Incomparability in local structures of s-degrees and Q-degrees

Abstract

We show that for every intermediate $\Sigma ^0_2$ s-degree (i.e. a nonzero s-degree strictly below the s-degree of the complement of the halting set) there exists an incomparable $\Pi ^0_1$ s-degree. (The same proof yields a similar result for other positive reducibilities as well, including enumeration reducibility.) As a consequence, for every intermediate $\Pi ^0_2$ Q-degree (i.e. a nonzero Q-degree strictly below the Q-degree of the halting set) there exists an incomparable $\Sigma ^0_1$ Q-degree. We also show how these results can be applied to provide proofs or new proofs (essentially already known, although some of them not explicitly noted in the literature) of upper density results in local structures of s-degrees and Q-degrees.