Relative to any non-arithmetic set

Abstract

Given a countable structure $\mathcal{A}$, the degree spectrum of $\mathcal{A}$ is the set of all Turing degrees which can compute an isomorphic copy of $\mathcal{A}$. One of the major programs in computable structure theory is to determine which (upwards closed, Borel) classes of degrees form a degree spectrum. We resolve one of the major open problems in this area by showing that the non-arithmetic degrees are a degree spectrum. Our main new tool is a new form of unfriendly jump inversions where the back-and-forth types are maximally complicated. This new tool has several other applications.

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