Abstract
Cholak, Groszek and Slaman proved in J Symb Log 66:881–901, 2001 that there is a nonzero computably enumerable (c.e.) degree cupping every low c.e. degree to a low c.e. degree. In the same paper, they pointed out that every nonzero c.e. degree can cup a low2 c.e. degree to a nonlow2 degree. In Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) improved the latter result by showing that every nonzero c.e. degree c is cuppable to a high c.e. degree by a low2 c.e. degree b. It is natural to ask in which subclass of low2 c.e. degrees can b in Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) be located. Wu proved in Math Log Quart 50:189–201, 2004 that b can be cappable. We prove in this paper that b in Jockusch, Li and Yang’s result can be noncuppable, improving both Jockusch, Li and Yang, and Wu’s results.
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References
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Cholak P., Groszek M., Slaman T.: An almost deep degree. J. Symb. Log. 66, 881–901 (2001)
Jockusch C.G. Jr, Li A., Yang Y.: A join theorem for the computably enumerable degrees. Trans. Am. Math. Soc. 356, 2557–2568 (2004)
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Wu G.: Quasi-complements of the cappable degrees. Math. Log. Quart. 50, 189–201 (2004)
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Wu is partially supported by a research grant No. RG58/06 from NTU.
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Liu, J., Wu, G. Joining to high degrees via noncuppables. Arch. Math. Logic 49, 195–211 (2010). https://doi.org/10.1007/s00153-009-0165-z
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DOI: https://doi.org/10.1007/s00153-009-0165-z