  In this paper, we prove that $\mathsf{WA}_1$ implies the consistency of
$\mathsf{WA}_0$, and $\mathsf{WA}_{n+2}$ implies the consistency of
$\mathsf{WA}_n$ for $n\ge 0$. We also prove that $\mathsf{ZFC+WA}_n$ is
finitely axiomatizable, and $\mathsf{ZFC+WA}$ is not finitely axiomatizable.
