Turing Degrees of Hyperjumps

TL;DR

This paper extends the Posner-Robinson Theorem to the hyperarithmetical setting, showing that nonhyperarithmetical degrees are hyperjumps relative to some oracle, thus broadening the understanding of Turing degrees and hyperjumps.

Contribution

It proves a hyperarithmetical analog of the Posner-Robinson Theorem, linking nonhyperarithmetical degrees to hyperjumps relative to some oracle.

Findings

Established a hyperarithmetical analog of the Posner-Robinson Theorem.

Showed that nonhyperarithmetical degrees are hyperjumps relative to some oracle.

Extended classical results to the hyperarithmetical hierarchy.

Abstract

The Posner-Robinson Theorem states that for any reals $Z$ and $A$ such that $Z \oplus 0' \leq_\mathrm{T} A$ and $0 <_\mathrm{T} Z$, there exists $B$ such that $A \equiv_\mathrm{T} B' \equiv_\mathrm{T} B \oplus Z \equiv_\mathrm{T} B \oplus 0'$. Consequently, any nonzero Turing degree $\operatorname{deg}_\mathrm{T}(Z)$ is a Turing jump relative to some $B$. Here we prove the hyperarithmetical analog, based on an unpublished proof of Slaman, namely that for any reals $Z$ and $A$ such that $Z \oplus \mathcal{O} \leq_\mathrm{T} A$ and $0 <_\mathrm{HYP} Z$, there exists $B$ such that $A \equiv_\mathrm{T} \mathcal{O}^B \equiv_\mathrm{T} B \oplus Z \equiv_\mathrm{T} B \oplus \mathcal{O}$. As an analogous consequence, any nonhyperarithmetical Turing degree $\operatorname{deg}_\mathrm{T}(Z)$ is a hyperjump relative to some $B$.