ordinal) of a well-ordering of A (that every set can be well-ordered follows from the Axiom of Choice in the usual way in both theories). The natural order on cardinal numbers is seen to be a well-ordering: that it is reflexive, antisymmetric (on abstract cardinals, which are now available) and transitive has been shown above. That it is a linear order follows from the Axiom of Choice: well-order two sets and an initial segment of one well-ordering will be isomorphic to the other, so one set will have cardinality smaller than that of the other. That it is a well-ordering follows from the Axiom of Choice in a similar way. With each infinite cardinal, many order types are associated for the usual reasons (in either set theory). Cantor's theorem shows (in both theories) that there are nontrivial distinctions between infinite cardinal numbers. In ZFC, one proves In NFU, the usual form of Cantor's theorem is false (consider the case A=V), but Cantor's theorem is an ill-typed statement. The correct form of the theorem in NFU is , where is the set of one-element subsets of A. shows that there are "fewer" singletons than sets (the obvious bijection from to V has already been seen not to be a set). It is actually provable in NFU + Choice that (where signals the existence of many intervening cardinals; there are many, many urelements!). Define a type-raising T operation on cardinals analogous to the T operation on ordinals: ; this is an external endomorphism of the cardinals just as the T operation on ordinals is an external endomorphism of the ordinals. A set A is said to be cantorian just in case ; the cardinal is also said to be a cantorian cardinal. A set A is said to be strongly cantorian (and its cardinal to be strongly cantorian as well) just in case the restriction of the singleton map to A () is a set. Well-orderings of strongly cantorian sets are always strongly cantorian ordinals; this is not always true of well-orderings of cantorian sets (though the shortest well-ordering of a cantorian set will be cantorian). A cantorian set is a set which satisfies the usual form of Cantor's theorem. The operations of cardinal arithmetic are defined in a set-theoretically motivated way in both theories. . One would like to define as , and one does this in ZFC, but there is