Complete theory

A consistent first-order theory is complete when it decides every sentence in its language.

A theory is $\kappa$-categorical when all of its models of cardinality $\kappa$ are isomorphic.

An aleph-zero-categorical theory has, up to isomorphism, exactly one countably infinite model.

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.

This theorem is the uncountable categoricity theorem for a categorical theory. If a countable first-order theory is categorical in one uncountable cardinal, then it is categorical in every uncountable cardinal.

Let a language partition the universe into two infinite unary predicates and include a bijection between them. Its complete theory is categorical in every infinite cardinal. The reduct forgetting the bijection is not categorical in any uncountable cardinal, because the two infinite parts can have different cardinalities.

Compactness gives a countable model elementarily equivalent to $(\mathbb{Z},+,0)$ that contains a nonzero element divisible by every positive standard integer. It is not isomorphic to $\mathbb{Z}$, so $\mathrm{Th}(\mathbb{Z},+,0)$ is not aleph-zero-categorical.

The complete theory of infinite-dimensional vector spaces over a fixed finite field is aleph-zero-categorical: every countably infinite model has countably infinite dimension and is therefore isomorphic to every other such model.