# Is $\aleph_1$-categoricity absolute for atomic models?

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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.