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

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 Čech-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 ${\omega }^{\ast }=\beta \omega \setminus \omega$. 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.