A computability theoretic equivalent to Vaught's conjecture

TL;DR

This paper establishes a computability-theoretic criterion equivalent to Vaught's conjecture, linking the number of countable models of a theory to the existence of computable copies in hyperarithmetic models relative to Turing degrees.

Contribution

It provides a new computability-theoretic characterization of Vaught's conjecture and describes the Turing-degree spectra of models for a counterexample.

Findings

Equivalence between the number of models and hyperarithmetic computability conditions.

A concrete description of Turing-degree spectra for a counterexample to Vaught's conjecture.

Characterization of theories with fewer than continuum many models via computability properties.

Abstract

We prove that, for every theory $T$ which is given by an ${\mathcal L}_{\omega_1,\omega}$ sentence, $T$ has less than $2^{\aleph_0}$ many countable models if and only if we have that, for every $X\in 2^\omega$ on a cone of Turing degrees, every $X$-hyperarithmetic model of $T$ has an $X$-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaught's conjecture.