## FANDOM

78 Pages

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.