ABSTRACT
The sets discussed in the preceeding chapter are all countable, that is they are all finite or denumerable. Indeed, we have found that the sets of integers and rational numbers are denumerable. We also found that the rational numbers are densely spaced on the line, so it may be tempting to think that the set of all real numbers (i.e., rational and irrational) is also denumerable. To contribute to this misconception, see Example 22.1 and Theorem 22.7. But we shall soon see that this is false. In other words, we are going to show that the set of real numbers is not cardinally equivalent to N. Thus, as indicated previously, there are infinite sets which are not cardinally equivalent to some other infinite sets. Cantor was among the first to recognize this. In fact, we shall see later in this chapter that there are infinitely many infinite sets, all of different "sizes."
