Relative to any non-arithmetic set

TL;DR

This paper proves that the non-arithmetic degrees form a degree spectrum for countable structures, advancing understanding of which degree classes can compute isomorphic copies of structures.

Contribution

It resolves a major open problem by showing non-arithmetic degrees constitute a degree spectrum, introducing a new unfriendly jump inversion technique with broad applications.

Findings

Non-arithmetic degrees form a degree spectrum.

Introduces a new unfriendly jump inversion method.

Technique has multiple applications.

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.