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

Tomáš Lávička; Jonathan L. Verner

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arXiv · 1709.06862

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

Tomáš Lávička·Jonathan L. Verner

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Abstract

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega$ implies that the modal 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 S4 is complete with respect to the space $\omega^*=\beta\omega\setminus\omega$. This improves the results of G. Bezhanishvili and J. Harding who prove these theorems under stronger assumptions ($\mathfrak{a}=\mathfrak{c}$). Our proof is also somewhat simpler.

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BibTeX RIS

Tomáš Lávička, Jonathan L. Verner. 2017-09-20. Completely separably MAD families and the modal logic of $\beta\omega$. https://arxiv.org/abs/1709.06862

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