On Shehtman's Two Problems

TL;DR

This paper addresses Shehtman's two problems on the modal logic of the Čech-Stone compactification of ordinal spaces, providing a finite axiomatization for eta( extsuperscript{2}) under CH and characterizations for certain ordinals.

Contribution

It offers the first finite axiomatization of the modal logic of eta( extsuperscript{2}) assuming CH and characterizes modal logics for specific ordinal compactifications.

Findings

Finite axiomatization of modal logic for eta( extsuperscript{2}) under CH

Characterization of modal logics for ordinal compactifications with specific Cantor normal forms

Partial solutions to Shehtman's two problems

Abstract

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.