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- W3005197822 abstract "It is a theorem of classical Computability Theory that the automorphism group of the enumeration degrees D_e embeds into the automorphism group of the Turing degrees D_T . This follows from the following three statements:1. D_T embeds to D_e ,2. D_T is an automorphism base for D_e,3. D_T is definable in D_e .The first statement is trivial. The second statement follows from the Selman’s theorem:A ≤e B ⇐⇒ ∀X ⊆ ω[B ≤e X ⊕ complement(X) implies A ≤e X ⊕ complement(X)].The third statement follows from the definability of a Kalimullin pair in the α-enumeration degrees D_e and the following theorem:an enumeration degree is total iff it is trivial or a join of a maximal Kalimullin pair.Following an analogous pattern, this thesis aims to generalize the results above to the setting of α-Computability theory. The main result of this thesis is Embedding Theorem:the automorphism group of the α-enumeration degrees D_αeembeds into the automorphism group of the α-degrees D_α if α is an infinite regular cardinal and assuming the axiom of constructibility V = L. If α is a general admissible ordinal, weaker results are proved involving assumptions on the megaregularity.In the proof of the definability of D_α in D_αe a helpful concept of α-rational numbers Q_α emerges as a generalization of the rational numbers Q and an analogue of hyperrationals. This is the most valuable theory development of this thesis with many potentially fruitful directions." @default.
- W3005197822 created "2020-02-14" @default.
- W3005197822 creator A5087060051 @default.
- W3005197822 date "2019-04-01" @default.
- W3005197822 modified "2023-09-26" @default.
- W3005197822 title "Embedding Theorem for the automorphism group of the α-enumeration degrees" @default.
- W3005197822 hasPublicationYear "2019" @default.
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