Theorem 6.2: It is possible to produce any number of pairwise disjoint sets of any size.

Proof: Let {n1, n2, . . . , nm} be any set of natural numbers. Recall that by Theorem 4.11, any finite collection of natural numbers is a set. Then the sets

n1 x {1}, n2 x {2} , . . . , nm x {m }

are pairwise disjoint sets where

#(ni x {i }) = ni

by Theorem 5.9. To see that they are disjoint, note that any elements in distinct sets will have different second coordinates, and so cannot be equal.