Neumann ordinals. The existence of order types for all well-orderings is not a theorem of Zermelo set theory: it requires the Axiom of replacement. Even Scott's trick cannot be used in Zermelo set theory without an additional assumption (such as the assumption that every set belongs to a rank which is a set, which does not essentially strengthen Zermelo set theory but is not a theorem of that theory). In NFU, the collection of all ordinals is a set by stratified comprehension. The Burali-Forti paradox is evaded in an unexpected way. There is a natural order on the ordinals defined by if and only if some (and so any) is similar to an initial segment of some (and so any) . Further, it can be shown that this natural order is a well-ordering of the ordinals and so must have an order type . It would seem that the order type of the ordinals less than with the natural order would be , contradicting the fact that is the order type of the entire natural order on the ordinals (and so not of any of its proper initial segments). But this relies on one's intuition (correct in ZFC) that the order type of the natural order on the ordinals less than is for any ordinal . This assertion is unstratified, because the type of the second is four higher than the type of the first (two higher if a type level pair is used). The assertion which is true and provable in NFU is that the order type of the natural order on the ordinals less than is for any ordinal , where is the order type of for any (it is easy to show that this does not depend on the choice of W; note that T raises type by one). Thus the order type of the ordinals less than with the natural order is , and . All uses of here can be replaced with if a type-level pair is used. This shows that the T operation is nontrivial, which has a number of consequences. It follows immediately that the singleton map is not a set, as otherwise restrictions of this map would establish the similarity of W and for any well-ordering W. T is (externally) bijective and order-preserving. Because of this, the fact establishes that is a "descending sequence" in the ordinals which cannot be a set.