type higher (assuming a type-level ordered pair). Now consider the product of the cardinals of these sets. The cardinality || is one type higher than the cardinals , so the correct definition of the infinite product of cardinals is (because the inverse of T is not total, it is possible that this may not exist). Repeat this for disjoint unions of families of sets and sums of families of cardinals. Again, let A be a set-valued function with domain : write for . The disjoint union is the set . This set is at the same type as the sets . The correct definition of the sum is thus , since there is no type displacement. It is possible to extend these definitions to handle index sets which are not sets of singletons, but this introduces an additional type level and is not needed for most purposes. In ZFC, define the disjoint union as , where abbreviates . Permutation methods can be used to show relative consistency with NFU of the assertion that for every strongly cantorian set A there is a set I of the same size whose elements are self-singletons: for each i in I. The cumulative hierarchy In ZFC, define the cumulative hierarchy as the ordinal-indexed sequence of sets satisfying the following conditions: ; ; for limit ordinals . This is an example of a construction by transfinite recursion. The rank of a set A is said to be if and only if . The existence of the ranks as sets depends on the axiom of replacement at each limit step (the hierarchy cannot be constructed in Zermelo set theory); by the axiom of foundation, every set belongs to some rank. The cardinal is called . This construction cannot be carried out in NFU because the power set operation is not a set function in NFU ( is one type higher than A for purposes of stratification). The sequence of cardinals can be implemented in NFU. Recall that is defined as , where is a convenient set of size 2, and . Let be the smallest set of cardinals which contains (the cardinality of the set of natural numbers), contains the cardinal whenever it contains , and which is closed under suprema of sets of cardinals. A convention for ordinal indexing of any well-ordering is defined as the element x of the field of such that the