= Solution
An <aleph-zero-categorical theory> has, up to <isomorphism>, exactly one <first-order model> of cardinality $\aleph_0$.
The <Ryll-Nardzewski theorem> says that, for a <complete theory> $T$ in a countable <first-order language>, with infinite <first-order models>, the following conditions are equivalent: $T$ is aleph-zero-categorical; for every positive integer $n$, its <type space> $S_n(T)$ of complete types over the empty set is finite; and, for every $n$, there are only finitely many $L$-formulas in $n$ free variables modulo equivalence over $T$. Here $\varphi$ and $\psi$ are equivalent over $T$ when $T\models\forall\mathbf x(\varphi\leftrightarrow\psi)$.
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