Archive for Mathematical Logic

, Volume 54, Issue 5–6, pp 711–723 | Cite as

A definable E 0 class containing no definable elements

  • Vladimir Kanovei
  • Vassily Lyubetsky


A generic extension \({\mathbf{L}[x]}\) by a real x is defined, in which the \({\mathsf{E}_0}\)-class of x is a lightface \({{\it \Pi}^1_2}\) (hence, ordinal-definable) set containing no ordinal-definable reals.


Jensen reals Definable equivalence class Definable elements 

Mathematics Subject Classification

03E15 03E35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Enayat A.: On the Leibniz-Mycielski axiom in set theory. Fundam. Math. 181(3), 215–231 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Friedman, Sy D.: The \({\Pi^1_2}\) -singleton conjecture. J. Am. Math. Soc. 3(4), 771–791 (1990)Google Scholar
  3. 3.
    Groszek M., Laver R.: Finite groups of OD-conjugates. Period. Math. Hung. 18, 87–97 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Harrington L.A., Kechris A.S., Louveau A.: A Glimm-Effros dichotomy for Borel equivalence relations. J. Am. Math. Soc. 3(4), 903–928 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jech T.: Set Theory, the Third Millennium Revised and Expanded Edition. Springer, Berlin (2003)zbMATHGoogle Scholar
  6. 6.
    Jensen, R.: Definable sets of minimal degree. In: Mathematical Logic and Foundations of Set Theory, Proceedings of an International Colloquium, Jerusalem, 1968, pp. 122–128 (1970)Google Scholar
  7. 7.
    Kanovei, V., Lyubetsky, V.: A countable definable set of reals containing no definable elements. ArXiv e-prints, arXiv:1408.3901 (2014)
  8. 8.
    Kanovei, V.: Borel Equivalence Relations. Structure and Classification. AMS, Providence, RI (2008)Google Scholar
  9. 9.
    Kanovei V., Sabok M., Zapletal J.: Canonical Ramsey theory on Polish Spaces. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kechris, A., Woodin, W.: On thin \({\Pi^1_2}\) sets. Handwritten note (1983), cited in [2]Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems (IITP)MoscowRussia
  2. 2.Moscow State University of Railway Engineering (MIIT)MoscowRussia

Personalised recommendations