Abstract
Todorčević (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is \({{\bf \it{\Sigma}}^{1}_{2}}\) ”, and equivalent to \({\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}\). We also consider real-valued games corresponding to Hausdorff gaps, and show that \({\mathsf{AD}_\mathbb{R}}\) for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if \({{\it{A}} \in {\bf \it{\Gamma}}}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hadamard J.: Sur les caractères de convergence des séries à termes positifs et sur les fonctions indéfiniment croissantes. Acta Math. 18, 319–336 (1894)
Hausdorff F.: Summen von \({\aleph_1}\) Mengen. Fund. Math. 26, 241–255 (1936)
Khomskii, Y.: Regularity properties and definability in the real number continuum. Idealized forcing, polarized partitions, Hausdorff gaps and mad families in the projective hierarchy. Ph.D. thesis, University of Amsterdam, ILLC Dissertations DS-2012-04 (2012)
Miller A.W.: Infinite combinatorics and definability. Ann. Pure Appl. Logic 41(2), 179–203 (1989)
Todorčević S.: Analytic gaps. Fund. Math. 150(1), 55–66 (1996)
Todorčević, S.: Definable ideals and gaps in their quotients. In: Di Prisco, C. A., Larson, J. A., Bagaria, J., Mathias, A. R. D. (eds.) Set theory (Curaçao, 1995; Barcelona, 1996), pp. 213–226. Kluwer Academic Publishers, Dordrecht (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khomskii, Y. Projective Hausdorff gaps. Arch. Math. Logic 53, 57–64 (2014). https://doi.org/10.1007/s00153-013-0355-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-013-0355-6