= Solution
\b[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 $N\geq1$ 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: \b[finite axiomatizability would force one finite fragment of the explicit axioms to suffice].
Back to article page
