# Abstract elementary classes stable in $\aleph_0$

## TL;DR

This paper investigates the stability and structural properties of abstract elementary classes in countable cardinality, demonstrating superstable-like behavior and the existence of superlimit models under certain conditions.

## Contribution

It establishes that AECs with specific properties in leph_0 exhibit superstable-like behavior, including the existence of superlimit models and a good leph_0-frame, under a locality assumption.

## Key findings

- Existence of a superlimit model of size leph_0
- Presence of a good leph_0-frame in the generated class
- Superlimit model of size leph_1

## Abstract

We study abstract elementary classes (AECs) that, in $\aleph_0$, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at $\aleph_0$. More precisely, there is a superlimit model of cardinality $\aleph_0$ and the class generated by this superlimit has a type-full good $\aleph_0$-frame (a local notion of nonforking independence) and a superlimit model of cardinality $\aleph_1$. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.