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The modal logic of forcing

Author(s): Joel David Hamkins; Benedikt Löwe
Journal: Trans. Amer. Math. Soc. 360 (2008), 1793-1817.
MSC (2000): Primary 03E40, 03B45
Posted: October 2, 2007
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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}$.

References:

[BdRV01]
Patrick Blackburn, Maarten de Rijke, and Yde Venema.
Modal Logic, volume 53 of Cambridge Tracts in Theoretical Computer Science.
Cambridge University Press, Cambridge, 2001. MR 1837791 (2003b:03001)

[Bla90]
Andreas Blass.
Infinitary combinatorics and modal logic.
J. Symbolic Logic, 55(2):761-778, 1990. MR 1056387 (91k:03123)

[Boo93]
George Boolos.
The Logic of Provability.
University of Cambridge, 1993. MR 1260008 (95c:03038)

[CZ97]
Alexander Chagrov and Michael Zakharyaschev.
Modal Logic, volume 35 of Oxford Logic Guides, Oxford Science Publications.
The Clarendon Press, Oxford University Press, New York, 1997. MR 1464942 (98e:03021)

[dJ70]
Dick de Jongh.
The maximality of the intuitionistic predicate calculus with respect to Heyting's arithmetic (abstract).
Journal of Symbolic Logic, 35:606, 1970.

[Ham03]
Joel David Hamkins.
A simple maximality principle.
Journal of Symbolic Logic, 68(2):527-550, June 2003. MR 1976589 (2005a:03094)

[HW05]
Joel David Hamkins and W. Hugh Woodin.
The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal.
Mathematical Logic Quarterly, 51(5):493-498, 2005. MR 2163760 (2006f:03082)

[JdJ98]
Giorgi Japaridze and Dick de Jongh.
The logic of provability.
In Samuel R. Buss, editor, Handbook of Proof Theory, volume 137 of Studies in Logic and the Foundations of Mathematics, chapter VII, pages 475-546. North-Holland Publishing Co., Amsterdam, 1998. MR 1640331 (2000a:03107)

[Jec03]
Thomas Jech.
Set Theory.
Springer Monographs in Mathematics, 3rd edition, 2003. MR 1940513 (2004g:03071)

[Lei04]
George Leibman.
Consistency Strengths of Maximality Principles.
Ph.D. thesis, The Graduate Center of the City University of New York, June 2004.

[SF96]
Raymond M. Smullyan and Melvin Fitting.
Set Theory and the Continuum Problem, volume 34 of Oxford Logic Guides.
Oxford University Press, 1996. MR 1433595 (98a:03071)

[Smo70]
Craig A. Smorynski.
Applications of Kripke models.
In Anne S. Troelstra, editor, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, volume 344 of Lecture Notes in Mathematics, chapter V, pages 324-391. Springer-Verlag, Berlin-New York, 1970.
(Also available as University of Amsterdam ILLC Prepublication Series X-93-05, 1993). MR 0444442 (56:2795)

[Sol76]
Robert M. Solovay.
Provability interpretations of modal logic.
Israel Journal of Mathematics, 25:287-304, 1976. MR 0457153 (56:15369)

[Vop65]
Petr Vopenka.
On $ \nabla$-model of set theory.
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 13:267-272, 1965. MR 0182571 (32:54)

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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
Email: jhamkins@gc.cuny.edu

Benedikt Löwe
Affiliation: Institute for Logic, Language and Computation, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Email: bloewe@science.uva.nl

DOI: 10.1090/S0002-9947-07-04297-3
PII: S 0002-9947(07)04297-3
Keywords: Forcing, modal logic, S4.2
Received by editor(s): September 29, 2005
Posted: October 2, 2007
Additional Notes: In addition to partial support from PSC-CUNY grants and other CUNY support, the first author was a \textit{Mercator-Gastprofessor} at the Westfälische Wilhelms-Universität Münster during May--August 2004, when this collaboration began, and was partially supported by \textit{NWO Bezoekersbeurs} \textsf{B 62-612} at Universiteit van Amsterdam during May--August 2005, when it came to fruition. The second author was partially supported by \textit{NWO Reisbeurs} \textsf{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.
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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