User blog comment:Pellucidar12/Peng-Pellucid-Armstrong Number/@comment-29915175-20170102011204/@comment-1605058-20170105155134

There are at least two more issue - first, the axiomatization given in Emlightened's link is not precise - things like \(\emptyset,\{x,y\},P(x),\cup x,\omega\), let alone \(wellorders\), although understood perfectly fine, are not of use when describing ZFC as a formal system. By the way, traditional treatment of set theory also doesn't include anything like a symbol for \(V\) - the universe of sets is not thought of as an entity to which things belong; we assert these things' existence.

Second, ZFC has, actually, infinitely many axioms, which are grouped into finitely many families (A3 and A9 from that file are actually infinite families of axioms, depending on a formula \(\phi\)). Because of that, there are infinitely many proofs of given length, so it's far from clear why your number should be well-defined (in fact, I am tempted to believe it isn't as it stands).

One more minor issue (probably more of a nitpick) is that it has to be specified exactly what it means that "ZFC proves \(\phi\) defines a unique number". One can easily formalize what this means, but I can imagine there being multiple ways to do that.