Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 144 / 1 / b / Solution

Define a parameter-free equivalence relation byIts classes correspond exactly to the distinct sets $\phi (a,\mathcal{U})$. If there are exactly $n<\infty$ classes, that fact is a first-order sentence, so the elementary substructure $M$ contains representatives $b_1,\dots ,b_n$ of all of them. Every $a\in \mathcal{U}$ is $E$-equivalent to some $b_i$, and $\phi (a,\mathcal{U})=\phi (b_i,\mathcal{U})$ is definable over $M$.

The converse also holds because $M$ is small and $\mathcal{U}$ is a monster model. If there were infinitely many $E$-classes, the partial complete typewould be finitely satisfiable. Saturation of $\mathcal{U}$ would realize $p$, producing a class with no representative in $M$, contrary to the hypothesis. Therefore there are only finitely many sets $\phi (a,\mathcal{U})$.