Building models in small cardinals in local abstract elementary classes

Abstract

There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that \emph{stability} is enough to construct larger models for small cardinals assuming a mild locality condition for Galois types. $\mathbf{Theorem.}$ Suppose $λ<2^{\aleph_0}$. Let $\mathbf{K}$ be an abstract elementary class with $λ\geq LS(\mathbf{K})$. Assume $\mathbf{K}$ has amalgamation in $λ$, no maximal model in $λ$, and is stable in $λ$. If $\mathbf{K}$ is $(<λ^+, λ)$-local, then $\mathbf{K}$ has a model of cardinality $λ^{++}$. The set theoretic assumption that $λ<2^{\aleph_0}$ and model theoretic assumption of stability in $λ$ can be weakened to the model theoretic assumptions that $|\mathbf{S}^{na}(M)|< 2^{\aleph_0}$ for every $M \in \mathbf{K}_λ$ and stability for $λ$-algebraic types in $λ$. This is a significant improvement of Theorem 0.1., as the result holds on some unstable abstract elementary classes.