Fix some countable theory $T$ . Let $S_{n}$ be the space of complete $n$ -types over the empty set. A type $p(x)\in S_{n}$ is said to be isolated if there is a formula $\phi (x)$ (over the empty set) such that every element satisfying $\phi (x)$ realizes $p(x)$ , and vice versa. Equivalently, $p(x)$ is an isolated point in the stone topology on $S_{n}$ .

The countable omitting types theorem says that if $Z$ is a countable set of non-isolated types (perhaps with varying $n$ ), then there is a countable model of $T$ in which no type in $Z$ is realized. One proof proceeds by model-theoretic forcing (as described in Hodges' book Building Models by Games). If $Z$ is finite, more elementary proofs exist. For example, one can take a model of $T$ , and begin building up a subset in which no type in $Z$ is realized, using the Tarski-Vaught criterion as a guide to determine what to add to this set, to make the set eventually be a model.

The omitting types theorem provides one direction in the Ryll-Nardzewski theorem on countably categorical theories. Specifically, if $T$ is a countable theory in which there is a non-isolated $n$ -type $p$ , then by the omitting types theorem, there is some countable model of $T$ in which $p$ is not realized. On the other hand, by compactness and Löwenheim-Skolem, there is a countable model in which $p$ is realized. The two countable models just described cannot be isomorphic, so $T$ is not countably categorical. ::: ::: :::