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CHARACTERIZING EXISTENCE OF A MEASURABLE CARDINAL VIA MODAL LOGIC

Published online by Cambridge University Press:  01 February 2021

GURAM BEZHANISHVILI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITYLAS CRUCES, NM, USA E-mail:guram@nmsu.edu
NICK BEZHANISHVILI
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAMAMSTERDAM, THE NETHERLANDS E-mail:N.Bezhanishvili@uva.nl
JOEL LUCERO-BRYAN
Affiliation:
DEPARTMENT OF MATHEMATICS KHALIFA UNIVERSITY OF SCIENCE AND TECHNOLOGYABU DHABI, UNITED ARAB EMIRATES E-mail:joel.bryan@ku.ac.ae
JAN VAN MILL
Affiliation:
KORTEWEG-DE VRIES INSTITUTE FOR MATHEMATICS UNIVERSITY OF AMSTERDAMAMSTERDAM, THE NETHERLANDS E-mail:j.vanMill@uva.nl

Abstract

We prove that the existence of a measurable cardinal is equivalent to the existence of a normal space whose modal logic coincides with the modal logic of the Kripke frame isomorphic to the powerset of a two element set.

Type
Article
Copyright
© The Association for Symbolic Logic 2021

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References

REFERENCES

Aczel, P., Non-Well-Founded Sets , CSLI Lecture Notes, vol. 14, Stanford University, Center for the Study of Language and Information, Stanford, CA, 1988.Google Scholar
Baltag, A., STS: A structural theory of sets . Logic Journal of the IGPL , vol. 7 (1999), no. 4, pp. 481515.CrossRefGoogle Scholar
Barwise, J. and Moss, L., Vicious Circles , CSLI Lecture Notes, vol. 60, CSLI Publications, Stanford, CA, 1996.Google Scholar
van Benthem, J., Bezhanishvili, G., and Gehrke, M., Euclidean hierarchy in modal logic . Studia Logica , vol. 75 (2003), no. 3, pp. 327344.CrossRefGoogle Scholar
Bezhanishvili, G., Bezhanishvili, N., Lucero-Bryan, J., and van Mill, J., Krull dimension in modal logic , this Journal, vol. 82 (2017), no. 4, pp. 13561386.Google Scholar
Bezhanishvili, G., Bezhanishvili, N., Lucero-Bryan, J., and van Mill, J., Tychonoff HED-spaces and Zemanian extensions of S4.3 . The Review of Symbolic Logic , vol. 11 (2018), no. 1, pp. 115132.CrossRefGoogle Scholar
Bezhanishvili, G., Bezhanishvili, N., Lucero-Bryan, J., and van Mill, J., On modal logics arising from scattered locally compact Hausdorff spaces . Annals of Pure and Applied Logic , vol. 170 (2019), no. 5, pp. 558577.CrossRefGoogle Scholar
Bezhanishvili, G. and Harding, J., The modal logic of  $\beta(\mathbb N)$ . Archive for Mathematical Logic , vol. 48 (2009), no. 3–4, pp. 231242.CrossRefGoogle Scholar
Chagrov, A. and Zakharyaschev, M., Modal Logic , Oxford University Press, Oxford, 1997.Google Scholar
Comfort, W. W. and Negrepontis, S., The Theory of Ultrafilters , Springer-Verlag, New York, 1974.CrossRefGoogle Scholar
Engelking, R., General Topology , second ed., Heldermann Verlag, Berlin, 1989.Google Scholar
Fine, K., An ascending chain of S4 logics . Theoria , vol. 40 (1974), pp. 110116.CrossRefGoogle Scholar
Hamkins, J. and Löwe, B., The modal logic of forcing . Transactions of the American Mathematical Society , vol. 360 (2008), no. 4, pp. 17931817.CrossRefGoogle Scholar
Jech, T., Set Theory , Academic Press [Harcourt Brace Jovanovich], New York, London, 1978.Google Scholar
Kunen, K., Set Theory. An Introduction to Independence Proofs , Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1983.Google Scholar