Omitting types

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.