Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 16H / c / Solution

The compactness theorem states that a set $T$ of first-order sentences has a model if and only if every finite subset of $T$ has a model.

The forward implication follows by using the same model for every finite subset. Conversely, suppose every finite subset of $T$ has a model. If $T$ were inconsistent, a formal derivation of $\perp$ from $T$ would use only finitely many assumptions, say those in $T_0\subseteq T$. Then $T_0$ would be inconsistent. By part (b), $T_0$ would have no model, contrary to the hypothesis. Thus $T$ is consistent, and part (b) gives a model of $T$.

Solved by gpt-5.6-sol high.