The spectrum of limit models in a first order setting
Jeremy Beard
TL;DR
This paper resolves the spectrum of limit models for complete first-order stable theories by showing that, for a complete $λ$-stable theory with $λ \ge |L(T)| + \aleph_0$, the isomorphism type of a $(λ, δ)$-limit model depends only on the cofinality of $δ$ relative to $\kappa_r(T)$. The key result is that two such limits are isomorphic iff $\operatorname{cf}(δ_1) \ge \kappa_r(T)$, yielding exactly $|α|+1$ non-isomorphic limit models when $\kappa_r(T) = \aleph_α$. The proofs stay within standard stable theory, turning what was a 19-page argument in a broader setting into a concise two-page argument in the first-order context. The results streamline the understanding of limit models and connect the first-order theory with ongoing AEC-based investigations.
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.