Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T00:49:31.369Z Has data issue: false hasContentIssue false

On a problem of Cooper and Epstein

Published online by Cambridge University Press:  12 March 2014

Shamil Ishmukhametov*
Affiliation:
42 Tolstoy St., Dept. Math., Ulyanovsk State University, 432700, Ulyanovsk, Russia, E-mail: ishm@vens.ru

Abstract

In “Bounding minimal degrees by computably enumerable degrees” by A. Li and D. Yang, (this Journal, [1998]), the authors prove that there exist non-computable computably enumerable degrees c > a > 0 such that any minimal degree m being below c is also below a. We analyze the proof of their result and show that the proof contains a mistake. Instead we give a proof for the opposite result.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2003

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

Cooper, S. B. [1989], The strong anticupping property for recursively enumerable degrees, this Journal, vol. 54, pp. 527539.Google Scholar
Cooper, S. B. [1994], Rigidity and definability in the noncomputable universe, Logic, methodology and philosophy of science IX (Prawitz, D., Skyrms, B., and Westerdahl, D., editors), North-Holland, Amsterdam, pp. 209236.Google Scholar
Cooper, S. B. and Epstein, R. [1987], Complementing below recursively enumerable degrees, Annals of Pure and Applied Logic, vol. 34, pp. 1532.CrossRefGoogle Scholar
Downey, R. and Stob, M. [1997], Minimal pairs in initial segments of the recursively enumerable degrees, Israel Journal of Mathematics, vol. 100, pp. 727.CrossRefGoogle Scholar
Epstein, R.L. [1975], Minimal degrees of unsolvability and the full approximation construction, Memoirs of the American Mathematical Society, no. 162.CrossRefGoogle Scholar
Epstein, R.L. [1979], Degrees ofunsolvability: structure and theory, Lecture Notes in Mathematics no. 759, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Epstein, R.L. [1981], Initial segments of degrees below 0′, Memoirs of the American Mathematical Society, no. 241.CrossRefGoogle Scholar
Li, A. and Yang, D. [1998], Bounding minimal degrees by computably enumerable degrees, this Journal, vol. 63, pp. 13191347.Google Scholar
Seetapun, D. and Slaman, T. [1992], Minimal complements, unpublished manuscript.Google Scholar
Slaman, T. and Steel, J.R. [1989], Complementation in the Turing degrees, this Journal, vol. 54, pp. 160176.Google Scholar
Soare, R.I. [1987], Recursively enumerable sets and degrees, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Spector, C. [1956], On degrees of recursive unsolvability, Annals of Mathematics. Second Series, vol. 64, pp. 581592.CrossRefGoogle Scholar
Yates, C. E. M. [1970], Initial segments of the degrees of unsolvability. part II. Minimal degrees, this Journal, vol. 35, pp. 243266.Google Scholar