Journal of Symbolic Logic

An almost-universal cupping degree

Jiang Liu and Guohua Wu

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

Abstract

Say that an incomplete d.r.e. degree has almost universal cupping property, if it cups all the r.e. degrees not below it to 0'. In this paper, we construct such a degree d, with all the r.e. degrees not cupping d to 0' bounded by some r.e. degree strictly below d. The construction itself is an interesting 0''' argument and this new structural property can be used to study final segments of various degree structures in the Ershov hierarchy.

Article information

Source
J. Symbolic Logic, Volume 76, Issue 4 (2011), 1137-1152.

Dates
First available in Project Euclid: 11 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1318338843

Digital Object Identifier
doi:10.2178/jsl/1318338843

Mathematical Reviews number (MathSciNet)
MR2895390

Zentralblatt MATH identifier
1247.03077

Citation

Liu, Jiang; Wu, Guohua. An almost-universal cupping degree. J. Symbolic Logic 76 (2011), no. 4, 1137--1152. doi:10.2178/jsl/1318338843. https://projecteuclid.org/euclid.jsl/1318338843


Export citation

References

  • M. M. Arslanov, Structural properties of the degrees below $\text\bfseries\upshape 0'$, Doklady Akademiya Nauk SSSR. New Series, vol. 283 (1985), pp. 270–273.
  • ––––, On structural properties of the degrees of Ershov's hierarchy, Izvestiya Vysshikh Uchebnykh Zavedeniĭ Matematika, vol. 7(1988), pp. 27–33.
  • S. B. Cooper, On a theorem of C. E. M. Yates, handwritten notes, 1974.
  • S. B. Cooper, L. Harrington, A. H. Lachlan, S. Lempp, and R. I. Soare, The d.r.e. degrees are not dense, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 125–151.
  • S. B. Cooper and X. Yi, Isolated d.r.e. degrees, preprint series, no. 17, 25pp., 1995, University of Leeds, Department of Pure Mathematics.
  • A. Li, Y. Song, and G. Wu, Universal cupping degrees, Theory and applications of models of computation (Jin yi Cai, S. Barry Cooper, and Angsheng Li, editors), Lecture Notes in Computer Science, 3959, 2006, pp. 721–730.
  • A. Li and X. Yi, Cupping the recursively enumerable degrees by d.r.e. degrees, Proceedings of the London Mathematical Society, vol. 78 (1999), pp. 1–21.
  • P. Odifreddi, Classical recursion theory, Studies in Logic and the Foundations of Mathematics 125, North-Holland, Amsterdam, 1989.
  • T. A. Slaman and J. R. Steel, Complementation in the Turing degrees, Journal of Symbolic Logic, vol. 54 (1989), pp. 160–176.
  • R. I. Soare, Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.
  • G. Wu, Isolation and the jump operator, Mathematical Logic Quarterly, vol. 47 (2001), pp. 525–534.
  • ––––, Isolation and lattice embeddigs, Journal of Symbolic Logic, vol. 67(2002), pp. 1055–1064.
  • ––––, Bi-isolation in the d.c.e. degrees, Journal of Symbolic Logic, vol. 69(2004), pp. 409–420.
  • ––––, Jump operator and Yates degrees, Journal of Symbolic Logic, vol. 71(2006), pp. 252–264.
  • C. E. M. Yates, Recursively enumerable degrees and the degrees less than $0\sp(1)$, Sets, models and recursion theory, Proceedings of the summer school in mathematical logic and tenth logic colloquium, Leicester, 1965 (J. N. Crossley, editor), North-Holland, Amsterdam, 1967, pp. 264–271.