The assumption here is clearly that the (transfinite) numbers will have to be added to set theory as new primitives. Kuratowski however undertakes to prove that the transfinite numbers can be dispensed with for a significant class of   applications.[34]  Application of the ordinal numbers in analysis, topology, etc. often focuses on some process of definition by transfinite recursion over these numbers. Kuratowski succeeds in showing that in a significant class of cases of this kind, the ordinals can be avoided by using purely set-theoretic methods which are reproducible in Zermelo's system. As he notes: