We investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILM$^\omega$. 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 partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).