real numbers are defined just as one would expect from the algebraic rules for adding and multiplying differences. The treatment of order is also as in elementary algebra. This is the briefest sketch of the constructions. Note that the constructions are exactly the same in ZFC and in NFU, except for the difference in the constructions of the natural numbers: since all variables are restricted to strongly cantorian sets, there is no need to worry about stratification restrictions. Without the Axiom of Counting, it might be necessary to introduce some applications of T in a full discussion of these constructions. Operations on indexed families of sets In this class of constructions it appears that ZFC has an advantage over NFU: though the constructions are clearly feasible in NFU, they are more complicated than in ZFC for reasons having to do with stratification. Throughout this section assume a type-level ordered pair. Define as . The definition of the general n-tuple using the Kuratowski pair is trickier, as one needs to keep the types of all the projections the same, and the type displacement between the n-tuple and its projections increases as n increases. Here, the n-tuple has the same type as each of its projections. General cartesian products are defined similarly: The definitions are the same in ZFC but without any worries about stratification (the grouping given here is opposite to that more usually used, but this is easily corrected for). Now consider the infinite cartesian product . In ZFC, this is defined as the set of all functions f with domain I such that (where A is implicitly understood as a function taking each i to ). In NFU, this is requires attention to type. Given a set I and set valued function A whose value at in is written , Define as the set of all functions f with domain I such that : notice that is stratified because of our convention that A is a function with values at singletons of the indices. Note that the very largest families of sets (which cannot be indexed by sets of singletons) will not have cartesian products under this definition. Note further that the sets are at the same type as the index set I (since one type higher than its elements); the product, as a set of functions with domain I (so at the same type as I) is one