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

Ur Ya'ar

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

Logic 2019-12-12 v2

Authors: Ur Ya'ar

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.

Keywords

set theory forcing

Cite

@article{arxiv.1701.05036,
  title  = {The modal logic of $\sigma$-centered forcing and related forcing classes},
  author = {Ur Ya'ar},
  journal= {arXiv preprint arXiv:1701.05036},
  year   = {2019}
}

Comments

22 pages. Revised version - all results and proofs remain essentially as they were. Some proofs were shortened and the presentation of some revised for clarity. The order of presentation changed somewhat, and a few minor additional results were added (section 4.1 and theorem 4.10). Additionally - the author's last name has changed (from Ben-Ari-Tishler to Ya'ar)

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

Abstract

Keywords

Cite

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