Ryll-Nardzewski theorem

For a complete theory with infinite models in a countable language, the following are equivalent: the theory is aleph-zero-categorical; every type space $S_n(T)$ is finite; and for every $n$ there are only finitely many formulas in $n$ variables modulo the theory.

Every reduct of an aleph-zero-categorical theory to a sublanguage is aleph-zero-categorical. There can only be fewer formulas in each arity, so the finite-formula characterization in the Ryll-Nardzewski theorem is preserved.