Degree invariance in the Π01 classes

Rebecca Weber
Source: J. Symbolic Logic Volume 76, Issue 4 (2011), 1184-1210.

Abstract

Let ℰΠ denote the collection of all Π01 classes, ordered by inclusion. A collection of Turing degrees 𝒞 is called invariant over ℰΠ if there is some collection 𝒮 of Π01 classes representing exactly the degrees in 𝒞 such that 𝒮 is invariant under automorphisms of ℰΠ. Herein we expand the known degree invariant classes of ℰΠ, 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 ℰΠ, 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 ℰΠ to show that this definability in G gives the desired degree invariance over ℰΠ.

First Page:
Primary Subjects: 03D25, 03D28
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1318338845
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Mathematical Reviews number (MathSciNet): MR2895392

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