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For a first-order theory $T$ in a first-order language $L$, a complete $n$-type is a maximal $T$-consistent set $p(x_1,\dots ,x_n)$ of formulas whose free variables lie among $x_1,\dots ,x_n$. Equivalently, it chooses exactly one of $\phi$ and $\neg \phi$ for every such formula while remaining consistent with $T$.

An isolated type $p$ is isolated by a formula $\theta (\bar{x})$ when $T\cup \{\exists \bar{x}\theta (\bar{x})\}$ is consistent andfor every $\phi \in p$. The type is an omitted type in an $L$-structure $M$ when no tuple $\bar{a}\in M^n$ satisfies every formula in $p$.

The omitting types theorem states that if $T$ is a consistent theory in a countable language and $(p_i{)}_{i\in \mathbb{N}}$ is a countable family of nonisolated finite-arity types, then $T$ has a countable model omitting every $p_i$.

Suppose first that $T$ is aleph-zero-categorical. If a type in some $S_n(T)$ were nonisolated, the omitting types theorem would produce a countable model omitting it, while a countable elementary submodel of a model realizing it would be another countable model. This contradicts categoricity. Thus every type is isolated. The compact Stone space $S_n(T)$ is then discrete and therefore finite.

Conversely, if every $S_n(T)$ is finite, every type is isolated. Every countable model is consequently atomic, and part i says that any two countable models are isomorphic. This proves the Ryll-Nardzewski theorem.