Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [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)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03E40, 03B45

Retrieve articles in all journals with MSC (2000): 03E40, 03B45

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.

American Mathematical Society