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

The countable omitting types theorem says that if {\displaystyle Z} is a countable set of non-isolated types (perhaps with varying {\displaystyle n}), then there is a countable model of {\displaystyle T} in which no type in {\displaystyle Z} is realized. One proof proceeds by model-theoretic forcing (as described in Hodges' book Building Models by Games). If {\displaystyle Z} is finite, more elementary proofs exist. For example, one can take a model of {\displaystyle T}, and begin building up a subset in which no type in {\displaystyle 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 {\displaystyle T} is a countable theory in which there is a non-isolated {\displaystyle n}-type {\displaystyle p}, then by the omitting types theorem, there is some countable model of {\displaystyle T} in which {\displaystyle p} is not realized. On the other hand, by compactness and Löwenheim-Skolem, there is a countable model in which {\displaystyle p} is realized. The two countable models just described cannot be isomorphic, so {\displaystyle T} is not countably categorical. ::: ::: :::