The spectrum of limit models in a first order setting
Jeremy Beard · arXiv (Cornell University) · 2025
Originally introduced by Shelah & Kolman (1996) as a surrogate for saturated models, limit models (also known as brimmed models) have been established as natural and useful objects when studying abstract elementary classes. Shelah (2012) began the study of when (multiple similar 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 (VanDieren, 2006), 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. Essentially, long limit models are isomorphic, and short limit models are distinct: Let $T$ be a complete $\lambda$-stable theory where $\lambda \geq |{\mathrm{L}(T)}| + \aleph_0$. Let $\delta_1, \delta_2 < \lambda^+$ be limit ordinals where $\mathrm{cof}(\delta_1)< \mathrm{cof}(\delta_2)$. Let $N_l$ be a $(\lambda, \delta_l)$-limit model for $l = 1, 2$. Then $N_1$ and $N_2$ are isomorphic if and only if $\mathrm{cof}(\delta_1) \geq \kappa_{\mathrm r}(T)$. Moreover, if $\kappa_{\mathrm r}(T) = \aleph_\alpha$, there are exactly $|\alpha| + 1$ limit models up to isomorphism. In the context of first-order stable theories, this reduces the proof of the main result of Beard & Mazari-Armida (2025) from 19 pages to 2 pages. We hope this will make limit models a more comprehensible and accessible tool in first-order model theory.