Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-24T10:00:59.824Z Has data issue: false hasContentIssue false
  • English
  • Français

A minimal degree not realizing least possible jump

Published online by Cambridge University Press:  12 March 2014

Leonard P. Sasso Jr
Leonard P. Sasso Jr*
Affiliation:
University of California, Irvine, California 92664
Get access Rights & Permissions [Opens in a new window]

Extract

The least possible jump for a degree of unsolvability a is its join a ∪ 0′ with 0′. Friedberg [1] showed that each degree b ≥ 0′ is the jump of a degree a realizing least possible jump (i.e., satisfying the equation a′ = a ∪ 0′). Sacks (cf. Stillwell [8]) showed that most (in the sense of Lebesgue measure) degrees realize least possible jump. Nevertheless, degrees not realizing least possible jump are easily found (e.g., any degree b ≥ 0′) even among the degrees <0′ (cf. Shoenfield [5]) and the recursively enumerable (r.e.) degrees (cf. Sacks [3]).

A degree is called minimal if it is minimal in the natural partial ordering of degrees excluding least element 0. The existence of minimal degrees <0” was first shown by Spector [7]; Sacks [3] succeeded in replacing 0” by 0′ using a priority argument. Yates [9] asked whether all minimal degrees <0′ realize least possible jump after showing that some do by exhibiting minimal degrees below each r.e. degree. Cooper [2] subsequently showed that each degree b > 0′ is the jump of a minimal degree which, as corollary to his method of proof, realizes least possible jump. We show with the aid of a simple combinatorial device applied to a minimal degree construction in the manner of Spector [7] that not all minimal degrees realize least possible jump. We have observed in conjunction with S. B. Cooper and R. Epstein that the new combinatorial device may also be applied to minimal degree constructions in the manner of Sacks [3], Shoenfield [6] or [4] in order to construct minimal degrees <0′ not realizing least possible jump. This answers Yates' question in the negative. Yates [10], however, has been able to draw this as an immediate corollary of the weaker result by carrying out the proof in his new system of prioric games.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 3 , September 1974 , pp. 571 - 574
Copyright
Copyright © Association for Symbolic Logic 1974

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]Friedberg, R. M., A criterion for completeness of degrees of unsolvability, this Journal, vol. 22 (1957), pp. 159–160.Google Scholar
[2]Cooper, S. B., Minimal degrees and the jump operator, this Journal, vol. 38 (1973), pp. 249–271.Google Scholar
[3]Sacks, G. E., Degrees of unsolvability, Annals of Mathematics Studies, Princeton, N.J., 1963, 2nd edition, 1966.Google Scholar
[4]Sasso, L. P. Jr., A cornucopia of minimal degrees, this Journal, vol. 35 (1970), pp. 383–388.Google Scholar
[5]Shoenfield, J. R., On degrees of unsolvability, Annals of Mathematics (2), vol. 69 (1959), pp. 644–653.CrossRefGoogle Scholar
[6]Shoenfield, J. R., A theorem on minimal degrees, this Journal, vol. 31 (1966), pp. 539–544.Google Scholar
[7]Spector, C. M., On degrees of recursive unsolvability, Annals of Mathematics (2), vol. 64 (1956), pp. 581–592.CrossRefGoogle Scholar
[8]Stillwell, J., Decidability of the “almost all” theory of degrees, this Journal, vol. 37 1972), pp. 501–506.Google Scholar
[9]Yates, C. E. M., Initial segments of the degrees of unsolvability. Part II: Minimal degrees, this Journal, vol. 35 (1970), pp. 243–266.Google Scholar
[10]Yates, C. E. M., Prioric games and minimal degrees below 0′, Fundamenta Mathematicae (to appear).Google Scholar