Nonfinite axiomatizability of torsion-free abelian groups

The class of torsion-free abelian groups has an infinite first-order axiomatization but no finite axiomatization. Add $\forall x(nx=0\to x=0)$ for every positive integer $n$ to the abelian group axioms. If an equivalent finite theory existed, compactness would make it follow from finitely many of these conditions. A cyclic group of prime order larger than the finitely many integers mentioned satisfies those conditions but has torsion, a contradiction.