Advertisement

Mathematical Notes

, Volume 102, Issue 3–4, pp 338–349 | Cite as

A countable definable set containing no definable elements

  • V. G. Kanovei
  • V. A. Lyubetsky
Article
  • 31 Downloads

Abstract

The consistency of the existence of a countable definable set of reals, containing no definable elements, is established. The model, where such a set exists, is obtained by means of a countable product of Jensen’s forcing with finite support.

Keywords

countable sets definable elements Jensen’s forcing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Hadamard, R. Baire, H. Lebesgue, and E. Borel, “Cinq lettres sur la théorie des ensembles,” Bull. Soc. Math. France 33, 261–273 (1905).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    A. Tarski, “DerWahrheitsbegriff in den formalisierten Sprachen,” Stud. Philos. 1, 261–405 (1936).Google Scholar
  3. 3.
    J. Myhill and D. Scott, “Ordinal definability,” in 1971 Axiomatic Set Theory (Amer. Math. Soc., Providence, RI, 1971), pp. 271–278.CrossRefGoogle Scholar
  4. 4.
    V. G. Kanovei and V. A. Lyubetsky, Modern Set Theory: Absolutely Unsolvable Classical Problems (MTsNMO, Moscow, 2013) [in Russian].Google Scholar
  5. 5.
    R. M. Solovay, “A model of set theory in which every set of reals is Lebesgue measurable,” Ann. of Math. (2) 92, 1–56 (1970).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    A Question About Ordinal Definable Real Numbers, in Mathoverflow (2010), http://mathoverflow. net/questions/17608.Google Scholar
  7. 7.
    A. Enayat, Ordinal Definable Numbers, in Foundations of Mathematics (2010), http://cs.nyu.edu/ pipermail/fom/2010-July/014944.html.Google Scholar
  8. 8.
    V. G. Kanovei and V. A. Lyubetsky, Modern Set Theory: Borel and Projective Sets (MTsNMO, Moscow, 2010) [in Russian].zbMATHGoogle Scholar
  9. 9.
    V. Kanovei, Borel Equivalence Relations. Structure and Classification, in Univ. Lecture Ser. (Amer. Math. Soc., Providence, RI, 2008), Vol.44.Google Scholar
  10. 10.
    A. Enayat, “On the Leibniz–Mycielski axiom in set theory,” Fund. Math. 181 (3), 215–231 (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    R. Jensen, “Definable sets of minimal degree,” in Mathematical Logic and Foundations of Set Theory (North-Holland, Amsterdam, 1970), pp. 122–128.Google Scholar
  12. 12.
    V. G. Kanovei and V. A. Lyubetsky, “Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes,” Izv. Ross. Akad. Nauk, Ser. Mat. 82 (1) (2018) (in press) [Izv. Math. 82 (1) (2018), in press].Google Scholar
  13. 13.
    V. Kanovei and V. Lyubetsky, “Counterexamples to countable-section?1 2 uniformization and?1 3 separation,” Ann. Pure Appl. Logic 167 (3), 262–283 (2016).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    M. Golshani, V. Kanovei, and V. Lyubetsky, “AGroszek–Laver pair of undistinguishable E0 classes,” MLQ Math. Log. Q. 63 (1-2), 19–31 (2017).CrossRefMathSciNetGoogle Scholar
  15. 15.
    V. G. Kanovei, “The projective hierarchy of N. N. uzin: The current state-of-the-art in the theory,” in Handbook of Mathematical Logic, Part II: Set Theory, Supplement to the Russian translation (Nauka, Moscow, 1982), pp. 273–364 [in Russian].Google Scholar
  16. 16.
    V. G. Kanovei and V. A. Lyubetsky, “An effective minimal encoding of uncountable sets,” Sib. Math. Zh. 52 (5), 1074–1086 (2011) [Sib. Math. J. 52 (5), 854–863 (2011)].zbMATHMathSciNetGoogle Scholar
  17. 17.
    J. Bagaria and V. Kanovei, “On coding uncountable sets by reals,” MLQ Math. Log. Q. 56 (4), 409–424 (2010).CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Kharkevich Institute for Information Transmission Problems (IITP) of the Russian Academy of SciencesMoscowRussia
  2. 2.Moscow State University of Railway Engineering (MIIT)MoscowRussia

Personalised recommendations