A Groszek-Laver pair of undistinguishable $E_0$ 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 ${\Pi }_2^1$ set, but neither of these two ${\mathsf{E}}_0$-classes is separately ordinal-definable.