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Noncappable enumeration degrees below 0e

Published online by Cambridge University Press:  12 March 2014

S. Barry Cooper
Affiliation:
School of Mathematics, University of Leeds, LS2 9JT, England, E-mail: s.b.cooper@leeds.ac.uk
Andrea Sorbi
Affiliation:
Department of Mathematics, University of Siena, 53100 Siena, Italy, E-mail: sorbi@unisi.it

Abstract

We prove that there exists a noncappable enumeration degree strictly below 0e.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1996

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References

REFERENCES

[1]Ahmad, S., Embedding the diamond in the Σ2 enumeration degrees, this Journal, vol. 56 (1991), pp. 195212.Google Scholar
[2]Ambos-Spies, K., On the structure of the recursively enumerable degrees, Ph.D. thesis, University of Munich, 1980.Google Scholar
[3]Ambos-Spies, K., On pairs of recursively enumerable degrees, Transactions of the American Mathematical Society, vol. 283 (1984), pp. 507531.CrossRefGoogle Scholar
[4]Case, J., Enumeration reducibility and partial degrees, Annals of Mathematical Logic, vol. 2 (1971), pp. 419439.CrossRefGoogle Scholar
[5]Cooper, S. B., Partial degrees and the density problem, this Journal, vol. 47 (1982), pp. 854859.Google Scholar
[6]Cooper, S. B., Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense, this Journal, vol. 49 (1984), pp. 503513.Google Scholar
[7]Cooper, S. B., Enumeration reducibility using bounded information: counting minimal covers, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), pp. 537560.CrossRefGoogle Scholar
[8]Cooper, S. B., Enumeration reducibility, non-deterministic computations and relative computability of partial functions, Recursion Theory Week (Heidelberg) (Ambos-Spies, K., Müller, G. H., and Sacks, G. E., editors), Lecture Notes in Mathematics, no. 1432, Springer-Verlag, year?, Proceedings Oberwolfach 1989, pp. 57110.Google Scholar
[9]Cooper, S. B. and Copestake, C. S., Properly Σ2 enumeration degrees, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 34 (1988), pp. 491522.CrossRefGoogle Scholar
[10]Cooper, S. B., Slaman, T. A., and Yi, X., The Σ2 theory of the recursively enumerable degrees, to appear.Google Scholar
[11]Friedberg, R. M. and Rogers, H. Jr., Reducibility and completeness for sets of integers, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 117125.CrossRefGoogle Scholar
[12]Gutteridge, L., Some results on enumeration reducibility, Ph.D. thesis, Simon Fraser University, 1971.Google Scholar
[13]Jockush, C. G. Jr., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420436.CrossRefGoogle Scholar
[14]Lagemann, J., Embedding theorems in the reducibility ordering of the partial degrees, Ph.D. thesis, MIT, 1972.Google Scholar
[15]Lerman, M., Degrees of Unsolvability, Perspectives in Mathematical Logic, Omega series, Springer-Verlag, Berlin, 1983.CrossRefGoogle Scholar
[16]McEvoy, K., Jumps of quasi-minimal enumeration degrees, this Journal, vol. 50 (1985), pp. 839848.Google Scholar
[17]McEvoy, K. and Cooper, S. B., On minimal pairs of enumeration degrees, this Journal, vol. 50 (1985), pp. 983–100.Google Scholar
[18]Odifreddi, P., Classical recursion theory, North-Holland, Amsterdam, 1989.Google Scholar
[19]Posner, D., The upper semilattice of degrees ≤ 0′ is complemented, this Journal, vol. 46 (1981), pp. 705713.Google Scholar
[20]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[21]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[22]Yates, C. E. M., A minimal pair of recursively enumerable degrees, this Journal, vol. 31 (1966), pp. 159168.Google Scholar