Zermalo-Frenkel Set theory with the Axiom of Choice is not proven to be consistent. One might argue that consistency is a natural requirement for a rule to be a

Zermalo-Frenkel Set theory with the Axiom of Choice is not proven to be consistent. One might argue that consistency is a natural requirement for a rule to be a rule. Especially for peano axioms, consistency is a given, so your comparison doesn't hold. Following the incomplefeness-theorem, the (dis-)proof might be external to any of ZFC's descriptions of the world, which believing in ZFC would be the world. disclaimer: I don't understand axiom of choice.