INTRODUCTION

In these lectures we survey some of the most important results and the fundamental methods concerning degrees of recursively enumerable (r. e.) sets. This material is similar to our C. I. M. E. lectures (Soare, 1980) except that there §7 was on the Renaissance in classical recursion theory, the Sacks density theorem and beyond, while here §7 deals with noncappable degrees and §8 with nonbranching degrees.

We begin §1 with Post's simple sets and a recent elegant generalization of the recursion theorem. In § 2 we give the finite injury priority method, the solution of Post's problem, and the Sacks splitting theorem. In § 3 the infinite injury method is introduced and applied to prove the thickness lemma and the Sacks density theorem. In §4 and §5 we develop the minimal pair method for embedding distributive lattices in the r. e. degrees by maps preserving infimums as well as supremums. In §6 we present the non-diamond theorem which asserts that such embeddings cannot always preserve greatest and least elements. For background reading we suggest Rogers (1967), Shoenfield (1971), and Soare (1982).

Our notation is standard as in Rogers (1967), with a few additions. For sets A, B, C ω we say B is recursive in A (B ≤T A) if there is a Turing reduction Φ such that Φ(A) = B. Let A ≡T B denote that A ≤T B and B ≤T A. The degree of A, dg(A) = {B: B ≡T A}.