The Arithmetics of a Theory

Abstract

In this paper we study the interpretations of a weak arithmetic, like Buss’s theory ${\mathsf{S}}_2^1$, in a given theory $U$. We call these interpretations the arithmetics of $U$. We develop the basics of the structure of the arithmetics of $U$. We study the provability logic(s) of $U$ from the standpoint of the framework of the arithmetics of $U$. Finally, we provide a deeper study of the arithmetics of a finitely axiomatized sequential theory.