Definability in the Recursively Enumerable Degrees

§1. Introduction . Natural sets that can be enumerated by a computable function (the recursively enumerable or r.e. sets ) always seem to be either actually computable ( recursive ) or of the same complexity (with respect to Turing computability) as the Halting Problem, the complete r.e. set K . The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K ? Let be the r.e. degrees, i.e., the r.e. sets modulo the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order (indeed, an uppersemilattice or usl )with least element 0 , the degree (equivalence class) of the computable sets, and greatest element 1 or 0 ′, the degree of K . Post's problem then asks if there are any other elements of . The (positive) solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure was in some way well behaved: T heorem 1.1 (Embedding theorem; Muchnik [1958], Sacks [1963]). Every countable partial ordering or even uppersemilattice can be embedded into . T heorem 1.2 (Sacks Splitting Theorem [1963b]). For every nonrecursive r.e. degree a there are r.e. degrees b, c a such that b ∨ c = a . T heorem 1.3 (Sacks Density Theorem [1964]). For every pair of nonrecursive r.e. degrees a b there is an r.e. degree c such that a c b .