THE MODAL LOGIC OF -CENTERED FORCING AND RELATED FORCING CLASSES

Author:

YA’AR UR

Abstract

AbstractWe 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.

Publisher

Cambridge University Press (CUP)

Subject

Logic,Philosophy

Reference12 articles.

1. Modal Logic

2. The modal logic of forcing

3. [2] U. Ben-Ari-Tishler (now Ya’ar), The modal logic of $\boldsymbol{\sigma}$ - centered forcing and related forcing classes , M.Sc. thesis, Hebrew University of Jerusalem, 2016.

4. [12] Piribauer, J. , The modal logic of generic multiverses , M.Sc. thesis, Universiteit van Amsterdam, 2017.

5. [9] Inamdar, T. C. , On the modal logics of some set-theoretic constructions , M.Sc. thesis, Universiteit van Amsterdam, 2013.

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