Completely separably MAD families and the modal logic of $\beta\omega$

TL;DR

This paper establishes the completeness of certain modal logics with respect to the space ch-Stone compactification of ch, assuming the existence of completely separable maximal almost disjoint families, thus improving prior results.

Contribution

It proves the completeness of S4.1.2 and S4 modal logics for ch-Stone spaces under weaker assumptions and offers a simpler proof than previous work.

Findings

Completeness of S4.1.2 with respect to ch-Stone compactification.

Completeness of S4 with respect to ch space ch.

Results hold under the existence of completely separable MAD families.

Abstract

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega$ implies that the modal logic S4.1.2 is complete with respect to the \v{C}ech-Stone compactification of the natural numbers, the space $\beta\omega$. In the same fashion we prove that the modal logic S4 is complete with respect to the space $\omega^*=\beta\omega\setminus\omega$. This improves the results of G. Bezhanishvili and J. Harding who prove these theorems under stronger assumptions ($\mathfrak{a}=\mathfrak{c}$). Our proof is also somewhat simpler.