The weakness of the proof outlined lies precisely here. It is not proved that the whole series of numbers Ω can be  “projected into” any multiplicity V which does not have an aleph as a cardinal number, but this is rather taken from a somewhat vague “intuition”. Apparently Cantor imagines the numbers of Ω successively and arbitrarily assigned to elements of V in such a way that every element of V is only used once. Either this process must then come to an end, in that all elements of V are used up, in which case V would be then be coordinated with an initial segment of the number series, and its power consequently an aleph, contrary to assumption; or V would remain inexhaustible and would then contain a component equivalent to the whole of Ω, thus an inconsistent component. Here, the intuition of time [Zeitanschauung] is being applied to a process which goes beyond all intuition, and a being [Wesen] supposed which can make successive arbitrary choices and thereby define a subset V′ of V which is not definable by the conditions given. (Zermelo in Cantor 1932:  451)[19]