= Solution
The <Godel completeness theorem> says that for every first-order theory $\Gamma$ and sentence $\varphi$,
$$
\Gamma\models\varphi\quad\Longrightarrow\quad\Gamma\vdash\varphi.
$$
Together with <soundness theorem for first-order logic>[soundness], this is equivalence; it is also equivalent to saying that every syntactically consistent first-order theory has a model.
The <compactness theorem> states that a set $\Gamma$ of first-order sentences has a model if and only if every finite subset $\Gamma_0\subseteq\Gamma$ has a model. To deduce it, suppose every finite subset is satisfiable but $\Gamma$ is not. Then $\Gamma\models\bot$, so completeness gives a formal proof $\Gamma\vdash\bot$. A proof uses only finitely many assumptions, yielding some finite $\Gamma_0\vdash\bot$, contrary to that subset's satisfiability.
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