Ordinals fixed by T are called Cantorian ordinals, and ordinals which dominate only cantorian ordinals (which are easily shown to be cantorian themselves) are said to be strongly cantorian. There can be no set of cantorian ordinals or set of strongly cantorian ordinals. Digression: von Neumann ordinals in NFU It is possible to reason about von Neumann ordinals in NFU. Recall that a von Neumann ordinal is a transitive set A such that the restriction of membership to A is a strict well-ordering. This is quite a strong condition in the NFU context, since the membership relation involves a difference of type. A von Neumann ordinal A is not an ordinal in the sense of NFU, but belongs to an ordinal which may be termed the order type of (membership on) A. It is easy to show that the order type of a von Neumann ordinal A is cantorian: for any well-ordering W of order type , the induced well-ordering of initial segments of W by inclusion has order type (it is one type higher, thus the application of T): but the order types of the well-ordering of a von Neumann ordinal A by membership and the well-ordering of its initial segments by inclusion are clearly the same because the two well-orderings are actually the same relation, so the order type of A is fixed under T. Moreover, the same argument applies to any smaller ordinal (which will be the order type of an initial segment of A, also a von Neumann ordinal) so the order type of any von Neumann ordinal is strongly cantorian. The only von Neumann ordinals which can be shown to exist in NFU without additional assumptions are the concrete finite ones. However, the application of a permutation method can convert any model of NFU to a model in which every strongly cantorian ordinal is the order type of a von Neumann ordinal. This suggests that the concept "strongly cantorian ordinal of NFU" might be a better analogue to "ordinal of ZFC" than is the apparent analogue "ordinal of NFU". Cardinal numbers Cardinal numbers are defined in NFU in a way which generalizes the definition of natural number: for any set A, . In ZFC, these equivalence classes are too large as usual. Scott's trick could be used (and indeed is used in ZF), is usually defined as the smallest order type (here a von Neumann