Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T03:51:47.812Z Has data issue: false hasContentIssue false
  • English
  • Français

A 1-generic degree which bounds a minimal degree

Published online by Cambridge University Press:  12 March 2014

Masahiro Kumabe
Masahiro Kumabe*
Affiliation:
Department of Mathematics, Waseda University, Tokyo 160, Japan
*
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Get access Rights & Permissions [Opens in a new window]

Extract

Let ω be the set of natural numbers, i.e. {0,1,2,…}. A set A (≤ω) is called n-generic if it is Cohen-generic for n-quantifier arithmetic. As characterized by Jockusch [4], this is equivalent to saying that for every set of strings S, there is a σ < A such that σ ∈ S or ∀ν ≥ σ(ν ∉ S). When we say degree, we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. A nonrecursive degree a is called minimal if there is no nonrecursive degree b with b < a. Jockusch [4] exhibited various properties of generic degrees, and he showed that any 2-generic degree bounds no minimal degree. Chong and Jockusch [1] showed that any 1-generic degree below 0′ bounds no minimal degree. Haught [3] refuted one of the conjectures in [1] and showed that if a is a 1-generic degree and 0 < b < a < 0′ then b is also 1-generic. We show here that there is a 1-generic degree which bounds a minimal degree. This gives an affirmative answer to questions in [1] and [4], As any 1-generic degree below 0′ bounds no minimal degree, we see that our 1-generic degree which bounds a minimal degree is not below 0′, but can be constructed recursively in 0″. Furthermore we see that the initial segments below 1-generic degrees are not order isomorphic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 2 , June 1990 , pp. 733 - 743
Copyright
Copyright © Association for Symbolic Logic 1990

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]Chong, C. T. and Jockusch, C. G., Minimal degrees and l-generic degrees below 0′, Computation and proof theory, Lecture Notes in Mathematics, vol. 1104, Springer-Verlag, Berlin, 1983, pp. 63–77.CrossRefGoogle Scholar
[2]Epstein, R. L., Degrees of unsolvabilily. Structure and theory, Lecture Notes in Mathematics, vol. 759, Springer-Verlag, Berlin, 1979.CrossRefGoogle Scholar
[3]Haught, C., Turing and truth table degrees of 1-generic and recursively enumerable sets, Ph.D. thesis, Cornell University, Ithaca, New York, 1985.Google Scholar
[4]Jockusch, C. G., Degrees of generic sets, Recursion theory, its generalizations and applications, London Mathematical Society Lecture Note Series, vol. 45, Cambridge University Press, Cambridge, 1980, pp. 110–139.CrossRefGoogle Scholar
[5]Lerman, M., Degrees of unsolvability, Springer-Verlag, Berlin, 1983.CrossRefGoogle Scholar