Sara Carlberg: Oändliga mängder och kardinaltal

Abstract.

This essay reviews the basics of infinite sets and their cardinal numbers. It begins with a historical background and then reviews a few basics of set theory. Once the foundation is laid, we move on to a brief description of functions. This is followed by a proof of representations of real numbers. The main part of the paper focuses on countably infinite and uncountably infinite sets and how we distinguish them, for example through Cantor’s diagonal argument. This is followed by a chapter on the cardinal numbers of infinite sets and how we count and compare them, for example through Schröder-Bernstein’s theorem. Finally, we review the fascinating properties of the non-trivial Cantor set.