In $\mathsf{ ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \in L(\mathbb{ R})$ on $\mathbb{ R}$ with all $\mathbf{ \Delta}_1^1$ classes and every $\sigma$-ideal $I$ on $\mathbb{ R}$ so that the associated forcing $\mathbb{ P}_I$ of $I^+$ $\mathbf{ \Delta}_1^1$ subsets is proper, there exists some $I^+$ $\mathbf{ \Delta}_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbf{ \Delta}_1^1$ equivalence relation. In $\mathsf{ ZF} + \mathsf{ DC} + \mathsf{ AD}_\mathbb{ R} + V = L(\mathscr{ P}(\mathbb{ R}))$, for every equivalence relation $E$ on $\mathbb{ R}$ with all $\mathbf{ \Delta}_1^1$ classes and every $\sigma$-ideal $I$ on $\mathbb{ R}$ so that the associated forcing $\mathbb{ P}_I$ is proper, there is some $I^+$ $\mathbf{ \Delta}_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbf{ \Delta}_1^1$ equivalence relation.
Categories: math.LO