Jumping in Arithmetic

Abstract

In this paper we study a new relation between sentences: the jump relation. The idea of the jump relation is based on an analysis of Feferman's Theorem that the inconsistency of a theory U is interpretable over U. The jump relation is based on a converse of Feferman's Theorem: if a sentence is interpretable over a theory U, it is, in a sense, an inconsistency statement over U. We introduce an antipode of the inconsistency statement the presistency statement. The jump relation allows one to 'jump' from persistencies to inconsistencies. We show that for a wide classes of theories U the jump relation coincides with interpretability over U and for an even wider class it coincides with Π1-conservativity over U. Thus, the jump relation provides a new way of looking at interpretability and Π1-conservativity. On the other hand, we will show that the jump relation admits variations that are distinct from interpretability and Π1-conservativity. We show that the jump relation satisfies the interpretability logic ILM.