Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2009 / B / 6 / a / Solution

A complete $n$-type over $T$ is a maximal set $p(\bar{x})$ of formulas in $n$ variables consistent with $T$; it decides each such formula or its negation. A partial type drops maximality. A tuple realizes the type if it satisfies every member. A complete type is principal, or isolated, if some formula $\theta (\bar{x})$ consistent with $T$ implies every member of the type modulo $T$; then $\theta$ itself belongs to that type.

The omitting types theorem says that in a countable first-order language, a consistent theory has a countable model omitting a specified nonprincipal type; more generally a countable family of nonprincipal types can be omitted simultaneously. In the present complete-theory setting it applies directly to complete nonisolated types. “Omitted” means that no tuple in the model realizes all formulas of the type, even though each finite subset is compatible with the theory.