A Groszek-Laver pair of undistinguishable $E_0$ classes

TL;DR

This paper constructs a generic extension where two reals have E_0-classes whose union is complex, yet neither class is individually definable, revealing nuanced properties of definability and class union.

Contribution

It introduces a specific generic extension demonstrating that E_0-classes can have a union with complex definability properties without each class being individually definable.

Findings

Union of E_0-classes is a Pi^1_2 set

Neither E_0-class is individually ordinal-definable

Extension illustrates complex definability behavior of classes

Abstract

A generic extension $L[x,y]$ of $L$ by reals $x,y$ is defined, in which the union of $\mathsf E_0$-classes of $x$ and $y$ is a $\Pi^1_2$ set, but neither of these two $\mathsf E_0$-classes is separately ordinal-definable.