set. This would imply that the limitation of size doctrine is incapable of justifying the power set axiom because it requires that the power set of x is not "too much bigger" than x. For the case where size is measured by cardinal size, Hallett mentions Paul Cohen's work. Starting with a model of ZFC and , Cohen built a model in which the cardinality of the power set of ω is if the cofinality of is not ω; otherwise, its cardinality is . Since the cardinality of the power set of ω has no bound, there is no link between the cardinal size of ω and the cardinal size of P(ω). Hallett also discusses the case where size is measured by "comprehensiveness," which considers a collection "too big" if it is of "unbounded comprehension" or "unlimited extent." He points out that for an infinite set, we cannot be sure that we have all its subsets without going through the unlimited extent of the universe. He also quotes John L. Bell and Moshé Machover: "... the power set P(u) of a given [infinite] set u is proportional not only to the size of u but also to the 'richness' of the entire universe ..." After making these observations, Hallett states: "One is led to suspect that there is simply no link between the size (comprehensiveness) of an infinite a and the size of P(a)." Hallett considers the limitation of size doctrine valuable for justifying most of the axioms of set theory. His arguments only indicate that it cannot justify the axioms of infinity and power set. He concludes that "von Neumann's explicit assumption [of the smallness of power-sets] seems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden implicit assumption of the smallness of power-sets." History Von Neumann developed the axiom of limitation of size as a new method of identifying sets. ZFC identifies sets via its set building axioms. However, as Abraham Fraenkel pointed out: "The rather arbitrary character of the processes which are chosen in the axioms of Z [ZFC] as the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments." The historical development of the ZFC axioms began in 1908 when Zermelo chose axioms to eliminate the paradoxes and to support his proof of the well-ordering theorem. In 1922, Abraham Fraenkel and Thoralf Skolem pointed out that Zermelo's axioms cannot