The arithmetics of a theory

Abstract

In this paper we study the interpretations of a weak arithmetic, like Buss' theory S^1_2, 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.