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The modal logic of forcing

Authors:	Joel David Hamkins Benedikt Lö we

Institution:	The Graduate Center of The City University of New York, Mathematics, 365 Fifth Avenue, New York, New York 10016 -- and -- The College of Staten Island of The City University of New York, Mathematics, 2800 Victory Boulevard, Staten Island, New York 10314 ; Institute for Logic, Language and Computation, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands

Abstract:	A set theoretical assertion $\psi$ is forceable or possible, written $\mathop{\raisebox{-1pt}{$\Diamond$}}\psi$ , if $\psi$ holds in some forcing extension, and necessary, written $\mathop{\raisebox{-1pt}{$\Box$}}\psi$ , if $\psi$ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory $\mathsf{S4.2}$ .

Keywords:	Forcing modal logic S4 2

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