 We can define addition for cardinal numbers as the cardinality of a set that is the union of two disjoint sets of those cardinalities. We can define multiplication for a pair of cardinal numbers as the cardinality of the set of all ordered pairs whose first elements are drawn from a set of the first cardinality and whose second elements are drawn from a set of the second cardinality. But it turns out that these operations are relatively trivial once we get beyond the finite cardinals—just as we saw that the sum or product of two countable ordinals was still countable, the sum or product of two infinite cardinals is equal to whichever of the two is larger! (At least this operation is commutative.) Thus, one can’t get to \(\aleph_1\) from \(\aleph_0\) by means of addition or multiplication (as we saw when we considered order types achieved by addition and multiplication).