if there is a way to do this without invoking Choice (the use of representatives is hardly unknown in ZFC, either). In NFU, the use of equivalence class constructions to abstract properties of general sets is more common, as for example in the definitions of cardinal and ordinal number below. Ordinal numbers Two well-orderings and are similar and write just in case there is a bijection f from the field of to the field of such that for all x and y. Similarity is shown to be an equivalence relation in much the same way that equinumerousness was shown to be an equivalence relation above. In New Foundations (NFU), the order type of a well-ordering W is the set of all well-orderings which are similar to W. The set of ordinal numbers is the set of all order types of well-orderings. This does not work in ZFC, because the equivalence classes are too large. It would be formally possible to use Scott's trick to define the ordinals in essentially the same way, but a device of von Neumann is more commonly used. For any partial order , the corresponding strict partial order < is defined as . Strict linear orders and strict well-orderings are defined similarly. A set A is said to be transitive if : each element of an element of A is also an element of A. A (von Neumann) ordinal is a transitive set on which membership is a strict well-ordering. In ZFC, the order type of a well-ordering W is then defined as the unique von Neumann ordinal which is equinumerous with the field of W and membership on which is isomorphic to the strict well-ordering associated with W. (the equinumerousness condition distinguishes between well-orderings with fields of size 0 and 1, whose associated strict well-orderings are indistinguishable). In ZFC there cannot be a set of all ordinals. In fact, the von Neumann ordinals are an inconsistent totality in any set theory: it can be shown with modest set theoretical assumptions that every element of a von Neumann ordinal is a von Neumann ordinal and the von Neumann ordinals are strictly well-ordered by membership. It follows that the class of von Neumann ordinals would be a von Neumann ordinal if it were a set: but it would then be an element of itself, which contradicts the fact that membership is a strict well-ordering of the von