Chapter B Countability This chapter is about Cantor’s countability theory which is a standard prerequisite for elementary real analysis. Our treatment is incomplete in that we cover only those results that are immediately relevant for the present course. | Chapter B Countability This chapter is about Cantor s countability theory which is a standard prerequisite for elementary real analysis. Our treatment is incomplete in that we cover only those results that are immediately relevant for the present course. In particular we have little to say here about cardinality theory and ordinal We shall however cover two relatively advanced topics here namely the theory of order isomorphisms and the Schroder-Bernstein Theorem on the equivalence of infinite sets. If you are familiar with countable and uncountable sets you might want to skip Section 1 and jump directly to the discussion of these two topics. The former one is put to good use in the last section of the chapter which provides an introduction to ordinal utility theory a topic that we shall revisit a few more times later. The Schroder-Bernstein Theorem is in turn proved via the Tarski Fixed Point Theorem the first of the many fixed point theorems that are discussed in this book. This theorem should certainly be included in the tool kit of an economic theorist for it has recently found a number of important applications in game theory. 1 Countable and Uncountable Sets In this section we revisit set theory and begin to sketch a systematic method of thinking about the size of any given set. The issue is not problematic in the case of finite sets for we can simply count the number of members of a given finite set and use this number as a measure of its size. Thus quite simply one finite set is more crowded than another if it contains more elements than the other. But how can one extend this method to the case of infinite sets Or how can we decide whether or not a given infinite set is more crowded than another such set Clearly things get icy with infinite sets. One may even think at first that any two infinite sets are equally crowded or even that the question is meaningless. There is however an intuitive way to approach the problem of ranking the sizes of .
