Interpretability over Peano Arithmetic

We investigate the modal logic of interpretability over Peano arith metic PA Our main result is an extension of the arithmetical complete ness theorem for the interpretability logic ILM This extension concerns recursively enumerable sets of formulas of interpretability logic rather than single formulas As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a theorem answering a question of Orey from All these results also hold for Zermelo Fraenkel set theory ZF