 Solution

The local omission of a partial type $\Gamma (\mathbf{x})$ by a consistent first-order theory $T$ means that, for every formula $\theta (\mathbf{x})$ consistent with $T$, there is some $\gamma \in \Gamma$ such thatis consistent. Equivalently, no formula consistent with $T$ implies every member of $\Gamma$. Thus every possible local condition has a compatible refinement excluding at least one proposed type formula.

The omitting types theorem states: in a countable first-order language, if $T$ is consistent and locally omits each of countably many finite-arity families ${\Gamma }_j$, then it has a countable model omitting them all. Omission means that no tuple satisfies every formula in the family. In the case of a complete type, local omission is equivalent to the type being nonisolated. Countability of the language and of the list of omission requirements is essential to this formulation.