Analytic equivalence relations satisfying hyperarithmetic-is-recursive

TL;DR

This paper establishes equivalences between properties of analytic equivalence relations under certain set-theoretic assumptions, linking class structure, hyperarithmetic computability, and the existence of sharps.

Contribution

It proves the equivalence of three conditions for analytic equivalence relations under ZF and $f ext{Sigma}^1_2$-determinacy, connecting class size, hyperarithmetic-is-recursive behavior, and oracle computations.

Findings

Equivalence of non-perfect class size and hyperarithmetic-is-recursive on a cone.

Connection between class structure and the existence of sharps.

Implication from class size to hyperarithmetic-is-recursive is equivalent to sharps existence.

Abstract

We prove, in ZF+$\bf\Sigma^1_2$-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 $\om_1^X\geq\om_1^{[Y]}$. We also show that the implication from (1) to (2) is equivalent to the existence of sharps over $ZF$.