But there was a problem concerning cardinality which is independent of the problem of definitional reduction. It was pointed out by both Fraenkel and Skolem in the early 1920s that Zermelo's theory cannot provide an adequate account of cardinality. The axiom of infinity and the power set axiom together allow the creation of sets of cardinality ≥ ℵn for each natural number n, but this (in the absence of a result showing that 2ℵ0 &gt; ℵn for every natural number n) is not enough to guarantee a set whose power is ≥ ℵω, and a set of power ℵω is a natural next step (in the Cantorian theory) after those of power ℵn. Fraenkel proposed a remedy to this (as did Skolem independently) by proposing what was called the Ersetzungsaxiom, the Axiom of Replacement (see Fraenkel 1922: 231 and Skolem 1923: 225–226). This says, roughly, that the ‘functional image’ of a set must itself be a set, thus if a is a set, then {F(x) : x ∈ a} must also be a set, where ‘F’ represents a functional correspondence. Such an axiom is certainly sufficient; assume that a0 is the set of natural numbers {0, 1, 2, …}, and now assume that to each number n is associated an an with power ℵn. Then according to the replacement axiom, a = {a0, a1, a2, …} must be a set, too. This set is countable, of course, but (assuming that the an are all disjoint) the union set of a must have cardinality at least ℵω.