Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
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We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has $${\overline{K}\not\le_{\rm ss} B}$$ (respectively, $${\overline{K}\not\le_{\overline{\rm s}} B}$$ ): here $${\le_{\overline{\rm s}}}$$ is the finite-branch version of s-reducibility, 驴ss is the computably bounded version of $${\le_{\overline{\rm s}}}$$ , and $${\overline{K}}$$ is the complement of the halting set. Restriction to $${\Sigma^0_2}$$ sets provides a similar characterization of the $${\Sigma^0_2}$$ hyperhyperimmune sets in terms of s-reducibility. We also show that no $${A \geq_{\overline{\rm s}}\overline{K}}$$ is hyperhyperimmune. As a consequence, $${\deg_{\rm s}(\overline{K})}$$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.