Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The modal logic of forcing

Authors: Joel David Hamkins and Benedikt Löwe
Journal: Trans. Amer. Math. Soc. 360 (2008), 1793-1817
MSC (2000): Primary 03E40, 03B45
Published electronically: October 2, 2007
MathSciNet review: 2366963
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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 $ {ZFC}$ is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory $ \mathsf{S4.2}$.

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Additional Information

Joel David Hamkins
Affiliation: 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

Benedikt Löwe
Affiliation: Institute for Logic, Language and Computation, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands

Keywords: Forcing, modal logic, S4.2
Received by editor(s): September 29, 2005
Published electronically: October 2, 2007
Additional Notes: In addition to partial support from PSC-CUNY grants and other CUNY support, the first author was a Mercator-Gastprofessor at the Westfälische Wilhelms-Universität Münster during May–August 2004, when this collaboration began, and was partially supported by NWO Bezoekersbeurs B 62-612 at Universiteit van Amsterdam during May–August 2005, when it came to fruition. The second author was partially supported by NWO Reisbeurs R 62-605 during his visits to New York and Los Angeles in January and February 2005. The authors would like to thank Nick Bezhanishvili (Amsterdam), Dick de Jongh (Amsterdam), Marcus Kracht (Los Angeles, CA), and Clemens Kupke (Amsterdam) for sharing their knowledge of modal logic.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.