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.

Theorem

Let K be a ${\aleph }_0$-tame abstract elementary class stable in $\lambda \ge \mathrm{LS}\left(K\right)$ with amalgamation, joint embedding and no maximal models. Let $\kappa <{\lambda }^{+}$ be regular. Suppose

is an independence relation on the models of size λ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and $(\ge \kappa )$-local character.

Suppose ${\delta }_1,{\delta }_2<{\lambda }^{+}$ with $\mathrm{cf}\left({\delta }_1\right)<\mathrm{cf}\left({\delta }_2\right)$. Then for any $N_1,N_2,M\in K_{\lambda }$ where $N_l$ is a $(\lambda ,{\delta }_l)$-limit model over M for $l=1,2$,

Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the ${\aleph }_0$-tameness assumption and assuming

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.