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 | References | Similar Articles | Additional Information

Abstract: A set theoretical assertion is *forceable* or *possible*, written , if holds in some forcing extension, and *necessary*, written , if 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 .

**[BdRV01]**Patrick Blackburn, Maarten de Rijke, and Yde Venema,*Modal logic*, Cambridge Tracts in Theoretical Computer Science, vol. 53, Cambridge University Press, Cambridge, 2001. MR**1837791****[Bla90]**Andreas Blass,*Infinitary combinatorics and modal logic*, J. Symbolic Logic**55**(1990), no. 2, 761–778. MR**1056387**, 10.2307/2274663**[Boo93]**George Boolos,*The logic of provability*, Cambridge University Press, Cambridge, 1993. MR**1260008****[CZ97]**Alexander Chagrov and Michael Zakharyaschev,*Modal logic*, Oxford Logic Guides, vol. 35, The Clarendon Press, Oxford University Press, New York, 1997. Oxford Science Publications. MR**1464942****[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*, J. Symbolic Logic**68**(2003), no. 2, 527–550. MR**1976589**, 10.2178/jsl/1052669062**[HW05]**Joel D. Hamkins and W. Hugh Woodin,*The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal*, MLQ Math. Log. Q.**51**(2005), no. 5, 493–498. MR**2163760**, 10.1002/malq.200410045**[JdJ98]**Giorgi Japaridze and Dick de Jongh,*The logic of provability*, Handbook of proof theory, Stud. Logic Found. Math., vol. 137, North-Holland, Amsterdam, 1998, pp. 475–546. MR**1640331**, 10.1016/S0049-237X(98)80022-0**[Jec03]**Thomas Jech,*Set theory*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR**1940513****[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*, Oxford Logic Guides, vol. 34, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR**1433595****[Smo70]**C. A. Smoryński,*Applications of Kripke models*, Metamathematical investigation of intuitionistic arithmetic and analysis, Springer, Berlin, 1973, pp. 324–391. Lecture Notes in Mathematics, Vol. 344. MR**0444442****[Sol76]**Robert M. Solovay,*Provability interpretations of modal logic*, Israel J. Math.**25**(1976), no. 3-4, 287–304. MR**0457153****[Vop65]**P. Vopěnka,*On 𝑚1𝑥-model of set theory*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**13**(1965), 267–272 (English, with Russian summary). MR**0182571**

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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:
https://doi.org/10.1090/S0002-9947-07-04297-3

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.