Past exam of the logic and set theory course of the University of Oxford / 2007 / A / 4 / ii / Solution

The compactness theorem says that a set $T$ of first-order sentences has a first-order model if and only if every finite subset of $T$ has a first-order model. One direction follows by using the same first-order model for each subset.

For the nontrivial direction suppose every finite subset has a first-order model, but $T$ has none. Semantically $T\models \perp$, so completeness gives a formal logical derivation $T⊢\perp$. A formal logical derivation is finite and uses only finitely many assumptions, say $T_0\subseteq T$. Thus $T_0⊢\perp$. Soundness would make $T_0$ unsatisfiable, contradicting the hypothesis. Here $\perp$ can be any fixed contradictory first-order sentence, such as $\exists x(x\ne x)$, if it is not a primitive logical symbol. Therefore finite satisfiability implies satisfiability.