Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2008 / B / 5 / iii / 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 every formal proof uses only finitely many premises, even when the whole first-order theory is infinite.