The modal logic of $\sigma$-centered forcing and related forcing classes

Ur Ya'ar

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arXiv · 1701.05036

The modal logic of $\sigma$-centered forcing and related forcing classes

Ur Ya'ar

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Abstract

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.

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Ur Ya'ar. 2017-01-18. The modal logic of $\sigma$-centered forcing and related forcing classes. https://arxiv.org/abs/1701.05036

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