Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbert-style programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove

Abstract. Is it possible to give a coordinate free formulation of the Second Incompleteness Theorem? We pursue one possible approach to this question. We show that (i) cutfree consistency for finitely axiomatized theories can be uniquely characterized modulo EA-provable equivalence, (ii) consistency for finitely axiomatized sequential theories can be uniquely characterized modulo EA-provable equivalence. The case of infinitely axiomatized ce theories is more delicate. We carefully discuss this in the paper. 1.

Abstract. In this paper we give an informally semi-rigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion m-sequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.

Abstract. In this paper we study the interpretations of a weak arithmetic, like Buss ’ theory S1 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