Interpretability logics and large cardinals in set theory

This paper is a first exploration how the modal logic of interpretability can play a role in the study of large cardinals in set theory. First we reason that such a role is likely to be fruitful. Next, for two particular readings of the binary modality we study its repercussions for the corresponding logic. The first reading of A B is “Inside any Vκ where A holds we can find a Vκ′ where B holds”. Here κ and κ ′ range over inaccessible cardinals. We show how this reading can be related and reduced to previous results by Solovay. The second reading of A B is “Inside any Vκ where A holds we can find a Vλ where B holds”. Here κ ranges over one particular large (at least inaccessible) cardinal notion and λ ranges over a different (but also at least inaccessible) large cardinal. We see that in this case a bi-modal logic suffices. Moreover, we obtain that for an ample class of cardinal notions the corresponding logic will not include the basic modal logic IL.