# Connections between a forcing class and its modal logic

## George Leibman

### Bronx Community College, CUNY

Every definable forcing class $\Gamma$ gives rise to a corresponding modal logic. Mapping the ordered structure given by a ground model $W$ and its $\Gamma$-forcing extensions $W[G]$ to finite ordered structures (frames) in a class which can be characterized by a particular modal theory $\mathsf{S}$ reveals connections between the modal logic of $\Gamma$-forcing and the modal theory $\mathsf{S}$. For example, in 2008, Hamkins and Loewe showed that if ${\rm ZFC}$ is consistent, then the ${\rm ZFC}$-provably valid principles of the class of all forcing are precisely the assertions of the modal theory $\mathsf{S4.2}$. In this talk I describe how specific statements of set theory are used as `controls’ to achieve this mapping for various classes of forcing, giving new results for modal logics of forcing classes such as Cohen forcing, collapse forcing and ${\rm CH}$-preserving forcing.

George Leibman is a professor in the Mathematics and Computer Science department at Bronx Community College, CUNY. He received his doctorate from the CUNY Graduate Center in 2004, under the direction of Joel Hamkins, and he conducts research in set theory, with a particular interest in the modal logic of forcing.