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On the jump classes of noncuppable enumeration degrees

Published online by Cambridge University Press:  12 March 2014

Charles M. Harris
Charles M. Harris*
Affiliation:
University of Leeds, Department of Mathematics, Leeds, UK, E-mail: [email protected], URL: http://www.maths.leeds.ac.uk/~charlie
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Abstract

We prove that for every Σ20 enumeration degree b there exists a noncuppable Σ20 degree a > 0e such that and . This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding , that there exist Σ20 noncuppable enumeration degrees at every possible—i.e., above low1—level of the high/low jump hierarchy in the context of .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 1 , March 2011 , pp. 177 - 197
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[Co90]Cooper, S. B., Enumeration reducibility, nondeterministic computations and relative computability of partial functions, Recursion theory week, Oberwolfach 1989 (Ambos-Spies, K., Müller, G., and Sacks, G. E., editors), Lecture Notes in Mathematics, vol. 1432, Springer Verlag, Heidelberg, 1990, pp. 57–110.Google Scholar
[Co04]Cooper, S. B., Computability theory, Chapman and Hall, 2004.Google Scholar
[CMc85]Cooper, S. B. and McEvoy, K., On minimal pairs of enumeration degrees, this Journal, vol. 50(4) (1985), pp. 983–1001.Google Scholar
[CSY]Cooper, S. B, Sorbi, A., and Yi, X., Cupping and noncupping in the enumeration degrees of Σ20 sets, Annals of Pure and Applied Logic, vol. 82 (1996), pp. 317–342.CrossRefGoogle Scholar
[Gi08]Giorgi, M., A high noncuppable Σ20 e-degree, Archive for Mathematical Logic, vol. 47 (2008), pp. 181–191.CrossRefGoogle Scholar
[GSY]Giorgi, M., Sorbi, A., and Yang, Y., Properly Σ20 enumeration degrees and the high/low hierarchy, this Journal, vol. 71(4) (2006), pp. 1125–1144.Google Scholar
[Gr03]Griffith, E. J., Limit lemmas and jump inversion in the enumeration degrees, Archive for Mathematical Logic, vol. 42 (2003), pp. 553–562.CrossRefGoogle Scholar
[Har10]Harris, C. M., Goodness in the enumeration and singleton degrees, Archive for Mathematical Logic, vol. 49 (2010), no. 6, pp. 673–691.CrossRefGoogle Scholar
[Har]Harris, C. M., Noncuppable enumeration degrees via finite injury, Journal of Logic and Computation, to appear.Google Scholar
[Jo68]Jockusch, C. G., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420–436.CrossRefGoogle Scholar
[LS92]Lachlan, H. and Shore, R. A., The n-rea enumeration degrees are dense, Archive for Mathematical Logic, vol. 31 (1992), pp. 277–285.CrossRefGoogle Scholar
[Mc84]McEvoy, K., The structure of the enumeration degrees, Ph.D. thesis, The University of Leeds, UK, 10 1984.Google Scholar
[NS00]Nies, A. and Sorbi, A., Structural properties and Σ20 enumeration degrees, this Journal, vol. 65(1) (2000), pp. 285–289.Google Scholar
[Od89]Odifreddi, P. G., Classical recursion theory, Elsevier, Amsterdam, 1989.Google Scholar
[Sa63a]Sacks, G. E., Recursive enumerability and the jump operator, Transactions of the American Mathematical Society, vol. 108 (1963), pp. 223–239.CrossRefGoogle Scholar
[Sa67]Sacks, G. E., On a theorem of Lachlan and Martin, Proceedings of the American Mathematical Society, vol. 18 (1967), pp. 140–141.CrossRefGoogle Scholar
[So87]Soare, R. I., Recursively enumerable sets and degrees, Springer, Berlin, 1987.CrossRefGoogle Scholar
[So97]Sorbi, A., The enumeration degrees of the Σ20 sets, Complexity, logic and recursion theory (Sorbi, A., editor), Marcel Dekker, New York, 1997, pp. 303–330.Google Scholar