The theorem von Neumann states is the central result of Cantor's mentioned here in the second paragraph of this section. As von Neumann goes on to point out here (also p. 374), it is the possibility of definition by transfinite induction which is key, and a rigorous treatment of this requires being able to prove at each stage in a transfinite inductive process that the collection of functional correlates to a set is itself a set which can thus act as a new argument at the next stage. It is just this which the replacement axiom guarantees. Once justified, definition by transfinite induction can be used as the basis for completely general definitions of the arithmetic operations on ordinal numbers, for the definition of the aleph numbers, and so on. It also allows a fairly direct transformation of Zermelo's first (1904) proof of the WOT into a proof that every set can be represented by (is equipollent with) an ordinal number, which shows that in the Zermelo system with the Axiom of Replacement added there are enough ordinal numbers.[38]