Abstract
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, ≤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.
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The research of the second author was supported by the Georgian National Science Foundation (Grants #GNSF/ST07/3-178 and #GNSF/ST08/3-391).
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Chitaia, I.O., Omanadze, R.S. & Sorbi, A. Immunity properties and strong positive reducibilities. Arch. Math. Logic 50, 341–352 (2011). https://doi.org/10.1007/s00153-010-0216-5
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DOI: https://doi.org/10.1007/s00153-010-0216-5