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

Abstrakt

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega$ implies that the modal logic $\logic{S4.1.2}$ 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 $\logic{S4}$ is complete with respect to the space {$\omega^*=\beta\omega\setminus\omega$}.This improves the results of G.~Bezhanishvili and J.~Harding.

Klíčová slova

• completely separable MAD faimilies
• modal logic
• completness
• topological semantics
• ultrafilters on natural numbers