Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T04:34:04.910Z Has data issue: false hasContentIssue false
  • English
  • Français

Working below a high recursively enumerable degree

Published online by Cambridge University Press:  12 March 2014

Richard A. Shore  and
Theodore A. Slaman
Richard A. Shore
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853, E-mail: [email protected]
Theodore A. Slaman
Affiliation:
Department of Mathematics, The University of Chicago, Chicago, Illinois 60637, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

In recent work, Cooper [3, 1990] has extended results of Jockusch and Shore [6, 1984] to show that the Turing jump is definable in the structure given by the Turing degrees and the ordering of Turing reducibility. In his definition of x′ from x, Cooper identifies an order-theoretic property shared by all of the degrees that are recursively enumerable in x and above x. He then shows that x′ is the least upper bound of all the degrees with this property. Thus, the jump of x is identified by comparing the recursively enumerable degrees with other degrees which are not recursively enumerable. Of course, once the jump operator is known to be definable, the relation of jump equivalence x′ = y′ is also known to be a definable relation on x and y. If we consider how much of the global theory of the Turing degrees is sufficient for Cooper's methods, it is immediately clear that his methods can be implemented to show that the jump operator and its weakening to the relation of jump equivalence are definable in any ideal closed under the Turing jump. However, his methods do not localize to , the degrees, or to the recursively enumerable degrees.

This paper fits, as do Shore and Slaman [16, 1990] and [17, to appear], within the general project to develop an understanding of the relationship between the local degree-theoretic properties of a recursively enumerable set A and its jump class. For an analysis of the possibility of defining jump equivalence in , consult Shore [15, to appear] who shows that the relation x(3) = y(3) is definable. In this paper, we will restrict our attention to definitions expressed completely in ℛ (Note: All sets and degrees discussed for the remainder of this paper will be recursively enumerable.) Ultimately, one would like to find some degree-theoretic properties definable in terms of the ordering of Turing reducibility and quantifiers over the recursively enumerable degrees that would define the relation of jump equivalence or define one or more of the jump classes Hn = {w∣ wn = 0n+1} or Ln = {w ∣ wn = 0n}. Such a result could very likely then be used as a springboard to other general definability results for the recursively enumerable degrees. It would be especially interesting to know whether every recursively enumerable degree is definable and whether every arithmetical degree-invariant property of the recursively enumerable sets is definable in .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 3 , September 1993 , pp. 824 - 859
Copyright
Copyright © Association for Symbolic Logic 1993

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]Cooper, S. B., Minimal degrees and the jump operator, this Journal, vol. 38 (1973), pp. 249–271.Google Scholar
[2]Cooper, S. B., Minimal pairs and high recursively enumerable degrees, this Journal, vol. 39 (1974), pp. 655–660.Google Scholar
[3]Cooper, S. B., The jump is definable within the structure of the Turing degrees, Bulletin of American Mathematical Societyg, vol. 23 (1990), pp. 151–158.CrossRefGoogle Scholar
[4]Epstein, R. L., Initial segments of degrees below 0′, Memoirs of the American Mathematical Society, vol. 30, American Mathematical Society, Providence, Rhode Island, 1981.Google Scholar
[5]Harrington, L. A. and Shelah, S., The undecidability of the recursively enumerable degrees, Bulletin of the American Mathematical Society, vol. 6 (1982), pp. 79–80. (Research announcement)CrossRefGoogle Scholar
[6]Jockusch, C. and Shore, R. A., Pseudo jump operators II: Transfinite iterations, hierarchies and minimal covers, this Journal, vol. 49 (1984), pp. 1205–1236.Google Scholar
[7]Lachlan, A. H., A recursively enumerable degree which will not split over all lesser ones, Annals of Mathematical Logic, vol. 9 (1975), pp. 307–365.CrossRefGoogle Scholar
[8]Lerman, M., Some theorems on r-maximal sets and major subsets of recursively enumerable sets, this Journal, vol. 36(1971), pp. 193–215.Google Scholar
[9]Lerman, M., Degrees of unsolvability, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidleberg, New York, Tokyo, 1983.CrossRefGoogle Scholar
[10]Martin, D. A., Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für Mathematische Logik and Grundlagen der Mathematik, vol. 12 (1966), pp. 295–310.CrossRefGoogle Scholar
[11]Miller, D., High recursively enumerable degrees and the anti-cupping property, Logic Year 1979–1980: University of Connecticut (Lerman, M., Schmerl, J., and Soare, R. I., editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, Heidleberg, New York, and Tokyo, 1981, pp. 230–245.CrossRefGoogle Scholar
[12]Miller, D., The relationship between the structure and the degrees of recursively enumerable sets, Ph.D. Dissertation, The University of Chicago, Chicago, Illinois, 1981.Google Scholar
[13]Posner, D., A survey of non-r.e. degrees ≤ 0′, Recursion theory: its generalisations and applications, Proceedings of the logic Colloquium 1979 (Drake, F. and Wainer, S. S., editors), London Mathematical Society Lecture Notes Series, vol. 45, Cambridge University Press, Cambridge, 1980, pp. 52–109.Google Scholar
[14]Robinson, R. W., The inclusion lattice and degrees of unsolvability of the recursively enumerable sets, Ph.D. Dissertation, Cornell University, Ithaca, 1966.Google Scholar
[15]Shore, R. A., Defining jump classes in the degrees below 0′, Proceedings of the American Mathematical Society, vol. 104 (1988), pp. 287–292.Google Scholar
[16]Shore, R. A. and Slaman, T. A., Working below a low2 recursively enumerable degree, Archive for Mathematical Logic, vol. 29 (1990), pp. 201–211.CrossRefGoogle Scholar
[17]Shore, R. A. and Slaman, T. A., Splitting and density cannot be combined below a high recursively enumerable degree (to appear).Google Scholar
[18]Slaman, T. A., The recursively enumerable degrees as a substructure of the -degrees (to appear).Google Scholar
[19]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidleberg, New York, Tokyo, 1987.CrossRefGoogle Scholar