| Abstract: | We study the degrees of unsolvability of sets which are cohesive (or have weaker recursion-theoretic “smallness” properties). We answer a question raised by the first author in 1972 by showing that there is a cohesive set A whose degree a satisfies a' = 0″ and hence is not high. We characterize the jumps of the degrees of r-cohesive sets, and we show that the degrees of r-cohesive sets coincide with those of the cohesive sets. We obtain analogous results for strongly hyperimmune and strongly hyperhyperimmune sets in place of r-cohesive and cohesive sets, respectively. We show that every strongly hyperimmune set whose degree contains either a Boolean combination of ∑2 sets or a 1-generic set is of high degree. We also study primitive recursive analogues of these notions and in this case we characterize the corresponding degrees exactly. MSC: 03D30, 03D55. |
