# A strong failure of aleph_0-stability for atomic classes

## TL;DR

This paper demonstrates that for certain classes of atomic models in countable theories, the failure of aleph_0-stability leads to a vast number of non-isomorphic models of size aleph_1, highlighting a significant instability phenomenon.

## Contribution

It establishes a strong link between the non-pcl-smallness of atomic classes and the existence of many non-isomorphic models, revealing a failure of aleph_0-stability.

## Key findings

- Uncountably many types over pcl(a) imply 2^aleph1 non-isomorphic models.
- Failure of aleph_0-stability occurs in non-pcl-small atomic classes.
- The result connects atomic model properties with model-theoretic stability concepts.

## Abstract

We study classes of atomic models At_T of a countable, complete first-order theory T . We prove that if At_T is not pcl-small, i.e., there is an atomic model N that realizes uncountably many types over pcl(a) for some finite tuple a from N, then there are 2^aleph1 non-isomorphic atomic models of T, each of size aleph1.