Journal of Symbolic Logic

A hierarchy for the plus cupping Turing degrees

Angsheng Li and Yong Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We say that a computably enumerable (c. e.) degree a is plus-cupping, if for every c. e. degree x with 0 < xa, there is a c. e. degree y0’ such that xy=0’. We say that a is n-plus-cupping, if for every c. e. degree x, if 0 < xa, then there is a lown c. e. degree l such that xl=0’. Let PC and PCn be the set of all plus-cupping, and n-plus-cupping c. e. degrees respectively. Then PC1PC2PC3 = PC. In this paper we show that PC1PC2, so giving a nontrivial hierarchy for the plus cupping degrees. The theorem also extends the result of Li, Wu and Zhang [li-wu-zhang] showing that LC1LC2, as well as extending the Harrington plus-cupping theorem [harrington1978].

Article information

J. Symbolic Logic, Volume 68, Issue 3 (2003), 972- 988.

First available in Project Euclid: 17 July 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D25: Recursively (computably) enumerable sets and degrees 03D30: Other degrees and reducibilities
Secondary: 03D35: Undecidability and degrees of sets of sentences


Wang, Yong; Li, Angsheng. A hierarchy for the plus cupping Turing degrees. J. Symbolic Logic 68 (2003), no. 3, 972-- 988. doi:10.2178/jsl/1058448450.

Export citation


  • K. Ambos-Spies, C. G. Jockusch, Jr., R. A. Shore, and R. I. Soare An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees, Transactions of the American Mathematical Society, vol. 281 (1984), pp. 109--128.
  • P. Cholak, M. Groszek, and T. A. Slaman An almost deep degree, Journal of Symbolic Logic, vol. 66 (2001), no. 2, pp. 881--901.
  • S. B. Cooper Distinguishing the arithmetical hierarchy, preprint, Berkeley, October 1972.
  • S. B. Cooper and A. Li Splitting and nonsplitting, II: A $\textupLow_2$ computably enumerable degree above which $\textup\bfseries 0'$ is not splittable, Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1391--1430.
  • P. A. Fejer and R. I. Soare The plus cupping theorem for the recursively enumerable degrees, Logic year 1979--80: University of Connecticut (M. Lerman, J. H. Schmerl, and R. I. Soare, editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, Heidelberg, Tokyo, New York,1981.
  • L. Harrington On Cooper's proof of a theorem of Yates, handwritten notes,1976.
  • C. G. Jockusch, A. Li, and Y. Yang A join theorem for the computably enumerable degrees, to appear.
  • A. H. Lachlan Lower bounds for pairs of recursively enumerable degrees, Proceedings of the London Mathematical Society, vol. 16 (1966), pp. 537--569.
  • A. Li Elementary differences among jump hierarchies, to appear.
  • A. Li, G. Wu, and Z. Zhang A hierarchy for the cuppable degrees, Illinois Journal of Mathematics, vol. 44 (Fall 2000), no. 3, pp. 619--632.
  • R. W. Robinson Jump restricted interpolation in the recursively enumerable degrees, Annals of Mathematics, vol. 93 (1971), no. 2, pp. 586--596.
  • G. E. Sacks On the degrees less than $\textup\bfseries 0'$, Annals of Mathematics, vol. 77 (1963), no. 2, pp. 211--231.
  • R. I. Soare Automorphisms of the lattice of recursively enumerable sets, Bulletin of the American Mathematical Society, vol. 80 (1974), pp. 53--58.
  • C. E. M. Yates A minimal pair of recursively enumerable degrees, Journal of Symbolic Logic, vol. 31 (1966), pp. 159--168.