Past exam of the model theory course of the University of Oxford / 2011 / 2 / c / ii

No. Let $S$ be the displayed infinite axiom set. Any finite subset $S_0\subseteq S$ mentions only finitely many of the two infinitude requirements. Choose $N\ge 1$ at least as large as every index mentioned. A finite equivalence relation with exactly $N$ singleton classes and $N$ doubleton classes satisfies $S_0$, but is not a first-order model of the complete theory.

If a finite axiomatization existed, let $\sigma$ be the conjunction of its axioms. Then $S\models \sigma$, so the compactness theorem gives a finite $S_0\subseteq S$ with $S_0\models \sigma$. Every first-order model of $S_0$ would then satisfy the full first-order theory, contradicting the finite first-order model above. This is the finite subaxiomatization from an equivalent finite theory argument: finite axiomatizability would force one finite fragment of the explicit axioms to suffice.