The number of countable models of first-order theories

TL;DR

This paper surveys the classification of countable models of first-order theories, discussing key conjectures and theories with few models, highlighting the complexity of Vaught's and Martin's conjectures.

Contribution

It provides a comprehensive survey of work on the number of countable models, including Vaught's conjecture, Martin's conjecture, and Ehrenfeucht theories.

Findings

Classification of theories with exactly one countable model

Discussion of theories with finitely many countable models

Overview of open problems like Vaught's conjecture

Abstract

Throughout, $T$ denotes a complete first-order theory in a countable language $L$ that has infinite models and $I(\aleph_0,T)$ denotes the number of countable models of $T$, up to an isomorphism. To determine $I(\aleph_0,T)$, it suffices to consider only countable models of $T$ with domain $\omega$; since there are at most continuum many $L$-structures with domain $\omega$, $I(\aleph_0,T)\leqslant 2^{\aleph_0}$ holds. Theories with $I(\aleph_0,T)=1$ are the $\aleph_0$-categorical theories. These include the theory of an infinite set, theories of infinite-dimensional vector spaces over a finite field, and the theory of dense linear orders. Theories with $I(\aleph_0,T)<2^{\aleph_0}$ are said to have few countable models. In this paper we discuss and survey work done on Vaught's conjecture, Martin's conjecture, and Ehhrenfeuch theories (theories with more than one but only finitely many, countable models).