Finite and Infinite Sets

Abstract

Learning to count is one of our first great intellectual triumphs. But what exactly are we doing when we count? Can the process be generalized in a way that lets us compare sets that are larger than numbers can describe? What is meant by an infinite set? Are all infinite sets essentially the same size? Exploring these mysteries (the topic of cardinality) is our central goal in this chapter. As we proceed we will examine many of the fundamental principles of counting that we have viewed as intuitively obvious since childhood. (Example: You have n objects, and you give some of them to a friend; then you have given your friend at most n objects.) This will lay the foundation for the many practical counting procedures that we will develop here and in Chapter 5. In Section 4.5 we will use our work on functions and cardinality to initiate the study of languages and automata, fundamental topics in the theory of computation.

“No room! No room!” they cried out when they saw Alice coming. “There’s plenty of room!” said Alice indignantly, and she sat down in a large armchair at one end of the table.

Lewis Carroll

Alices Adventures in Wonderland