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

Contents 1. Introduction, Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 360 2. Modal logic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 3. Proof of Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 4. Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 5. Propositional theories and Magari-algebras . . . . . . . . . . . . . . . . . . . . . 368 6. The extent of Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 369 7. Classification of provability logics . . . . . . . . . . . . . . . . . . . . . . . . . . 371 8. Bimodal and polymodal provability logics . . . . . . . . . . . . . . . . . . . . . 374 9. Rosser orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 10.

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.

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