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On a problem of Cooper and Epstein

Published online by Cambridge University Press:  12 March 2014

Shamil Ishmukhametov
Shamil Ishmukhametov*
Affiliation:
42 Tolstoy St., Dept. Math., Ulyanovsk State University, 432700, Ulyanovsk, Russia, E-mail: [email protected]
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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
Information
The Journal of Symbolic Logic , Volume 68 , Issue 1 , March 2003 , pp. 52 - 64
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

Cooper, S. B. [1989], The strong anticupping property for recursively enumerable degrees, this Journal, vol. 54, pp. 527–539.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. 209–236.Google Scholar
Cooper, S. B. and Epstein, R. [1987], Complementing below recursively enumerable degrees, Annals of Pure and Applied Logic, vol. 34, pp. 15–32.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. 7–27.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. 1319–1347.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Seetapun, D. and Slaman, T. [1992], Minimal complements, unpublished manuscript.Google Scholar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. 160–176.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. 581–592.CrossRefGoogle Scholar
Yates, C. E. M. [1970], Initial segments of the degrees of unsolvability. part II. Minimal degrees, this Journal, vol. 35, pp. 243–266.Google Scholar