Index sets and degrees of unsolvability

The characterization of classes of r.e. sets by their index sets has proved valuable in producing new results about the r.e. sets and degrees. The classic example is Yates' proof [5, Theorem 7] of Sacks' density theorem for r.e. degrees using his classification of {e: We ≤TD) as Σ3(D) whenever D is r.e. Theorem 1 of this paper is a refinement of this index set theorem of Yates which has already proved to have interesting consequences about the r.e. degrees. This theorem was originally announced by Kallibekov [1, Theorem 1]. Kallibekov there proposed a new and ingenious method for doing priority arguments which has also since been used by Kinber [2]. Unfortunately his proof to this particular theorem contains an error. We have a totally different proof using standard techniques which is of independent interest.The proof to Theorem 1 is an infinite injury priority argument. In §1 therefore we give a short summary of the infinite injury priority method. We draw heavily on the exposition of Soare [4] where a complete description of the method is given along with many examples. In §2 we prove the main theorem and also give what we think are the most interesting corollaries to this theorem announced by Kallibekov. In §3 we prove a theorem about Σ3 sets of indices of r.e. sets. This theorem is a strengthening of a theorem of Kinber [2, Theorem 1] which was proved using a modification of Kallibekov's technique. As application, we use our theorem to show that an r.e. set A has supersets of every r.e. degree iff A is not simple.