Some Infinite Surprises—in which some wild sets are tamed, and some nearly escape

Abstract

In this chapter we look at the properties, many of them paradoxical, of infinite sets. The subject is closely associated with the name of Georg Cantor (1845–1918). Unlike many, perhaps most, major theories in mathematics, Cantor’s ideas on infinite sets owe little to foundations built over previous centuries by other mathematicians. He was the source of most of the ideas, and for this reason the subject is relatively easy to tie down to its origins. We shall be adding a few remarks to give some historical colour to our story, but by the end of the chapter you will probably agree that the subject is quite colourful enough anyway!

‘Je le vois, mais je ne le crois pas.’

Georg Cantor, 1877

‘Infinity is just so big that by comparison, bigness itself looks really titchy.’

Douglas Adams, The Hitch-hiker’s Guide to the Galaxy