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

Abstract

We consider the modality "$\phi$ is true in every $\sigma$-centered forcing extension", denoted $\square \phi$, and its dual "$\phi$ is true in some $\sigma$-centered forcing extension", denoted $◊\phi$ (where $\phi$ 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 $S4.2$. We also generalize this result to other related classes of forcing.

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)