= Solution
The <compactness theorem> states that a <set> of <first-order sentences> has a <first-order model> if and only if every finite <subset> has a <first-order model>.
For the nontrivial direction suppose every finite <subset> of $\Sigma$ is <satisfiable>. If $\Sigma$ were inconsistent, a finite formal derivation of a <contradiction> would use only finitely many premises $\Sigma_0\subseteq\Sigma$. <Soundness of an arithmetic theory> would then exclude a <first-order model> of $\Sigma_0$, contradicting finite satisfiability. Therefore $\Sigma$ is <first-order consistent>. The completeness theorem, proved by adjoining witnesses and constructing a Henkin <term model>, supplies a <first-order model> of $\Sigma$.
The reverse direction is immediate by restriction of the list of required <first-order sentences>. The key reason <compactness theorem> follows from syntactic completeness is that \b[every <formal proof> uses only finitely many premises], even when the whole <first-order theory> is infinite.
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