3 - The Cardinality of Sets

Abstract

This chapter discusses the properties characteristic of infinite sets. Any infinite subset of the set of natural numbers is countable. This means that there can be no infinite set whose cardinality is less than the cardinality of a countable set. One can conclude from this that the cardinality of a countable set is not greater than the cardinality of any infinite set. It is usually not easy to prove that a set is uncountable. To prove that a set is countable means simply to invent a method of enumerating its elements. However, to prove that a set is uncountable, one has to prove that no such method exists. In other words, no matter what method one applied, some element of the set would fail to be counted. The concept of cardinality is a generalization of the concept of number of elements in a finite set. The cardinalities of sets do only half the work of the natural numbers. However, the cardinality tells nothing about the order of the elements. And even though the set of natural numbers has as many elements as the set of integers, they are ordered in quite different ways. The set of natural numbers has a first element, while the set of integers has no first element. The cardinal numbers therefore yield insufficient knowledge for the study of the order of arrangement of the elements in a set.