Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Relative to any non-arithmetic set

Matthew Harrison-Trainor

TL;DR

The paper resolves the long-standing question of whether the non-arithmetic degrees form a degree spectrum by constructing a countable structure $\mathcal{M}$ such that $X$ computes an isomorphic copy of $\mathcal{M}$ if and only if $X$ is not arithmetic, thereby showing that non-arithmetic degrees form a degree spectrum. Central to the result is a novel technique, unfriendly jump inversions, which produce maximally complex back-and-forth structures and enable tight control over the complexity of Scott sentences and definability of back-and-forth types. The authors further develop a broad toolkit of applications, including computable Scott sentences, definability of back-and-forth types, and separating structures, and provide constructions of maximally unfriendly structures with highly complete $n$-back-and-forth relations. These contributions deepen our understanding of degree spectra in computable structure theory and offer new methods for encoding computability-theoretic properties into infinitary-logical descriptions, with potential wide-ranging implications for definability, categoricity, and interpretability in logical structures.

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.

Relative to any non-arithmetic set

TL;DR

The paper resolves the long-standing question of whether the non-arithmetic degrees form a degree spectrum by constructing a countable structure such that computes an isomorphic copy of if and only if is not arithmetic, thereby showing that non-arithmetic degrees form a degree spectrum. Central to the result is a novel technique, unfriendly jump inversions, which produce maximally complex back-and-forth structures and enable tight control over the complexity of Scott sentences and definability of back-and-forth types. The authors further develop a broad toolkit of applications, including computable Scott sentences, definability of back-and-forth types, and separating structures, and provide constructions of maximally unfriendly structures with highly complete -back-and-forth relations. These contributions deepen our understanding of degree spectra in computable structure theory and offer new methods for encoding computability-theoretic properties into infinitary-logical descriptions, with potential wide-ranging implications for definability, categoricity, and interpretability in logical structures.

Abstract

Given a countable structure , the degree spectrum of is the set of all Turing degrees which can compute an isomorphic copy of . 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.

Paper Structure

This paper contains 15 sections, 26 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Other applications of unfriendly jump inversions
  3. Computability of Scott sentences
  4. Definability of back-and-forth types
  5. Separating structures
  6. Computing optimal Scott sentences and separating sentences
  7. Unfriendly jump inversions
  8. Construction and properties of unfriendly jump inversions
  9. Constructing the required structures
  10. Non-arithmetic is a degree spectrum
  11. Constructions for other applications
  12. Computable Scott sentences
  13. Back-and-forth types
  14. Separating structures
  15. Maximally unfriendly structures

Key Result

Theorem 1.1

There is a countable structure $\mathcal{M}$ such that for any set $X$, $X$ computes an isomorphic copy of $\mathcal{M}$ if and only if $X$ is not computable.

Figures (1)

  • Figure 1: An element $x \in U_k$ and all of its associated elements. The element $x$ shown here satisfies $\neg R_1(x)$ because each $D_1(x)$-structure is isomorphic to $\mathcal{A}$ and each $\overline{D}_1(x)$-structure is isomorphic to $\mathcal{B}$. It satisfies $R_2(x)$ as witnessed by the fourth equivalence class in $D_2(x)$ and $\overline{D}_2(x)$.

Theorems & Definitions (52)

  • Theorem 1.1: Slaman Slaman and Wehner Wehner
  • Theorem 1.2: Greenberg, Montalbán, Slaman nonhyp
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Ash and Knight
  • Theorem 2.4: Ash and Knight
  • Theorem 2.5
  • Definition 2.6: Montalbán MonSR, see also MonBook1
  • Theorem 2.7: Montalbán MonSR
  • ...and 42 more