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Abstract

This thesis analyzes the structure of the Medvedev lattice of non-empty Π 0 1 classes in 2 ω from the viewpoint of branching and non-branching degrees. This lattice is a countable distributive lattice with least and greatest element, which describes the relative information content of certain subsets of 2 ω. Chapter 1 is an introduction, providing background history, notation, and an overview of necessary concepts. Chapter 2 is essentially my paper “Non-Branching Degrees in the Medvedev Lat-tice of Π 0 1 classes.”[1]. The chapter adds an additional theorem which strengthens the theorem on inseparable and not hyperinseparable classes. The chapter is also slightly more verbose. We begin by taking an existing condition, homogeneous, which implies non-branching and define two successively weaker conditions, hyperinseparable and inseparable. We then demonstrate that inseparable is equivalent to non-branching and is invariant under Medvedev equivalence. Finally, we prove separation theorems, namely the existence of an inseparable and not hyperinseparable degree and the existence of a hyperinseparable and not homogeneous degree. Chapter 3 defines a combinatorial method for constructing Π 0 1 classes by priority arguments. This section does not contain any difficult proofs but abstracts many of the common elements of such constructions. The definitions and results are used i