Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-25T05:13:48.012Z Has data issue: false hasContentIssue false
  • English
  • Français

Notions of weak genericity

Published online by Cambridge University Press:  12 March 2014

Stuart A. Kurtz
Stuart A. Kurtz*
Affiliation:
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 University of Chicago, Chicago, Illinois 60637
Get access Rights & Permissions [Opens in a new window]

Extract

This paper deals with forcing in arithmetic (as first introduced by Feferman [2]) and its connections with recursive function theory. We define for each n ≥ 1 the class of weakly n-generic sets. We prove that these classes merge with the classes of n-generic sets to form the hierarchy suggested by the terminology. Our notation is the same as that of Jockusch [5].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 3 , September 1983 , pp. 764 - 770
Copyright
Copyright © Association for Symbolic Logic 1983

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]Dekker, J.C.E. and Myhill, J., Retraceable sets, Canadian Journal of Mathematics, vol. 10 (1958), pp. 357–373.CrossRefGoogle Scholar
[2]Feferman, S., Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae, vol. 56 (1965), pp. 325–345.CrossRefGoogle Scholar
[3]Hinman, P.G., Some applications of forcing to hierarchy problems in arithmetic, Zeitschrift für Mathematishe Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 341–352.CrossRefGoogle Scholar
[4]Jockusch, C.G., The degrees of bi-immune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 135–140.CrossRefGoogle Scholar
[5]Jockusch, C.G., Degrees of generic sets, Recursion theory: Its generalizations and applications (Drake, F.R. and Wainer, S.S., Editors), Cambridge University Press, Cambridge, 1980.Google Scholar
[6]Kurtz, S.A., Randomness and genericity in the degrees of unsolvability, Ph.D Thesis, University of Illinois at Urbana-Champaign, Urbana, 1981.Google Scholar
[7]Miller, W. and Martin, D.A., The degrees of hyperimmune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 159–166.CrossRefGoogle Scholar
[8]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1968.Google Scholar
[9]Shoenfield, J.R., Unramified forcing, Axiomatic set theory. I, Proceedings of Symposia in Pure Mathematics, vol. 13, part 1 (Scott, D., Editor), American Mathematical Society, Providence, R.I., 1971.Google Scholar
[10]Selman, A.L., Applications of forcing to the degree theory of the arithmetical hierarchy, Proceedings of the London Mathematical Society (3), vol. 25 (1972), pp. 586–602.CrossRefGoogle Scholar