ANALYTIC EQUIVALENCE RELATIONS SATISFYING HYPERARITHMETIC-IS-RECURSIVE

Abstract

We prove, in $\text{ZF}+\boldsymbol{{\it\Sigma}}_{2}^{1}$-determinacy, that, for any analytic equivalence relation $E$, the following three statements are equivalent: (1) $E$ does not have perfectly many classes, (2) $E$ satisfies hyperarithmetic-is-recursive on a cone, and (3) relative to some oracle, for every equivalence class $[Y]_{E}$ we have that a real $X$ computes a member of the equivalence class if and only if ${\it\omega}_{1}^{X}\geqslant {\it\omega}_{1}^{[Y]}$. We also show that the implication from (1) to (2) is equivalent to the existence of sharps over $ZF$.