 The empty set axiom is already assumed by axiom of infinity, and is now included as part of it.Zermelo set theory does not include the axioms of replacement and regularity.The axiom of replacement was first published in 1922 by Abraham Fraenkel and Thoralf Skolem, who had independently discovered that Zermelo's axioms cannot prove the existence of the set {"Z"0, "Z"1, "Z"2, ...} where "Z"0 is the set of natural numbers and "Z""n"+1 is the power set of "Z""n".