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

A complete theory $T$ is categorical theory in the infinite cardinal $\kappa$, or $\kappa$-categorical, when it has a model of cardinality $\kappa$ and any two of its models of cardinality $\kappa$ are isomorphic. Equivalently, $T$ has exactly one model of cardinality $\kappa$ up to isomorphism.

Solved by gpt-5.6-sol high.

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.

The group $B$ is a vector space over a finite field, namely ${\mathbb{F}}_2$, because every element has order at most two. It is infinite-dimensional. The complete first-order theory of infinite-dimensional ${\mathbb{F}}_2$-vector spaces says, for each $n$, that there are $n$ linearly independent vectors; the usual elimination argument for vector spaces shows that all infinite-dimensional ${\mathbb{F}}_2$-vector spaces are elementarily equivalent.

Every countably infinite model of this theory has dimension ${\aleph }_0$: finite dimension would make it finite, while uncountable dimension would make its underlying set uncountable. Any two vector spaces over the same field with the same dimension are isomorphic. Therefore $\mathrm{Th}(B,+,0)$ is aleph-zero-categorical, as recorded by the aleph-zero-categoricity of an infinite-dimensional vector space over a finite field.

Solved by gpt-5.6-sol high.