into , if is the universal set! The resolution requires that one observes that makes no sense in the theory of types: the type of is one higher than the type of . The correctly typed version (which is a theorem in the theory of types for essentially the same reasons that the original form of Cantor's theorem works in ZF) is , where is the set of one-element subsets of . The specific instance of this theorem of interest is : there are fewer one-element sets than sets (and so fewer one-element sets than general objects, if we are in NFU). The "obvious" bijection from the universe to the one-element sets is not a set; it is not a set because its definition is unstratified. Note that in all known models of NFU it is the case that ; Choice allows one not only to prove that there are urelements but that there are many cardinals between and . One can now introduce some useful notions. A set which satisfies the intuitively appealing is said to be Cantorian: a Cantorian set satisfies the usual form of Cantor's theorem. A set which satisfies the further condition that , the restriction of the singleton map to A, is a set is not only Cantorian set but strongly Cantorian. The Burali-Forti paradox of the largest ordinal number goes as follows. Define (following naive set theory) the ordinals as equivalence classes of well-orderings under isomorphism. There is an obvious natural well-ordering on the ordinals; since it is a well-ordering it belongs to an ordinal . It is straightforward to prove (by transfinite induction) that the order type of the natural order on the ordinals less than a given ordinal is itself. But this means that is the order type of the ordinals and so is strictly less than the order type of all the ordinals — but the latter is, by definition, itself! The solution to the paradox in NF(U) starts with the observation that the order type of the natural order on the ordinals less than is of a higher type than . Hence a type level ordered pair is two types higher than the type of its arguments and the usual Kuratowski ordered pair four types higher. For any order type , we can define an order type one type higher: if , then is the order type of the order