Infinities, Big and Small

Abstract

The Cantor set and the Koch curve are two examples of fractals may be considered not only as geometrical objects, but also as a means to explore apparent paradoxes and difficulties in set theory and analysis. This chapter focuses on the Cantor set and the properties of infinite sets. More specifically, it considers how big the Cantor set is. It discusses infinities of different sizes, first by thinking about finite sets, counting, countable infinities, rational numbers and irrational numbers, and binary. It then examines the cardinality of the unit interval and the Cantor set.