cardinal size," then most of the axioms can be justified: The axiom of separation produces a subset of x that is not bigger than x. The axiom of replacement produces an image set f(x) that is not bigger than x. The axiom of union produces a union whose size is not bigger than the size of the biggest set in the union times the number of sets in the union. The axiom of choice produces a choice set whose size is not bigger than the size of the given set of nonempty sets. The limitation of size doctrine does not justify the axiom of infinity: which uses the empty set and sets obtained from the empty set by iterating the ordinal successor operation. Since these sets are finite, any set satisfying this axiom, such as ω, is much bigger than these sets. Fraenkel and Lévy regard the empty set and the infinite set of natural numbers, whose existence is implied by the axioms of infinity and separation, as the starting point for generating sets. Von Neumann's approach to limitation of size uses the axiom of limitation of size. As mentioned in , von Neumann's axiom implies the axioms of separation, replacement, union, and choice. Like Fraenkel and Lévy, von Neumann had to add the axiom of infinity to his system since it cannot be proved from his other axioms. The differences between von Neumann's approach to limitation of size and Fraenkel and Lévy's approach are: Von Neumann's axiom puts limitation of size into an axiom system, making it possible to prove most set existence axioms. The limitation of size doctrine justifies axioms using informal arguments that are more open to disagreement than a proof. Von Neumann assumed the power set axiom since it cannot be proved from his other axioms. Fraenkel and Lévy state that the limitation of size doctrine justifies the power set axiom. There is disagreement on whether the limitation of size doctrine justifies the power set axiom. Michael Hallett has analyzed the arguments given by Fraenkel and Lévy. Some of their arguments measure size by criteria other than cardinal size—for example, Fraenkel introduces "comprehensiveness" and "extendability." Hallett points out what he considers to be flaws in their arguments. Hallett then argues that results in set theory seem to imply that there is no link between the size of an infinite set and the size of its power