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 the Upward Löwenheim-Skolem Theorem 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.
$\blacksquare$
Source of Name
This entry was named for Jerzy Maria Michał Łoś and Robert Lawson Vaught.
This result is also known as Vaught's Test.