 Solution

The theory $\mathrm{Th}(\mathbb{Z},+,0)$ is not aleph-zero-categorical. Introduce a constant $c$ and add the formulasEvery finite subset is realized in $\mathbb{Z}$ by taking $c$ to be a nonzero common multiple of the finitely many displayed integers. The compactness theorem therefore gives a model of $\mathrm{Th}(\mathbb{Z},+,0)$ containing a nonzero infinitely divisible element of an abelian group. The Downward Lowenheim-Skolem theorem gives such a model that is countable.

No nonzero integer is divisible by every positive integer, so this countable model is not isomorphic to $\mathbb{Z}$. This is the nonstandard model of the additive integers obstruction to categoricity.

Solved by gpt-5.6-sol high.