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.
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Chitaia’s work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) (Grant number: YS-18-168). Ng was partially supported by the Grants MOE-RG131/17 and MOE2015-T2-2-055. Sorbi was partially supported by PRIN 2017 Grant “Mathematical Logic: models, sets, computability”. Sorbi is a member of INDAM-GNSAGA. Yang’s research was partially supported by NUS Grant R-146-000-231-114.
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Chitaia, I., Ng, K.M., Sorbi, A. et al. Incomparability in local structures of s-degrees and Q-degrees. Arch. Math. Logic 59, 777–791 (2020). https://doi.org/10.1007/s00153-020-00714-x
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DOI: https://doi.org/10.1007/s00153-020-00714-x