On the spectrum of limit models

Abstract

We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all `long' limit models are isomorphic, and all `short' limit models are non-isomorphic. $\text{Theorem.}$ Let $\mathbf{K}$ be a ${\aleph }_0$-tame abstract elementary class stable in $\lambda \ge \mathrm{LS}(\mathbf{K})$ with amalgamation, joint embedding and no maximal models. Suppose there is an independence relation on the models of size $\lambda$ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and $(\ge \kappa )$-local character in a minimal regular $\kappa <{\lambda }^{+}$. Suppose ${\delta }_1,{\delta }_2<{\lambda }^{+}$ with $\mathrm{cf}({\delta }_1)<\mathrm{cf}({\delta }_2)$. Then for any $N_1,N_2,M\in {\mathbf{K}}_{\lambda }$ where $N_l$ is a $(\lambda ,{\delta }_l)$-limit model over $M$ for $l=1,2$, \[N_1 \text{ is isomorphic to } N_2 \text{ over } M \iff \operatorname{cf}(δ_1) \geq κ\] Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the ${\aleph }_0$-tameness assumption and assuming the independence relation is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation. We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes.

arXiv comment40 pages. Key words and phrases: Limit models; Abstract Elementary Classes; Stability; Towers