{\alpha} degrees as an automorphism base for the {\alpha}-enumeration degrees

Dávid Natingga

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arXiv · 1812.11308

{\alpha} degrees as an automorphism base for the {\alpha}-enumeration degrees

Dávid Natingga

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Abstract

Selman's Theorem in classical Computability Theory gives a characterization of the enumeration reducibility for arbitrary sets in terms of the enumeration reducibility on the total sets: $A \le_e B \iff \forall X [X \equiv_{e} X \oplus \overline{X} \land B \le_{e} X \oplus \overline{X} \implies A \le_{e} X \oplus. \overline{X} ]$. This theorem implies directly that the Turing degrees are an automorphism base of the enumeration degrees. We lift the classical proof to the setting of the $\alpha$-Computability Theory to obtain the full generalization when $\alpha$ is a regular cardinal and partial results for a general admissible ordinal $\alpha$.

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BibTeX RIS

Dávid Natingga. 2018-12-29. {\alpha} degrees as an automorphism base for the {\alpha}-enumeration degrees. https://arxiv.org/abs/1812.11308

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