Local structure theory and the Ershov hierarchy

Abstract

This thesis is concerned with three special properties of Turing degree structure and the Ershov hierarchy. We study the distributions of the nonhemimaximal c.e. degrees, noncomputable left c.e. reals with only computable presentations, and the cupping property in the Ershov hierarchy. A general introduction is presented in Chapter 1. In Chapter 2, we study the distribution of nonhemimaximal c.e. degrees. Here, we give an alternative proof of the existence of a low2, but not low, nonhemimaximal c.e. degree, where the technique used is a 0′′ -priority argument. In Chapter 3, we investigate left c.e. reals and prove that below any high c.e. degree, there is a noncomputable left c.e. real with only computable presentations. The proof of this result utilizes the machinery developed by Shore and Slaman. In Chapter 4, we study the complements of cappable c.e. degrees. We prove that for any nonzero cappable c.e. degree c, there is an almost universal cupping d.c.e. degree d and a c.e. degree b < d such that (i) b and c form a minimal pair; and (ii) b bounds all c.e. degrees below d.