Although Kuratowski's work solved many of the representational problems for Zermelo's theory, and the Replacement Axiom shows how the most obvious cardinality gap can be closed, there still remained the issue (Kuratowski's view to one side) of representing accurately the full extent of the theory which Cantor had developed, with the transfinite numbers as fully fledged objects which ‘mirror’ the size/ordering of sets. Once the ordinal number-classes are present, the representation of the alephs is not a severe problem, which means that the representation of transfinite numbers amounts to assuring the existence of sufficiently many transfinite ordinal numbers. Indeed, as was stated above, the hypothesis that the scale of aleph numbers is sufficient amounts to the claim that any set can be ‘counted’ by some ordinal. There are then two interrelated problems for the ‘pure’ theory of sets: one is to show how to define ordinals as sets in such a way that the natural numbers generalise; the other problem is to make sure that there are enough ordinals to ‘count’ all the sets.