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On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom

Author

Listed:
  • Ali Enayat

    (Department of Philosophy, Linguistics, and Theory of Science, University of Gothenburg, 405 30 Gothenburg, Sweden
    These authors contributed equally to this work.)

  • Vladimir Kanovei

    (Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
    These authors contributed equally to this work.)

  • Vassily Lyubetsky

    (Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
    These authors contributed equally to this work.)

Abstract

Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface Π 2 1 equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes, neither one of which is ordinal definable, and therefore the classes are OD-indiscernible. A similar but somewhat weaker result is obtained for Silver extensions. The other main result is that for any n , starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface Π n 1 , does not imply the existence of such a pair with the associated relation in Σ n 1 or in a lower class.

Suggested Citation

  • Ali Enayat & Vladimir Kanovei & Vassily Lyubetsky, 2021. "On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom," Mathematics, MDPI, vol. 9(14), pages 1-19, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:14:p:1670-:d:595235
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