Definable ${\mathsf{E}}_0$ classes at arbitrary projective levels

Abstract

Using a modification of the invariant Jensen forcing, we define a model of ZFC, in which, for a given $n\ge 3$, there exists a lightface ${\Pi }_n^1$ 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 ${\Sigma }_n^1$ set of reals is necessarily constructible, hence contains only OD reals.

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Cited by in corpus (3)

• Definable minimal collapse functions at arbitrary projective levels
• A model in which the Separation principle holds for a given effective projective Sigma-class
• Indiscernible pairs of countable sets of reals at a given projective level