Analytic Equivalence Relations Satisfying Hyperarithmetic-Is-Recursive

Abstract

We prove, in ZF + Σ 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 ω1 . We also show that the implication from (1) to (2) is equivalent to the existence of sharps over Z F.

Bi‐embeddability spectra and bases of spectra

“…Definitions analogous to this were given by Liang and Montalbán . Under this notion the classical degree spectrum of a structure is .…”

A recursion theoretic characterization of the Topological Vaught Conjecture in the Zermelo‐Fraenkel set theory

“…Remark (a) Montalbán proved that under the axiom of ‐determinacy an analytic equivalence relation E on 2 ω does not have perfectly many classes exactly when there is some such that for all every is E ‐equivalent to some (the latter property is called HYP‐is‐recursive on a cone ). Moreover he showed that the converse direction is in fact provable in .…”

“…The proof of the converse direction when we have an arbitrary 1 1 equivalence relation E in a recursive Polish space X is exactly the same. [11] proved that under the axiom of 1 1 -determinacy an analytic equivalence relation E on 2 ω does not have perfectly many classes exactly when there is some ε ∈ N such that for all α ≥ T ε every x ≤ h α is E-equivalent to some y ≤ T α (the latter property is called HYP-is-recursive on a cone). Moreover he showed that the converse direction is in fact provable in ZF.…”

“…The question of characterizing the (Topological) Vaught Conjecture in terms of recursion theory has been investigated by Montalbán , where he provides the answer under some determinacy hypothesis. In this note we give one more characterization of the topological version of the conjecture in a recursion theoretic language, which is actually provable in the Zermelo‐Fraenkel set theory .…”

“…In this note we give one more characterization of the topological version of the conjecture in a recursion theoretic language, which is actually provable in the Zermelo-Fraenkel set theory ZF. Another distinctive aspect of our characterization is that, unlike [11], it refers to the Polish group actions rather than to the more general case of analytic equivalence relations, although one of the directions holds in the latter case. We do not know if the other direction is also true in the general case.…”