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Degree invariance in the Π10classes

Published online by Cambridge University Press:  12 March 2014

Rebecca Weber
Rebecca Weber*
Affiliation:
Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, NH 03755, USA, E-mail: rweber@math.dartmouth.edu
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Abstract

Let denote the collection of all Π10 classes, ordered by inclusion. A collection of Turing degrees in is called invariant over if there is some collection of Π10 classes representing exactly the degrees such that is invariant under automorphisms of . Herein we expand the known degree invariant classes of , previously including only {0} and the array noncomputable degrees, to include all highn and non-lown degrees for n > 2. This is a corollary to a very general definability result. The result is carried out in a substructure G of , within which the techniques used model those used by Cholak and Harrington [6] to obtain the same definability for the c.e. sets. We work back and forth between G and to show that this definability in G gives the desired degree invariance over .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1184 - 1210
Copyright
Copyright © Association for Symbolic Logic 2011

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References

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