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

This paper is a presentation of a status quaestionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.

Vardanyan’s Theorem states that the set of PA-valid principles of Quantified Modal Logic, QML, is complete Π 0 2. We generalize this result to a wide class of theories. The crucial step in the generalization is avoiding the use of Tennenbaum’s Theorem.

In the first part of the paper we discuss some conceptual problems related to the notion of proof. In the second part we survey five major open problems in Provability Logic as well as possible directions for future research in this area. 1

Abstract. For finitely axiomatized sequential theories F and reflexive theories R, we give a characterization of the relation `F interprets R ' in terms of provability of restricted consistency statements on cuts. This characterization is used in a proof that the set of IIl (as well as E1) sentences yr s.t. GB interprets ZF + Tr is E3-complete. Relative interpretability among formal theories has been particularly well studied in two specific cases: that of finitely axiomatized sequential theories (see Smorynski [14], Pudlak [11], Visser [16] etc.), and of reflexive, esp. essentially reflexive theories (see Lindstrom [7], [8] etc.). We have nice characterizations of the interpretability relation between a pair of theories

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.