Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 3 / c / Solution

Use the Downward Lowenheim-Skolem theorem to choose a small elementary submodel $M⪯N$. By hypothesis, $M$ meets every $E$-class. If there were infinitely many classes, the typewould be finitely satisfiable: finitely many parameters meet only finitely many classes, so choose $x$ from another class. Its parameter set is small, so saturation would realize $q$ in $N$. That realization would lie in an $E$-class disjoint from $M$, contradicting the hypothesis. This proves the small elementary submodel meeting every definable equivalence class criterion: $E$ has only finitely many classes.