We prove that every abstract elementary class (a.e.c.) with Löwenheim–Skolem–Tarski (LST) number $\kappa$ and vocabulary $\tau$ of cardinality $\leq \kappa$ can be axiomatized in the logic ${\mathbb L}_{\beth _2(\kappa )^{+++},\kappa ^+}(\tau )$. An a.e.c. $\mathcal {K}$ in vocabulary $\tau$ is therefore an EC class in this logic, rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the