Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-24T23:24:05.816Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized r-cohesiveness and the arithmetical hierarchy: a correction to “Generalized cohesiveness”

Published online by Cambridge University Press:  12 March 2014

Carl G. Jockusch Jr.  and
Tamara J. Lakins
Carl G. Jockusch Jr.
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801, USA, E-mail: jockusch@math.uiuc.edu
Tamara J. Lakins
Affiliation:
Department of Mathematics, Allegheny College, 520 N. Main St., Meadville, PA 16335, USA, E-mail: tlakins@allegheny.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

For X ⊆ ω, let [X]n denote the class of all n-element subsets of X. An infinite set A ⊆ ω is called n-r-cohesive if for each computable function f: [ω]n → {0, 1} there is a finite set F such that f is constant on [A − F]n. We show that for each n > 2 there is no Πn0 set A ⊆ ω which is n-r-cohesive. For n = 2 this refutes a result previously claimed by the authors, and for n ≥ 3 it answers a question raised by the authors.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 3 , September 2002 , pp. 1078 - 1082
Copyright
Copyright © Association for Symbolic Logic 2002

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]Hummel, T. and Jockusch, C. G. Jr., Generalized cohesiveness, this Journal, vol. 64 (1999), pp. 489–516.Google Scholar
[2]Hummel, T. and Jockusch, C. G. Jr., Ramsey's theorem for computably enumerable colorings, this Journal, vol. 66 (2001), pp. 873–880.Google Scholar
[3]Jockusch, C. G. Jr., Ramsey's theorem and recursion theory, this Journal, vol. 37 (1972), pp. 268–280.Google Scholar
[4]Ramsey, F. P., On a problem in formal logic, Proceedings of the London Mathematical Society, vol. 30, 1930, pp. 264–286.CrossRefGoogle Scholar
[5]Soare, R. I., Recursively enumerable sets and degrees, Springer–Verlag, Berlin, 1987.CrossRefGoogle Scholar