Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 144 / 3

A model $M$ is aleph-zero-homogeneous model when, for finite tuples $\bar{a},\bar{b}$ with the same complete type and every $c\in M$, there is $d\in M$ such thatEquivalently, every finite partial elementary map extends by one more element.

Solved by gpt-5.6-sol high.

Start with the countable model $M_0$. There are countably many finite tuples and formulas. For every pair $\bar{a},\bar{b}\in M_i$ having the same type and every $c\in M_i$, use compactness to realize over $\bar{b}$ the transported type $\mathrm{tp}(c/\bar{a})$. Realize all these countably many requirements in an elementary extension and use the Downward Lowenheim-Skolem theorem to choose it countable; call it $M_{i+1}$.

The elementary union $M_{\omega }={\bigcup }_{i<\omega }{M}_i$ is countable. Any finite tuples and element in it occur at one stage, and their required matching element appears at the next. Thus $M_{\omega }$ is an aleph-zero-homogeneous elementary extension of $M_0$.

Solved by gpt-5.6-sol high.

Letwith the lexicographic order, where the initial $\mathbb{Z}$ is one discrete block. This is a countable model of $\mathrm{Th}(\mathbb{Z},<)$: it is a discrete order without endpoints, and every interval is either of its prescribed finite length or contains arbitrarily long finite chains.

An element $a$ in the initial block and an element $b$ in a later block have the same one-type. There is, however, a $c<b$ with infinitely many points between $c$ and $b$, whereas no such $c<a$ exists because every predecessor of $a$ lies at finite distance within the initial block. The type of $(b,c)$ therefore cannot be transported over $a$, so $M$ is not aleph-zero-homogeneous.

Solved by gpt-5.6-sol high.