Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T12:30:30.990Z Has data issue: false hasContentIssue false

The cupping theorem in R/M

Published online by Cambridge University Press:  12 March 2014

Sui Yuefei
Affiliation:
Institute of Software, Academia Sinica, P.O. Box 8718, Beijing, China E-mail: [email protected]
Zhang Zaiyue
Affiliation:
Department of Mathematics, Yangzhou Teaching College, Yangzhou, Jiangsu, China

Abstract

It will be proved that the Shoenfield cupping conjecture holds in R/M, the quotient of the recursively enumerable degrees modulo the cappable r.e. degrees. Namely, for any [a], [b] ∈ R/M such that [0] ≺ [b] ≺ [a] there exists [c] ∈ R/M such that [c] ≺ [a] and [a] = [b] ∨ [c].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Ambos-Spies, K., Jockusch, C.G. Jr., Shore, R.A., and Soare, R.I., An algebraic decomposition of the recursively enumerable degrees and the coincidence of several classes with the promptly simple degrees, Transactions of the American Mathematical Society, vol. 281 (1984), pp. 109–128.CrossRefGoogle Scholar
[2] Jockusch, C.G. Jr., Review of Schwarz [1984], Mathematical Reviews, 851:3777.Google Scholar
[3] Lachlan, A.H., The impossibility of finding relative complements for recursively enumerable degrees, this Journal, vol. 31 (1966), pp. 434–454.Google Scholar
[4] Ladner, R.E. and Sasso, L.P., The weak truth table degrees of recursively enumerable sets, Annals of Mathematical Logic, vol. 4 (1975), pp. 429–448.Google Scholar
[5] Lempp, S. and Sui, Y., An Extended Lachlan splitting theorem, Annals of Pure and Applied Logic, to appear.Google Scholar
[6] Miller, D., High recursively enumerable degrees and the anti-cupping property, Proceedings of logic year 1979–80, Springer Lecture Notes in Math, vol. 859, Springer-Verlag, 1981, pp. 230–267.Google Scholar
[7] Sacks, G.E., Recursive enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), pp. 300–312.CrossRefGoogle Scholar
[8] Schwarz, S., The quotient semilattice of the recursively enumerable degrees modulo the cappable degrees, Transactions of the American Mathematical Society, vol. 283 (1984), pp. 315–328.CrossRefGoogle Scholar
[9] Shoenfield, J.R., Applications of model theory to degrees of unsolvability, Symposium on the theory of Models (1965), pp. 359–363.Google Scholar
[10] Soare, R.I., Recursively enumerable sets and degrees, Ω-series, Springer-Verlag, 1987.CrossRefGoogle Scholar
[11] Yates, C.E.M., A minimal pair of recursively enumerable sets, this Journal, vol. 31 (1966), pp. 159–168.Google Scholar