The theory $\operatorname{Th}(\mathbb Z,+,0)$ is not <aleph-zero-categorical theory>[aleph-zero-categorical]. Introduce a constant $c$ and add the formulas
$$
c\ne0,
\qquad
\exists y\;(ny=c)quad(n=1,2,\ldots).
$$
Every 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 $\operatorname{Th}(\mathbb Z,+,0)$ 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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 144 / 1 / b / i / Solution

{title2=$\aleph_0$-categorical}

An aleph-zero-categorical theory has, up to isomorphism, exactly one countably infinite model.


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