User blog comment:Pellucidar12/Peng-Pellucid-Armstrong Number/@comment-29915175-20170102011204/@comment-27513631-20170105210234

All of those except wellorders (use a choice function or ZF is probably best) should be able to be fairly easily fixed.

There aren't, as far as I understand it, infinitely many proofs of \(\phi\) of a given length (if we take 'length' to be in terms of applications of inference rules). Suppose \(\phi\) has quantifier rank of \(n\). Each application of an inference rule can reduce the quantifier rank of a statement by at most \(1\). Therefore, only axioms with quantifier rank at most \(m+n\) can be used in the inference, and there are only finitely many of these (probably about \(2\uparrow\uparrow(m+n)\)).