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

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.