User blog comment:Emlightened/Little Bigeddon/@comment-24061664-20170117003234/@comment-1605058-20170117094645

That there are finitely many n-ary formulae with bounded rank can be proven by induction on rank. When the rank increases by one, the formula is built out of subformulas which have at most one more free variable and smaller rank, so you have finitely many "building blocks" for these formulas. I don't quite see where Emli got the bound \(F_{n,k+1}\leq 2^{F_{n+1,k}}\), but we still get finitely many (I believe).