Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T08:58:54.734Z Has data issue: false hasContentIssue false
  • English
  • Français

Recursively enumerable generic sets

Published online by Cambridge University Press:  12 March 2014

Wolfgang Maass
Wolfgang Maass*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes. This method leads to a new class of recursively enumerable sets: r.e. generic sets. All r.e. generic sets are low and simple and therefore of Turing degree strictly between 0 and 0′. Further they supply the first example of a class of low recursively enumerable sets which are automorphic in the lattice ℰ of recursively enumerable sets with inclusion. We introduce the notion of a promptly simple set. This describes the essential feature of r.e. generic sets with respect to automorphism constructions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 4 , December 1982 , pp. 809 - 823
Copyright
Copyright © Association for Symbolic Logic 1982

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]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[2]Lerman, M. and Soare, R. I., d-simple sets, small sets, and degree classes, Pacific Journal of Mathematics, vol. 87(1980), pp. 135–155.CrossRefGoogle Scholar
[3]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[4]Soare, R. I., Automorphisms of the lattice of recursively enumerable sets. Part I. Maximal sets, Annals of Mathematics, vol. 100 (1974), pp. 80–120.CrossRefGoogle Scholar
[5]Soare, R. I., Computational complexity, speedable and levelable sets, this Journal, vol. 42 (1977), pp. 545–563.Google Scholar
[6]Soare, R. I., Automorphisms of the lattice of recursively enumerable sets. Part II. Low sets (to appear).Google Scholar