This paper from 2008 is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the foundations are laid for later results. These foundations consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic ${ \textbf{ IL}}$. Next, the method is applied to obtain new results: the modal completeness of the logic ${ \textbf{ IL}}{ \sf M_0}$, and modal completeness of ${ \textbf{ IL}}({ \sf W^*})$.
Categories: math.LO