The modal logic of forcing

Abstract

A set theoretical assertion ψ \psi is forceable or possible, written ◊ ψ \Diamond \psi , if ψ \psi holds in some forcing extension, and necessary, written ◻ ψ \Box \psi , if ψ \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 S 4.2 \mathsf {S4.2} .