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 (Formula presented.), thus resolving Shehtman's first problem for (Formula presented.). We also characterize modal logics arising from the Čech–Stone compactification of an ordinal (Formula presented.) provided the Cantor normal form of (Formula presented.) satisfies an additional condition. This gives a partial solution of Shehtman's second problem.