A recursion theoretic property of $\mathbf {\Sigma ^1_1}$ equivalence relations

Abstract:

Let $E$ be a $\mathbf {\Sigma ^1_1}$ equivalence relation on $2^\omega$ which does not have perfectly many equivalence classes. For $a \in 2^\omega$, define $L^a_E$ to be the set $\{[x]_E: (\exists y) (y \in [x]_E$ and $\omega _1^{\langle a,y \rangle } = \omega _1^a)\}$. For a Turing cone of $a$’s, $L^a_E$ is countable. This is proved assuming $\mathbf {\Pi }^1_2$-determinacy.