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.

In this paper we expose a method for building models for interpretability logics. The method can be compared to the method of taking unions of chains in classical model theory. Many applications of the method share a common part. We isolate this common part in a main lemma. Doing so, many of our results become applications of this main lemma. We also briefly describe how our method can be generalized to modal logics with a different signature. With the general method, we prove completeness for the interpretability logics IL, ILM, ILM0 and ILW ∗. We also apply our method to obtain a classification of the essential Σ1-sentences of essentially reflexive theories. We briefly comment on such a classification for finitely axiomatizable theories. As a digression we proof some results on self-provers. Towards the end of the paper we concentrate on modal matters concerning IL(All), the interpretability logic of all reasonable arithmetical theories. We prove the modal incompleteness of the logic ILW ∗ P0. We put forward a new principle R, and show it to be arithmetically sound in any reasonable arithmetical theory. Finally we make some general remarks on the logics ILRW and IL(All).