User blog comment:Emlightened/Little Bigeddon/@comment-24061664-20170117003234/@comment-5529393-20170117094924

There are infinitely many n-ary formulas of quantifier rank k, but what Emlightened noted was that there are only finitely many inequivalent n-ary formulas of rank k. She proved this inductively on the rank, by noting that any n,k+1 formulas can be built up from n+1,k formulas, so if there are only finitely many of the latter there are only finitely many of the former (again, up to equivalence).