On Shehtman's two problems

We provide partial solutions to two problems posed by Shehtman concerning the modal logic of the Čech–Stone compactification of an ordinal space. We use the Continuum Hypothesis to give a finite axiomatization of the modal logic of $\beta \left({\omega }^2\right)$, thus resolving Shehtman's first problem for $n=2$. We also characterize modal logics arising from the Čech–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.