Vaught's conjecture for theories of discretely ordered structures

TL;DR

This paper proves that countable theories with a definable infinite discrete linear order have continuum many countable models, using a purely first-order approach, and discusses the Borel complexity of such theories.

Contribution

It establishes a new result linking definable discrete orders to the number of models, and explores the Borel complexity implications.

Findings

Countable theories with definable discrete orders have continuum many models.

The proof is purely first-order.

Raises questions about Borel completeness of such theories.

Abstract

Let $T$ be a countable complete first-order theory with a definable, infinite, discrete linear order. We prove that $T$ has continuum-many countable models. The proof is purely first-order, but raises the question of Borel completeness of $T$.