Abstract In this paper, we show that $\mathsf {ZFC}+\mathsf {WA}_{n+1}$ implies the consistency of $\mathsf {ZFC}+\mathsf {WA}_n$ for $n\ge 0$ . We also prove that $\mathsf {ZFC}+\mathsf {WA}_n$ is finitely axiomatizable, and $\mathsf {ZFC}+\mathsf {WA}$ is not finitely axiomatizable unless it is inconsistent.