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Degree invariance in the Π10classes
Published online by Cambridge University Press: 12 March 2014
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
.
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- Research Article
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- Copyright
- Copyright © Association for Symbolic Logic 2011
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