Structural connections between a forcing class and its modal logic

  • J. D. Hamkins, G. Leibman, and B. Löwe, “Structural connections between a forcing class and its modal logic,” Israel J. Math., vol. 207, iss. 2, pp. 617-651, 2015.  
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    AUTHOR = {Hamkins, Joel David and Leibman, George and L{\"o}we,
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    TITLE = {Structural connections between a forcing class and its modal
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    JOURNAL = {Israel J. Math.},
    FJOURNAL = {Israel Journal of Mathematics},
    VOLUME = {207},
    YEAR = {2015},
    NUMBER = {2},
    PAGES = {617--651},
    ISSN = {0021-2172},
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    DOI = {10.1007/s11856-015-1185-5},
    url = {http://jdh.hamkins.org/a-forcing-class-and-its-modal-logic},
    eprint = {1207.5841},
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The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, interpreting $\square$ as “in all forcing extensions” and $\Diamond$ as “in some forcing extension”. In this modal language one may easily express sweeping general forcing principles, such as $\Diamond\square\varphi\to\square\Diamond\varphi$, the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or $\Diamond\square\varphi\to\varphi$, the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC (see A simple maximality principle).

Every definable forcing class similarly gives rise to the corresponding forcing modalities, for which one considers extensions only by forcing notions in that class. In previous work, we proved that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2 (see The modal logic of forcing). In this article, we prove that the provably valid principles of collapse forcing, Cohen forcing and other classes are in each case exactly S4.3; the provably valid principles of c.c.c. forcing, proper forcing, and others are each contained within S4.3 and do not contain S4.2; the provably valid principles of countably closed forcing, CH-preserving forcing and others are each exactly S4.2; and the provably valid principles of $\omega_1$-preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise, including the connection between various control statements, such as buttons, switches and ratchets, and their corresponding forcing validities. These structural connections therefore support a forcing-only analysis of other diverse forcing classes.

Preprints available at:  ar$\chi$iv | NI12055-SAS | UvA ILLC PP-2012-19 | HBM 446

The modal logic of forcing

  • J. D. Hamkins and B. Löwe, “The modal logic of forcing,” Trans.~Amer.~Math.~Soc., vol. 360, iss. 4, pp. 1793-1817, 2008.  
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    AUTHOR = {Hamkins, Joel David and L{\"o}we, Benedikt},
    TITLE = {The modal logic of forcing},
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    FJOURNAL = {Transactions of the American Mathematical Society},
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    MRREVIEWER = {Andreas Blass},
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    URL = {http://dx.doi.org/10.1090/S0002-9947-07-04297-3},
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What are the most general principles in set theory relating forceability and truth? As with Solovay’s celebrated analysis of provability, both this question and its answer are naturally formulated with modal logic. We aim to do for forceability what Solovay did for provability. A set theoretical assertion $\psi$ is forceable or possible, if $\psi$ holds in some forcing extension, and necessary, 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 known as S4.2.

Follow-up article:  Structural connections between a forcing class and its modal logic