Definable E0 classes at arbitrary projective levels

Using a modification of the invariant Jensen forcing, we define a model of ZFC, in which, for a given $n\ge3$, there exists a lightface $\varPi^1_n$ set of reals, which is a ${\mathsf E}_0$ equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable $\varSigma^1_n$ set of reals is necessarily constructible, hence contains only OD reals.

[1]  Alain Louveau,et al.  A Glimm-Effros dichotomy for Borel equivalence relations , 1990 .

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

[3]  N. Lusin,et al.  Sur les ensembles analytiques , 2022 .

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

[5]  V G Kanoveĭ.  On the Nonemptiness of Classes in Axiomatic Set Theory , 1978 .

[6]  F. Stephan,et al.  Set theory , 2018, Mathematical Statistics with Applications in R.

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

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

[9]  S. Griffis.  EDITOR , 1997, Journal of Navigation.

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

[11]  Vladimir Kanovei,et al.  Counterexamples to countable-section Π12 uniformization and Π13 separation , 2015 .

[12]  V. G. Kanovei,et al.  Cardinality of the set of Vitali equivalence classes , 1991 .

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

[14]  Serge Grigorieff,et al.  Combinatorics on ideals and forcing , 1971 .

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

[16]  Peter Aczel etc.  HANDBOOK OF MATHEMATICAL LOGIC , 1999 .

[17]  Vassily A. Lyubetsky,et al.  Counterexamples to countable-section uniformization and separation , 2016, Ann. Pure Appl. Log..