1. Sets and Size Basic Question: Compare the “size” of sets. First distinction finite or infinite. –What is a finite set? –How can one compare finite sets? Number of elements Inclusion: A proper subset has less elements –What about infinite sets? By definition they all have infinitely many elements. Does a proper subset have less elements?

2. Basic questions and techniques Good tool to compare sets is a 1-1 correspondence which is sometimes called a bijection. This works well for finite sets and also for infinite sets. It gives a way to talk about the size, magnitude or power of infinite sets. Basic question: compare the infinite sets of the natural numbers N, the rational numbers Q and the real numbers R.

3. Set Theory Greek philosophers: the infinite as a source for paradox. –Aristotle (384-322 BC) –Zeno (495?-435? B.C.) Bernard Bolzano (1781-1848): a first mathematical study of the nature of infinite sets. Georg Cantor (1845-1918): A comprehensive study of infinite sets. The beginning of modern set theory. Introduces cardinals, ordinals the continuum hypothesis. Bertrand Russel (1872-1970). Gives his paradox which shows that the naïve approach to set theory like that of Frege will not work. Ernst Zermelo (1871-1953). Introduces the Axiom of Choice and shows that with it all sets are can be well ordered. Together with Adolf Fraenkel (1891-1965) gives axioms of set-theory.

4. Axioms of choice and the continuum hypothesis Cantor introduced the continuum hypothesis and believed that he can prove it. Zermelo introduced the axiom of choice. Zermelo and Fraenkel introduced the axioms of set theory ZF and ZFC. Kurt Gödel (1906-1978) showed that –It is impossible to prove the consistency of set theory within set theory. –The axiom of choice is relatively consistent, i.e. if ZF is consistent, so is ZFC. –The continuum hypothesis is relatively consistent w.r.t. ZFC. Paul Cohen (*1934): –The negation of the continuum hypothesis is relatively consistent w.r.t. ZFC, i.e. it is independent. –The axiom of choice is independent of ZF.