User blog comment:Pellucidar12/Peng-Pellucid-Armstrong Number/@comment-29915175-20170102011204/@comment-1605058-20170105213935

I'm not saying a single statement can be proven in infinitely many ways using n inferences. I'm saying that there are infinitely many statements which can be proven using n inferences. For example, all the axioms can be proven using just one inference. I can imagine that using a fixed a number of inferences, we can prove an infinite number of statements of the form "there is a unique natural number...". This is a threat to well-definedness of a number.

I am not saying either of these are difficult to fix. I am just saying these have to be fixed. I agree that stating AC might be easier, but even stating "wellorders" shouldn't be terribly hard. Going for ZF sounds like a good alternative as well.