= Solution
Define a parameter-free equivalence relation by
$$
E(x,y)\;\Longleftrightarrow\;\forall z\bigl(\varphi(x,z)\leftrightarrow\varphi(y,z)\bigr).
$$
Its classes correspond exactly to the distinct sets $\varphi(a,\mathcal U)$. If there are exactly $n<\infty$ classes, that fact is a first-order sentence, so the <elementary substructure> $M$ contains representatives $b_1,\ldots,b_n$ of all of them. Every $a\in\mathcal U$ is $E$-equivalent to some $b_i$, and $\varphi(a,\mathcal U)=\varphi(b_i,\mathcal U)$ 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 type>
$$
p(x)=\{\neg E(x,b):b\in M\}
$$
would be finitely satisfiable. Saturation of $\mathcal U$ would realize $p$, producing a class with no representative in $M$, contrary to the hypothesis. Therefore \b[there are only finitely many sets $\varphi(a,\mathcal U)$].
Back to article page
