Interpretability in

In this paper, we study (), the interpretability logic of . As is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to : () is not or . () does, of course, contain all the principles known to be part of (All), the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of and see what their consequences in the modal logic () are. These properties are reflected in the so-called Beklemishev Principle , and Zambella’s Principle , neither of which is a part of (All). Both principles and their interrelation are submitted to a modal study. In particular, we prove a frame condition for . Moreover, we prove that follows from a restricted form of . Finally, we give an overview of the known relationships of () to important other interpretability principles.