Łoś-Vaught Test

Theorem
Let $T$ be a satisfiable $\mathcal{L}$-theory with no finite models. If $T$ is $\kappa$-categorical for some infinite cardinal $\kappa\geq |\mathcal{L}|$, then $T$ is complete.

Proof
We prove the contrapositive. The main idea is that if such a theory $T$ is incomplete, we can construct size $\kappa$ models which disagree on a sentence.

Suppose $T$ is not complete.

By the definition of complete, this means that there is some sentence $\phi$ such that both $T\not\models\phi$ and $T\not\models\neg\phi$. This in turn means that both $T\cup\{\neg\phi\}$ and $T\cup\{\phi\}$ have models. Since $T$ has no infinite models, this means that $T\cup\{\neg\phi\}$ and $T\cup\{\phi\}$ both have infinite models.

Using that $\kappa$ is infinite and greater than the cardinality of the language and that these theories have infinite models, one can prove that there are size $\kappa$ models $\mathcal{M}_{\neg\phi}$ and $\mathcal{M}_{\phi}$ of $T\cup\{\neg\phi\}$ and $T\cup\{\phi\}$ respectively (see Theory with one Infinite Model has Models of each Infinite Cardinal Larger than Language). In particular, $\mathcal{M}_{\neg\phi}$ and $\mathcal{M}_{\phi}$ are models of $T$ which disagree about the sentence $\phi$. Such models cannot be isomorphic since isomorphisms preserve the truth of sentences.

Thus, $T$ is not $\kappa$-categorical.

This result is also known as Vaught's Test.