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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
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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

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References

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