Constructions in the Recursively Enumerable Degrees

Abstract

In these lectures we survey some of the most important results and the fundamental methods concerning degrees of recursively enumerable (r. e.) sets. 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 Superscript>remums. 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 [17], Shoenfield [23], and Soare [25].