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COMPLETELY SEPARABLE MAD FAMILIES AND THE MODAL LOGIC OF βω

Part of: General logic Fairly general properties Set theory

Published online by Cambridge University Press:  15 June 2020

TOMÁŠ LÁVIČKA  and
JONATHAN L. VERNER
TOMÁŠ LÁVIČKA
Affiliation:
THE CZECH ACADEMY OF SCIENCES INSTITUTE OF INFORMATION THEORY AND AUTOMATION POD VODÁRENSKOU VĚŽÍ 4, 182 07 PRAHA, CZECH REPUBLICE-mail:lavickat@utia.cas.cz
JONATHAN L. VERNER
Affiliation:
DEPARTMENT OF LOGIC FACULTY OF ARTS CHARLES UNIVERSITY PALACHOVO NÁM. 2, 116 38 PRAHA 1, CZECH REPUBLICE-mail:jonathan.verner@matfyz.cz
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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 $\mathbf {S4.1.2}$ 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 ( $\mathfrak {a=c}$ ). Our proof is also somewhat simpler.

Keywords

modal logicČech-Stone compactification of natural numberscompletely separable mad families

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 03E17: Cardinal characteristics of the continuum 54D80: Special constructions of spaces (spaces of ultrafilters, etc.)
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 498 - 507
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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