Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T14:38:03.506Z Has data issue: false hasContentIssue false
  • English
  • Français

On the ranked points of a Π10 set

Published online by Cambridge University Press:  12 March 2014

Douglas Cenzer  and
Rick L. Smith
Douglas Cenzer
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611
Rick L. Smith
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper continues joint work of the authors with P. Clote, R. Soare and S. Wainer (Annals of Pure and Applied Logic, vol. 31 (1986), pp. 145–163). An element x of the Cantor space 2ω is said have rank α in the closed set P if x is in Dα(P)/Dα + 1(P), where Dα is the iterated Cantor-Bendixson derivative. The rank of x is defined to be the least α such that x has rank a in some set. The main result of the five-author paper is that for any recursive ordinal λ + n (where λ is a limit and n is finite), there is a point with rank λ + n which is Turing equivalent to O(λ + 2n) All ranked points constructed in that paper are singletons. We now construct a ranked point which is not a singleton. In the previous paper the points of high rank were also of high hyperarithmetic degree. We now construct points with arbitrarily high rank. We also show that every nonrecursive RE point is Turing equivalent to an RE point of rank one and that every nonrecursive point is Turing equivalent to a hyperimmune point of rank one. We relate Clote's notion of the height of a singleton in the Baire space with the notion of rank. Finally, we show that every hyperimmune point x is Turing equivalent to a point which is not ranked.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 3 , September 1989 , pp. 975 - 991
Copyright
Copyright © Association for Symbolic Logic 1989

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]Cenzer, D., Clote, P., Smith, R., Soare, R. and Wainer, S., Members of countable Π10 classes, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 145–163.CrossRefGoogle Scholar
[2]Chen, K. H., Recursive well-founded orderings, Annals of Mathematical Logic, vol. 13 (1978), pp. 117–147.CrossRefGoogle Scholar
[3]Clote, P., On recursive trees with a unique infinite branch, Proceedings of the American Mathematical Society, vol. 93 (1985), pp. 335–342.CrossRefGoogle Scholar
[4]Friedman, H., Simpson, S. and Smith, R., Set existence axioms and countable algebra, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 141–181.CrossRefGoogle Scholar
[5]Harrington, L., McLaughlin's conjecture (handwritten notes).Google Scholar
[6]Hinman, P., Recursion-theoretic hierarchies, Springer-Verlag, Berlin, 1978.CrossRefGoogle Scholar
[7]Jockusch, C. Jr., and McLaughlin, T., Countable retracting functions and Π20 predicates, Pacific Journal of Mathematics, vol. 30 (1969), pp. 67–93.CrossRefGoogle Scholar
[8]Jockusch, C. Jr., and Shore, R., Pseudo-jump operators. II, this Journal, vol. 49 (1984), pp. 1205–1236.Google Scholar
[9]Jockusch, C. Jr., and Soare, R., Π10 classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[10]Jockusch, C. Jr., and Soare, R., Degrees of members of Π10 classes, Pacific Journal of Mathematics, vol. 40 (1972), pp. 605–616.CrossRefGoogle Scholar
[11]Kreisel, G., Analysis of the Cantor-Bendixson theorem by means of the analytic hierarchy, Bulletin de l'Académie Polonaise des Sciences, Series des Sciences Mathématiques, Astronomiques et Physiques, vol. 7 (1959), pp. 621–626.Google Scholar
[12]Kuratowski, K., Topology. Vol. I, PWN, Warsaw, and Academic Press, New York, 1966.Google Scholar
[13]Kurtz, S., Notions of weak genericity, this Journal, vol. 48 (1983), pp. 764–770.Google Scholar
[14]Metakides, G. and Nerode, A., Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289–320.CrossRefGoogle Scholar
[15]Miller, W. and Martin, D. A., Degrees of hyperimmune sets, Zeitschrift für Mathematische Logik and Grundlagen der Mathematik, vol. 14 (1968), pp. 159–166.CrossRefGoogle Scholar
[16]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar