study such statements for nonreflexive theories, especially for finitely axiomatizable theories (which are never reflexive). These theories behave differently, although they may be quite close to reflexive theories, as e.g. GB to ZF. An important fact is that in such theories one can define proper cuts. By a cut we mean a formula with one free variable which defines a nonempty initial segment of natural numbers closed under the successor function. The importance of cuts for interpretations in GB was realized already by Vopěnka and Hájek in [30]. Pioneering work was done by Solovay in [24]. There he developed the method of “shortening of cuts”. Using this method it is possible to replace any cut by a cut which is contained in it and has some desirable additional properties; in particular it can be closed under + and ·. This introduces ambiguity in the concept of arithmetic in theories which admit proper cuts, namely, which cut (closed under + and ·) should be called the arithmetic of the theory? Cuts played the crucial role also in [20].