Journal of Symbolic Logic

Degree invariance in the Π01 classes

Rebecca Weber

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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.

Article information

J. Symbolic Logic Volume 76, Issue 4 (2011), 1184-1210.

First available: 11 October 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 03D25: Recursively (computably) enumerable sets and degrees 03D28: Other Turing degree structures


Weber, Rebecca. Degree invariance in the Π 0 1 classes. Journal of Symbolic Logic 76 (2011), no. 4, 1184--1210. doi:10.2178/jsl/1318338845.

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