The modal logic of forcing

Hamkins, Joel; Löwe, Benedikt

doi:10.1090/S0002-9947-07-04297-3

The modal logic of forcing
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by Joel David Hamkins and Benedikt Löwe PDF

Trans. Amer. Math. Soc. 360 (2008), 1793-1817 Request permission

Abstract:

A set theoretical assertion $\psi$ is forceable or possible, written $\Diamond \psi$, if $\psi$ holds in some forcing extension, and necessary, written $\Box \psi$, if $\psi$ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory $\mathsf {S4.2}$.

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