The spectrum of limit models in a first order setting

Abstract

Originally introduced by Kolmann and Shelah as a surrogate for saturated models, limit models have been established as natural and useful objects when studying abstract elementary classes. Shelah began the study of when (multiple notions of) limit models exist for first order theories. In this paper we look at their structure. In superstable theories it is known that all limit models are isomorphic, but in the strictly stable case the number of non-isomorphic limit models was not well understood. Here we characterise the full spectrum of limit models in the first order stable setting, by a short and simple argument using only the familiar machinery of stable first order theories: $\textbf{Theorem.}$ Let $T$ be a complete $λ$-stable theory where $λ\geq |\operatorname{L}(T)| + \aleph_0$. Let $δ_1, δ_2 < λ^+$ be limit ordinals where $\operatorname{cf}(δ_1)< \operatorname{cf}(δ_2)$. Let $N_l$ be a $(λ, δ_l)$-limit model for $l = 1, 2$. Then $N_1$ and $N_2$ are isomorphic if and only if $\operatorname{cf}(δ_1) \geq κ_r(T)$. Moreover, if $κ_r(T) = \aleph_α$, there are exactly $|α| + 1$ limit models up to isomorphism. In the context of first order stable theories, this reduces the proof of the main result of arXiv:2503.11605 from 19 pages to 2. We hope this will make limit models a more comprehensible and accessible tool in first order model theory.