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Embedding Theorem for the automorphism group of the α-enumeration degrees

Natingga, David (2019) Embedding Theorem for the automorphism group of the α-enumeration degrees. PhD thesis, University of Leeds.

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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_αe embeds 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.

Item Type: Thesis (PhD)
Keywords: higher computability theory, α-Computability Theory, α-enumeration degrees, automorphism groups of degree structures, definability in degree structures
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.794173
Depositing User: David Natingga
Date Deposited: 13 Jan 2020 13:52
Last Modified: 18 Feb 2020 12:51
URI: http://etheses.whiterose.ac.uk/id/eprint/25517

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