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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2101.08818 |
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| _version_ | 1866914877440786432 |
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| author | Jananthan, Hayden R. Simpson, Stephen G. |
| author_facet | Jananthan, Hayden R. Simpson, Stephen G. |
| contents | 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 Here we prove the hyperarithmetical analog, based on an unpublished proof of Slaman, namely that for any reals $Z$ and $A$ such that $Z \oplus \mathcal{O} \leq_\mathrm{T} A$ and $0 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 $B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_08818 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Turing Degrees of Hyperjumps Jananthan, Hayden R. Simpson, Stephen G. Logic 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 Here we prove the hyperarithmetical analog, based on an unpublished proof of Slaman, namely that for any reals $Z$ and $A$ such that $Z \oplus \mathcal{O} \leq_\mathrm{T} A$ and $0 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 $B$. |
| title | Turing Degrees of Hyperjumps |
| topic | Logic |
| url | https://arxiv.org/abs/2101.08818 |