A note on the interpretability logic of finitely axiomatized theories

Ill [6] Albert Visser shows that ILP completely axiomatizes all schemata about provabihty and relative interpretability that are prov-able in finitely axiomatized theories. In this paper we introduce a system called ILP ~ that completely axiomatizes the arithmetically valid principles of provability in and interpretabihty over such theories. To prove the arith-metical completeness of ILP ~ we use a suitable kind of tail models; as a byproduct we obtain a somewhat modified proof of Visser's completeness result. 1. Introduction In [5] Albert Visser introduces a logic ILP in a modal language s ~>) with a unary operator [3, to be interpreted arithmetically as provability, and a binary operator ~>, to be interpreted arithmetically as relative inter-pretability over some fixed theory U. In [6] he shows that ILP completely axiomatizes all schemata about provability and relative interpretability that are provable in E~ finitely axiomatized sequential theories U that extend IA0 + SupExp. In this paper we present a complete axiomatization, called ILP ~, of all true such schemata; on the way we obtain a somewhat modified proof of Visser's completeness result. The main difference between Visser's proof of the arithmetical completeness of ILP and ours, is that we use infinite Kripke-like models, instead of finite ones, to find arithmetical interpretations for unprovable modal fornm-las. The models we use are variations on the tail models for provability logic as developed by Albert Visser (cf. [4]). We think that the use of tail models in this setting is rather natural. The advantage of using these models is twofold. First of all, it allows us to set up things in such a way, that we can prove the arithmetical completeness of ILP and ILP ~ (almost) in one go. To understand the second advantage, recall that the arithmetical sentences needed to prove the arithmetical completeness of some given logic A are usually found by embedding models of A into arithmetic. If these models are finite, the embedding will only be partial, in the following sense.