The University of Chicago Topics in Recursively Enumerable Sets and Degrees a Dissertation Submitted to the Faculty of the Division of the Physical Sciences in Candidacy for the Degree of Doctor of Philosophy Department of Mathematics by Steffen Lempp

In Chapter I, we exhibit a high strongly noncappable degree. Chapter II answers negatively the question whether a deep degree exists. It also shows a weak converse of this. Chapter III is devoted to index sets. We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy. We also classify the index set of quasimaximal sets, of coinfinite r.e. sets not having an atomless superset, and of r.e. sets major in a fixed nonrecursive r.e. set. Chapter IV investigates properties of the partial order of w-degrees. We show that the w-degree of 0 splits and that there is a minimal pair in the r.e. w-degrees. A forcing argument shows that the w-degrees below 0(w) do not form an upper semilattice.