Łoś–Vaught test

Remarks

Suppose $T$ is not complete, and pick a sentence $\varphi$ such that neither $\varphi$ nor $\neg \varphi$ follows from $T$. Then $T \cup \set{\varphi}$ and $T \cup \set{\neg \varphi}$ both have models, and these are infinite. By the Löwenheim–Skolem theorem, both have models of size $\kappa$. These two models are not isomorphic, since one satisfies $\varphi$ and the other does not, which contradicts $\kappa$-categoricity.

Example: the theory of dense linear orders without endpoints has no finite models and is $\aleph_0$-categorical, so it is complete. Similarly, the theory of algebraically closed fields of a fixed characteristic is complete, since it is categorical in every uncountable cardinal.

The hypothesis on finite models is necessary. Take the empty theory $T = \emptyset$ in the empty signature, whose models are the nonempty sets. For each infinite $\kappa$, any two sets of size $\kappa$ are in bijection, so $T$ is $\kappa$-categorical. But $T$ is not complete: the sentence $\forall x \, \forall y \, (x = y)$ holds in a model of size $1$ and fails in a model of size $2$.

Going deeper

For a student

The test turns a structural fact, uniqueness of models of one cardinality, into a logical fact, completeness. Completeness of a recursively axiomatized theory implies that it is decidable, so the test also gives decision procedures. It is often combined with a back-and-forth argument for $\aleph_0$-categoricity, or with a dimension argument, such as transcendence degree, for uncountable categoricity. Its converse fails: the theory of $(\Z, <)$ is complete but categorical in no infinite cardinal.