Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 16G / a / Solution

A set $S$ is a deductively closed set of formulae when every consequence of $S$ already belongs to it:It is a consistent set of formulae when it does not prove a contradiction,

If $S$ is inconsistent, there is a formal derivation of $\perp$ from assumptions in $S$. By the finite character of formal proofs, that derivation mentions only finitely many assumptions $s_1,\dots ,s_m\in S$. Henceso this finite subset of $S$ is already inconsistent.