Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T18:54:42.588Z Has data issue: false hasContentIssue false
  • English
  • Français

THE MCKINSEY–TARSKI THEOREM FOR LOCALLY COMPACT ORDERED SPACES

Part of: Special properties Fairly general properties Peculiar spaces General logic

Published online by Cambridge University Press:  29 April 2021

GURAM BEZHANISHVILI ,
NICK BEZHANISHVILI ,
JOEL LUCERO-BRYAN  and
JAN VAN MILL
GURAM BEZHANISHVILI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITYLAS CRUCES, NM, USAE-mail: guram@nmsu.edu
NICK BEZHANISHVILI
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAMAMSTERDAM, THE NETHERLANDSE-mail: N.Bezhanishvili@uva.nl
JOEL LUCERO-BRYAN
Affiliation:
DEPARTMENT OF MATHEMATICS KHALIFA UNIVERSITY OF SCIENCE AND TECHNOLOGY ABU DHABI, UNITED ARAB EMIRATESE-mail: joel.bryan@ku.ac.ae
JAN VAN MILL
Affiliation:
KORTEWEG-DE VRIES INSTITUTE FOR MATHEMATICS UNIVERSITY OF AMSTERDAMAMSTERDAM, THE NETHERLANDSE-mail: j.vanMill@uva.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the modal logic of a crowded locally compact generalized ordered space is $\textsf {S4}$ . This provides a version of the McKinsey–Tarski theorem for generalized ordered spaces. We then utilize this theorem to axiomatize the modal logic of an arbitrary locally compact generalized ordered space.

Keywords

modal logictopological semanticslinearly ordered spacegeneralized ordered spacelocally compact spacescattered space

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 54D45: Local compactness, $sigma$-compactness 54G12: Scattered spaces 03B55: Intermediate logics
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 27 , Issue 2 , June 2021 , pp. 187 - 211
Check for updates
Copyright
© The Association for Symbolic Logic 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S., Domain theory in logical form. Annals of Pure and Applied Logic, vol. 51 (1991), no. 1–2, pp. 1–77.CrossRefGoogle Scholar
Baltag, A., Bezhanishvili, N., Özgün, A., and Smets, S., Justified belief and the topology of evidence, Logic, Language, Information, and Computation—23rd International Workshop, WoLLIC (Väänänen, J. A., Hirvonen, Å., and de Queiroz, R. J. G. B., editors), Lecture Notes in Computer Science, vol. 9803, Springer, New York, 2016, pp. 83–103.CrossRefGoogle Scholar
Baltag, A., Gierasimczuk, N., and Smets, S., On the solvability of inductive problems: A study in epistemic topology, Proceedings of the 15th Conference on Theoretical Aspects of Rationality and Knowledge, ENTCS, 2015, pp. 81–98.Google Scholar
Bezhanishvili, G., Bezhanishvili, N., Lucero-Bryan, J., and van Mill, J., Krull dimension in modal logic. The Journal of Symbolic Logic, vol. 82 (2017), no. 4, pp. 1356–1386.CrossRefGoogle Scholar
Bezhanishvili, G., Bezhanishvili, N., Lucero-Bryan, J., and van Mill, J., A new proof of the McKinsey–Tarski theorem. Studia Logica, vol. 106 (2018), pp. 1291–1311.CrossRefGoogle Scholar
Bezhanishvili, G., Bezhanishvili, N., Lucero-Bryan, J., and van Mill, J., On modal logics arising from scattered locally compact Hausdorff spaces. The Annals of Pure and Applied Logic, vol. 170 (2019), no. 5, pp. 558–577.CrossRefGoogle Scholar
Bezhanishvili, G., Bezhanishvili, N., Lucero-Bryan, J., and van Mill, J., Tree-like constructions in topology and modal logic. Archive for Mathematical Logic, vol. 60 (2021), pp. 265–299.CrossRefGoogle Scholar
Bezhanishvili, G., Bezhanishvili, N., Lucero-Bryan, J., and van Mill, J., Characterizing existence of a measurable cardinal via modal logic. The Journal of Symbolic Logic, 2021, doi: 10.1017/jsl.2021.5.CrossRefGoogle Scholar
Bezhanishvili, G., Gabelaia, D., and Lucero-Bryan, J., Modal logics of metric spaces. The Review of Symbolic Logic, vol. 8 (2015), no. 1, pp. 178–191.CrossRefGoogle Scholar
Bezhanishvili, G. and Gehrke, M., Completeness of S4 with respect to the real line: Revisited. The Annals of Pure and Applied Logic, vol. 131 (2005), no. 1–3, pp. 287–301.CrossRefGoogle Scholar
Bezhanishvili, G. and Harding, J., Modal logics of Stone spaces. Order, vol. 29 (2012), no. 2, pp. 271–292.CrossRefGoogle Scholar
Brecht, M. and Yamamoto, A., Topological properties of concept spaces. Information and Computation, vol. 208 (2010), no. 4, pp. 327–340.CrossRefGoogle Scholar
Ceder, J. G., On maximally resolvable spaces. Fundamenta Mathematicae, vol. 55 (1964), pp. 87–93.CrossRefGoogle Scholar
Chagrov, A. and Zakharyaschev, M., Modal Logic, Oxford University Press, Oxford, 1997.Google Scholar
Eckertson, F. W., Resolvable, not maximally resolvable spaces. Topology and Its Applications, vol. 79 (1997), pp. 1–11.CrossRefGoogle Scholar
Engelking, R.. General Topology, Heldermann Verlag, Berlin, 1989.Google Scholar
Goubault, É., Ledent, J., and Rajsbaum, S., A simplicial complex model for dynamic epistemic logic to study distributed task computability, Proceedings of the Ninth International Symposium on Games, Automata, Logics, and Formal Verification, Electronic Proceedings in Theoretical Computer Science, vol. 277, 2018, pp. 73–87.Google Scholar
Herlihy, M., Kozlov, D., and Rajsbaum, S., Distributed Computing Through Combinatorial Topology, Elsevier/Morgan Kaufmann, Waltham, MA, 2014.Google Scholar
Herrlich, H., Ordnungsfähigkeit total-diskontinuierlicher Räume. Mathematische Annalen, vol. 159 (1965), pp. 77–80.CrossRefGoogle Scholar
Hewitt, E., A problem of set-theoretic topology. Duke Mathematical Journal, vol. 10 (1943), pp. 309–333.CrossRefGoogle Scholar
Illanes, A., Finite and $\omega$ -resolvability. Proceedings of the American Mathematical Society, vol. 124 (1996), no. 4, pp. 1243–1246.CrossRefGoogle Scholar
Johnstone, P. T., Stone Spaces, Cambridge University Press, Cambridge, 1982.Google Scholar
Lutzer, D. J., On Generalized Ordered Spaces, Dissertationes Mathematicae Rozprawy Matematyczny, vol. 89, Instytut Matematyczny Polskiej Akademi Nauk, Warsaw, 1971.Google Scholar
McKinsey, J. C. C. and Tarski, A., The algebra of topology. Annals of Mathematics, vol. 45 (1944), pp. 141–191.CrossRefGoogle Scholar
Özgün, A., Evidence in epistemic logic: A topological perspective , Ph.D. thesis, ILLC, University of Amsterdam and University of Lorraine, 2017.Google Scholar
Rasiowa, H. and Sikorski, R., The Mathematics of Metamathematics, Monografie Matematyczne, Tom 41, Państwowe Wydawnictwo Naukowe, Warsaw, 1963.Google Scholar
Semadeni, Z., Banach Spaces of Continuous Functions, vol. I, PWN—Polish Scientific Publishers, Warsaw, 1971.Google Scholar
Telgársky, R., Total paracompactness and paracompact dispersed spaces. Bulletin of the Polish Academy of Sciences, Series of Mathematics, Astronomy and Physics, vol. 16 (1968), pp. 567–572.Google Scholar
Vickers, S., Topology Via Logic, Cambridge University Press, Cambridge, 1989.Google Scholar