On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom

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.

[1]  A. Kechris.  Classical descriptive set theory , 1987 .

[2]  Vassily A. Lyubetsky,et al.  Definable Minimal collapse Functions at Arbitrary Projective Levels , 2019, J. Symb. Log..

[3]  A. Tarski,et al.  Sur les ensembles définissables de nombres réels , 1931 .

[4]  P. Howard,et al.  Consequences of the axiom of choice , 1998 .

[5]  Horst Herrlich,et al.  Axiom of Choice , 2006 .

[6]  Vladimir Kanovei.  On a Glimm -- Effros dichotomy and an Ulm--type classification in Solovay model , 1995 .

[7]  Vladimir Kanovei.  An Ulm-Type Classification Theorem for Equivalence Relations in Solovay Model , 1997, J. Symb. Log..

[8]  V. Kanovei,et al.  On the ‘Definability of Definable’ Problem of Alfred Tarski , 2020, Mathematics.

[9]  Vladimir Kanovei,et al.  A countable definable set containing no definable elements , 2017 .

[10]  Vassily A. Lyubetsky,et al.  Definable E0 classes at arbitrary projective levels , 2018, Ann. Pure Appl. Log..

[11]  Vladimir Kanovei,et al.  On some classical problems of descriptive set theory , 2003 .

[13]  Vassily A. Lyubetsky,et al.  A Groszek-Laver pair of undistinguishable E0-classes , 2017, Math. Log. Q..

[14]  Sy D. Friedman.  Constructibility and Class Forcing , 2010 .

[15]  Alfred Tarski,et al.  A problem concerning the notion of definability , 1948, Journal of Symbolic Logic.

[16]  P. J. Cohen.  Set Theory and the Continuum Hypothesis , 1966 .

[17]  Alfred Tarski,et al.  Der Wahrheitsbegriff in den formalisierten Sprachen , 1935 .

[18]  Joel David Hamkins,et al.  Ehrenfeucht's Lemma in Set Theory , 2018, Notre Dame J. Formal Log..

[19]  A. Tarski,et al.  What are logical notions , 1986 .

[20]  John W. Addison,et al.  Tarski's theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic , 2004, Ann. Pure Appl. Log..

[21]  Vladimir Kanovei,et al.  Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes , 2018 .

[22]  Andrei Alexandru,et al.  Foundations of Finitely Supported Structures: A Set Theoretical Viewpoint , 2020 .

[23]  V. Kanovei,et al.  Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level , 2020, Mathematics.

[24]  Ali Enayat.  Leibnizian models of set theory , 2004, J. Symb. Log..

[25]  Richard Laver,et al.  Finite groups ofOD-conjugates , 1987 .

[26]  James Halpern.  On a question of Tarski and a maximal theorem of Kurepa , 1972 .

[27]  Asaf Karagila,et al.  The Bristol model: An abyss called a Cohen real , 2017, J. Math. Log..

[28]  Vladimir Kanovei,et al.  A definable E0 class containing no definable elements , 2015, Arch. Math. Log..

[29]  Arnold W. Miller,et al.  Rational perfect set forcing , 1984 .

[30]  Marcin Sabok,et al.  Canonical Ramsey Theory on Polish Spaces , 2013 .

[31]  The Implicitly Constructible Universe , 2019, J. Symb. Log..

[32]  Ali Enayat.  On the Leibniz–Mycielski axiom in set theory , 2004 .

[33]  J. Hadamard,et al.  Cinq lettres sur la théorie des ensembles , 1905 .

[34]  Jan Mycielski.  New Set-Theoretic Axioms Derived from a Lean Metamathematics , 1995, J. Symb. Log..

[35]  Sy-David Friedman,et al.  A model of second-order arithmetic satisfying AC but not DC , 2019, J. Math. Log..

[36]  Ronald Jensen.  Definable Sets of Minimal Degree , 1970 .