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

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

Classical Theorems in Reverse Mathematics and Higher Recursion Theory

Ph.DDOCTOR OF PHILOSOPH