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THE MODAL LOGIC OF $\sigma $-CENTERED FORCING AND RELATED FORCING CLASSES

Part of: Set theory General logic

Published online by Cambridge University Press:  03 December 2020

UR YA’AR
UR YA’AR*
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY OF JERUSALEM EDMOND J. SAFRA CAMPUS GIVAT RAM, JERUSALEM91904, ISRAELE-mail: ur.yaar@mail.huji.ac.il
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Abstract

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We consider the modality “ $\varphi $ is true in every $\sigma $ -centered forcing extension,” denoted $\square \varphi $ , and its dual “ $\varphi $ is true in some $\sigma $ -centered forcing extension,” denoted $\lozenge \varphi $ (where $\varphi $ is a statement in set theory), which give rise to the notion of a principle of $\sigma $ -centered forcing. We prove that if ZFC is consistent, then the modal logic of $\sigma $ -centered forcing, i.e., the ZFC-provable principles of $\sigma $ -centered forcing, is exactly $\mathsf {S4.2}$ . We also generalize this result to other related classes of forcing.

Keywords

forcingmodal logicmodal logic of forcingsigma-centeredS4.2

MSC classification

Primary: 03E40: Other aspects of forcing and Boolean-valued models
Secondary: 03B45: Modal logic (including the logic of norms)
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 1 , March 2021 , pp. 1 - 24
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Copyright
© The Association for Symbolic Logic 2020

References

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