 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 $\phi$ and $\psi$ are equivalent over $T$ when $T\models \forall \mathbf{x}(\phi \leftrightarrow \psi )$.