Transductions in Arithmetic

Abstract

In this paper we study a new relation between sentences: transducibility. The idea of transducibility is based on an analysis of Feferman's Theorem that the inconsistency of a theory U is interpretable over U. Transducibility is based on a converse of Feferman's Theorem: if a sentence is interpretable over a theory U, it is, in a sense that we will explain, an inconsistency statement for U over U. We show that, for a wide class of theories U, transducibility coincides with interpretability over U and, for an even wider class, it coincides with Pi_1-conservativity over U. Thus, transducibility provides a new way of looking at interpretability and Pi_1-conservativity. On the other hand, we will show that transducibility admits variations that are distinct from interpretability and Pi_1-conservativity. We show that transducibility satisfies the interpretability logic ILM.