Degree invariance in the Π01 classes

Abstract

Let ℰ_Π denote the collection of all Π⁰₁ classes, ordered by inclusion. A collection of Turing degrees 𝒞 is called invariant over ℰ_Π if there is some collection 𝒮 of Π⁰₁ classes representing exactly the degrees in 𝒞 such that 𝒮 is invariant under automorphisms of ℰ_Π. Herein we expand the known degree invariant classes of ℰ_Π, previously including only {0} and the array noncomputable degrees, to include all high_n and non-low_n degrees for n≥2. This is a corollary to a very general definability result. The result is carried out in a substructure G of ℰ_Π, 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 ℰ_Π to show that this definability in G gives the desired degree invariance over ℰ_Π.