User blog comment:Emlightened/Little Bigeddon/@comment-1605058-20170114100429/@comment-27513631-20170115203941

Yes, but there are only boundedly many \(n\)-ary formulae of quantifier rank \(k\). Let \(F_{n,k}\) be the set of first-order formulae of quantifier rank \(\leq n\) and \(k\) free variables in the language \(\{\in\}\). Clearly, \(|F_{0,k}|\leq 2^{k^2}\). For each formula \([\phi] \in F_{n,k+1}\), there is a corresponding formula \([\exists x_n \phi] \in F_{n+1,k}\), and each formula \(\in F_{n+1,k}\) can be built up from such formulae using conjunction, disjunction and negation. In particular, \(F_{n,k+1}\leq 2^{F_{n+1,k}}\).