Let $L$ be a first order language and let $S$ be a set of $L$ -sentences. If $\kappa$ is a cardinal, then $S$ is said to be $\kappa$ -categorical if $S$ has a model of cardinality $\kappa$ and any two such models are isomorphic.

In other words, $S$ is categorical iff it has a unique model of cardinality $\kappa$ , to within isomorphism.