On Shehtman's Two Problems

We provide partial solutions to two problems posed by Shehtman concerning the modal logic of the \v{C}ech-Stone compactification of an ordinal space. We use the Continuum Hypothesis to give a finite axiomatization of the modal logic of $\beta(\omega^2)$, thus resolving Shehtman's first problem for $n=2$. We also characterize modal logics arising from the \v{C}ech-Stone compactification of an ordinal $\gamma$ provided the Cantor normal form of $\gamma$ satisfies an additional condition. This gives a partial solution of Shehtman's second problem.