Abstract
Let \({\beta(\mathbb{N})}\) denote the Stone–Čech compactification of the set \({\mathbb{N}}\) of natural numbers (with the discrete topology), and let \({\mathbb{N}^\ast}\) denote the remainder \({\beta(\mathbb{N})-\mathbb{N}}\). We show that, interpreting modal diamond as the closure in a topological space, the modal logic of \({\mathbb{N}^\ast}\) is S4 and that the modal logic of \({\beta(\mathbb{N})}\) is S4.1.2.
Similar content being viewed by others
References
Aiello M., van Benthem J., Bezhanishvili G.: Reasoning about space: the modal way. J. Logic Comput. 13(6), 889–920 (2003)
van Benthem J., Bezhanishvili G., Gehrke M.: Euclidean hierarchy in modal logic. Studia Logica 75(3), 327–344 (2003)
van Benthem J., Bezhanishvili G., ten Cate B., Sarenac D.: Multimodal logics of products of topologies. Studia Logica 84(3), 369–392 (2006)
Bezhanishvili G., Gehrke M.: Completeness of S4 with respect to the real line: revisited. Ann. Pure Appl. Logic 131(1–3), 287–301 (2005)
Bezhanishvili G., Mines R., Morandi P.J.: Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces. Topol. Appl. 132(3), 291–306 (2003)
Chagrov, A., Zakharyaschev, M.: Modal logic, Oxford Logic Guides, vol. 35. The Clarendon Press Oxford University Press, New York (1997)
Engelking R.: General Topology. PWN—Polish Scientific Publishers, Warsaw (1977)
Gabelaia, D.: Modal definability in topology, Master’s Thesis (2001)
Hewitt E.: A problem of set-theoretic topology. Duke Math. J. 10, 309–333 (1943)
Koppelberg S.: Handbook of Boolean Algebras, vol. 1. North-Holland Publishing Co., Amsterdam (1989)
Kunen, K.: Set theory. An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, vol. 102. North-Holland Publishing Co., Amsterdam (1980)
McKinsey J.C.C., Tarski A.: The algebra of topology. Ann. Math. 45, 141–191 (1944)
McKinsey J.C.C., Tarski A.: Some theorems about the sentential calculi of Lewis and Heyting. J. Symbolic Logic 13, 1–15 (1948)
Mints G.: A completeness proof for propositional S4 in Cantor Space. In: Orlowska, E. (eds) Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa, Physica-Verlag, Heidelberg (1998)
Shehtman V.B.: Modal logics of domains on the real plane. Studia Logica 42(1), 63–80 (1983)
Simon P.: A compact Fréchet space whose square is not Fréchet. Comment. Math. Univ. Carolin. 21(4), 749–753 (1980)
van Mill, J.: An introduction to βω. Handbook of Set-theoretic Topology, pp. 503–567. North-Holland, Amsterdam (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Lazo Zambakhidze (1942–2008).
Rights and permissions
About this article
Cite this article
Bezhanishvili, G., Harding, J. The modal logic of \({\beta(\mathbb{N})}\) . Arch. Math. Logic 48, 231–242 (2009). https://doi.org/10.1007/s00153-009-0123-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-009-0123-9