In first order logic, the Baldwin-Lachlan characterization of $\aleph_1$-categorical
theories implies that the notion is absolute between transitive models of set theory.  
Here, we seek a similar characterization for having a unique atomic model of size $\aleph_1$.
At present, we have several conditions that imply many non-isomorphic atomic models of size $\aleph_1$.
Curiously, even though the results are in ZFC, their proofs rely on forcing.
This is joint work with John Baldwin and Saharon Shelah.