TURNING DEGREES CONSTRUCTIONS AND THEIR INDUCTION STRENGTH IN REVERSE MATHEMATICS

Abstract

In this thesis, we study Turing degrees in the context of classical computability theory (recursion Theory) and reverse recursion theory. In the first part, we prove that P + \Sigma_1 Induction proves the existence of isolated d.c.e. degrees and upper isolated d.c.e. degrees. We also prove that P + \Sigma_3 Induction verifies the existence of \Omega-c.e. universal cupping degrees.

In the second part, we study another algebraic decomposition of R where R denotes the class of computably enumerable degrees. Ambos-Spies et al. (1984) proved that R = M \cup PS. We first define another subclass of R, STB, which consists of 0 together with all degrees for the base (set) in any Slaman triple. We will show R = STB \cup PS in the sense that given any c.e. set A, either there exist c.e. sets B and C such that (A,B,C) is a Slaman triple or A has promptly simple degree.