On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom

• Main Authors: Ali Enayat, Vladimir Kanovei, Vassily Lyubetsky
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Mathematics
• Subjects: indiscernible sets; Leibniz-Mycielski axiom; projective hierarchy; generic models; ordinal definability; Miller forcing
• Online Access: https://www.mdpi.com/2227-7390/9/14/1670

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Π</mi><mn>2</mn><mn>1</mn></msubsup></semantics></math></inline-formula> 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 <i>n</i>, starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Π</mi><mi>n</mi><mn>1</mn></msubsup></semantics></math></inline-formula>, does not imply the existence of such a pair with the associated relation in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Σ</mi><mi>n</mi><mn>1</mn></msubsup></semantics></math></inline-formula> or in a lower class.