Turing Degrees of Hyperjumps

Abstract

The Posner-Robinson Theorem states that for any reals $Z$ and $A$ such that $Z\oplus 0^{\prime }{\le }_{\mathrm{T}}A$ and $0{<}_{\mathrm{T}}Z$, there exists $B$ such that $A{\equiv }_{\mathrm{T}}{B}^{\prime }{\equiv }_{\mathrm{T}}B\oplus Z{\equiv }_{\mathrm{T}}B\oplus 0^{\prime }$. Consequently, any nonzero Turing degree ${\mathrm{deg}}_{\mathrm{T}}(Z)$ 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 Here we prove the hyperarithmetical analog, based on an unpublished proof of Slaman, namely that for any reals $Z$ and $A$ such that $Z\oplus \mathcal{O}{\le }_{\mathrm{T}}A$ and $0{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 ${\mathrm{deg}}_{\mathrm{T}}(Z)$ is a hyperjump relative to some $B$.

arXiv comment16 pages, submitted for publication in Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, 2020