Completely separable mad families and the modal logic of βω

Journal of Symbolic Logic 87 (2):498-507 (2022)
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We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $. In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $. This improves the results of G. Bezhanishvili and J. Harding in [4], where the authors prove these theorems under stronger assumptions. Our proof is also somewhat simpler.



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A Model Theory of Topology.Paolo Lipparini - forthcoming - Studia Logica:1-35.

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References found in this work

Diodorean modality in Minkowski spacetime.Robert Goldblatt - 1980 - Studia Logica 39 (2-3):219 - 236.
The modal logic of {beta(mathbb{N})}.Guram Bezhanishvili & John Harding - 2009 - Archive for Mathematical Logic 48 (3-4):231-242.

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