Natingga, David (2019) Embedding Theorem for the automorphism group of the αenumeration 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 αenumeration 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 αComputability theory. The main result of this thesis is Embedding Theorem: the automorphism group of the αenumeration degrees D_αe embeds into the automorphism group of the αdegrees 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 αrational 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, αComputability Theory, αenumeration 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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