Completely separably MAD families and the modal logic of $βω$

Abstract

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of ω implies that the modal logic S4.1.2 is complete with respect to the Čech-Stone compactification of the natural numbers, the space βω. In the same fashion we prove that the modal logic S4 is complete with respect to the space ω * = βω \ω. This improves the results of G. Bezhanishvili and J. Harding in [4], where the authors prove these theorems under stronger assumptions (a = c). Our proof is also somewhat simpler.