 Solution

The Godel completeness theorem for first-order logic states that $\Gamma \models \varphi$ implies $\Gamma ⊢\varphi$; together with soundness, semantic consequence and derivability coincide. Equivalently, every syntactically consistent first-order theory has a model. The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset has a model. These statements use nonempty first-order structures.