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On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom

by 1,†, 2,*,† and 2,*,†
1
Department of Philosophy, Linguistics, and Theory of Science, University of Gothenburg, 405 30 Gothenburg, Sweden
2
Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
*
Authors to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editor: Juan Benigno Seoane-Sepúlveda
Mathematics 2021, 9(14), 1670; https://doi.org/10.3390/math9141670
Received: 28 May 2021 / Revised: 8 July 2021 / Accepted: 9 July 2021 / Published: 15 July 2021
(This article belongs to the Special Issue Mathematical Logic and Its Applications 2021)
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 Π21 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 Πn1, does not imply the existence of such a pair with the associated relation in Σn1 or in a lower class. View Full-Text
Keywords: indiscernible sets; Leibniz-Mycielski axiom; projective hierarchy; generic models; ordinal definability; Miller forcing; Laver forcing; Silver forcing indiscernible sets; Leibniz-Mycielski axiom; projective hierarchy; generic models; ordinal definability; Miller forcing; Laver forcing; Silver forcing
MDPI and ACS Style

Enayat, A.; Kanovei, V.; Lyubetsky, V. On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom. Mathematics 2021, 9, 1670. https://doi.org/10.3390/math9141670

AMA Style

Enayat A, Kanovei V, Lyubetsky V. On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom. Mathematics. 2021; 9(14):1670. https://doi.org/10.3390/math9141670

Chicago/Turabian Style

Enayat, Ali, Vladimir Kanovei, and Vassily Lyubetsky. 2021. "On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom" Mathematics 9, no. 14: 1670. https://doi.org/10.3390/math9141670

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