Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-12T17:27:40.596Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-distributive upper semilattice of Kleene degrees

Published online by Cambridge University Press:  12 March 2014

Hisato Muraki
Hisato Muraki*
Affiliation:
Department of Mathematics, School of Science, Nagoya University, Chikusa-Ku, Nagoya 464-8602, Japan, E-mail: muraki@math.nagoya-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

K denotes the upper semilattice of all Kleene degrees. Under ZF + AD + DC. K is well-ordered and deg(XSJ) is the next Kleene degree above deg(X) for X ⊆ ωω (see [4] and [5, Chapter V]). While, without AD, properties of K are not always clear. In this note, we prove the non-distributivity of K under ZFC (§1), and that of Kleene degrees between deg(X) and deg(XSJ) for some X under ZFC + CH (§2.3).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 1 , March 1999 , pp. 147 - 158
Copyright
Copyright © Association for Symbolic Logic 1999

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]Ambos-Spies, K., On the structure of the polynomial time degrees of recursive sets, Habilitationsschrift, Universität Dortmund, 1984.Google Scholar
[2]Devlin, K. J., Constructibility, Springer-Verlag, Berlin.Google Scholar
[3]Muraki, H., Non-complementedness and non-distributivity of Kleene degrees, to appear.Google Scholar
[4]Solovay, R., Determinacy and type 2 recursion, this Journal, vol. 36 (1971), p. 374, abstract.Google Scholar
[5]Weitbcamp, G., Kleene recursion over the continuum, Ph.D. thesis, Pennsylvania State University, 1980.Google Scholar