Łoś-Vaught Test

Theorem
Let $T$ be a satisfiable $\mathcal L$-theory with no finite models.

Let $T$ be $\kappa$-categorical for some infinite cardinal $\kappa \ge \left|{\mathcal L}\right|$.

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 $T \cup \left\{{\neg \phi}\right\}$ and $T \cup \left\{{\phi}\right\}$ have models.

Since $T$ has no finite models, this means that $T \cup \left\{{\neg \phi}\right\}$ and $T \cup \left\{{\phi}\right\}$ both have infinite models.

We have that $\kappa$ is infinite and greater than the cardinality of the language.

We also have that these theories have infinite models.

From Theory with one Infinite Model has Models of each Infinite Cardinal Larger than Language one can prove that there are size $\kappa$ models $\mathcal M_{\neg \phi}$ and $\mathcal M_{\phi}$ of $T \cup \left\{{\neg \phi}\right\}$ and $T \cup \left\{{\phi}\right\}$ respectively.

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.