Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 121 / 1 / iii / Solution

Let $M$ be the given transitive model and suppose that its ordinal height of a model of set theory $\mathrm{Ord}\cap M$ were a countable set. For every $x\in M$, the internal Axiom of choice gives a bijection from $x$ to an ordinal of $M$; transitivity makes this an actual bijection, and the ordinal is externally countable. Thus every element of $M$ is externally countable.

For each $\alpha \in \mathrm{Ord}\cap M$, the internal rank $V_{\alpha }^M$ belongs to $M$ and is therefore countable. Every $x\in M$ lies in one of these ranks, so $M$ is a countable union of countable sets and is itself countable, contrary to the hypothesis. By uncountable transitive set model has uncountable ordinal height, $M$ contains uncountably many ordinals.