Omitting Types of Prenex Formulas

In the literature, a type Γ is a set of formulas ϕ ( ν 0 ) in the one free variable ν 0 . A model M realizes Γ if there is an element a of M such that M ╞ ϕ [ a ] for all ϕ ∈ Γ. A model M omits Γ if no element of M realizes Γ. Evidently, the two statements: (*) M realizes Γ and (**) M omits Γ