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- Title
On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom.
- Authors
Enayat, Ali; Kanovei, Vladimir; Lyubetsky, Vassily
- 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.
- Subjects
SET theory; REAL numbers; MODEL theory; AXIOMS; SILVER; EQUIVALENCE relations (Set theory)
- Publication
Mathematics (2227-7390), 2021, Vol 9, Issue 14, p1670
- ISSN
2227-7390
- Publication type
Article
- DOI
10.3390/math9141670