The Mathematics of Infinity: A Guide to Great Ideas, 2nd Edition

Chapter 5

Infinite Cardinals

As we defined in the previous chapter, the cardinality of a set A, card (A), is the class of all sets B that are equivalent to A. For a finite set A = {1,…, n}, card(A) is the collection of sets that contain exactly n elements. Then card(A) is a good way to describe the size of A, even when A is infinite. We might naively say that if A and B are infinite sets, then card(A) = card(B). However, we showed in Chapter 3 that Hilbert's Infinite Hotel, whose rooms are numbered 1, 2, 3,… cannot fit the Realists whose group is indexed by real numbers . In another light, there is no bijection from the set (0, 1) onto the set of natural numbers . If such a bijection does not exist, then we must conclude intuitively and accurately that

and card(0, 1) are different cardinalities.

This should come as a shock to your common sense. Both cardinals are infinite, so how can they be different values? The inequality is True enough, so we have learned that our common sense is wrong when we deal with infinity. This is an important point. We do not have any common sense when ...