A definable $\mathsf E_0$-class containing no definable elements

TL;DR

The paper constructs a generic extension of the constructible universe where an E_0-equivalence class contains no elements that are definable from ordinals, highlighting complex definability properties in set theory.

Contribution

It introduces a new generic extension where an E_0-class is a lightface Pi^1_2 set with no ordinal-definable reals, advancing understanding of definability in set-theoretic extensions.

Findings

E_0-class can be made non-definable in certain generic extensions.

Constructs a lightface Pi^1_2 set containing no OD reals.

Shows the existence of non-definable classes in generic extensions.

Abstract

A generic extension $L[x]$ of $L$ by a real $x$ is defined, in which the $\mathsf E_0$-class of $x$ is a lightface $\Pi^1_2$ set containing no ordinal-definable reals.