On Cantor and the Transfinite

A set in mathematics is just a collection of elements; an example is the set of natural numbers {1, 2, 3, ...}. Simplifying somewhat, the theory of sets can be regarded as the foundation on which the whole of mathematics is built; and the founder of set theory is the German logician and mathematician Georg Cantor (1845 1918). However, the aspect of Cantor’s work that’s most widely known—or most controversial, at any rate—isn’t so much set theory in general, but rather those parts of that theory that have to do with infinite sets in particular. Cantor claimed among other things that the infinite set of real numbers contains strictly more elements than the infinite set of natural numbers. From this result, he concluded that there’s more than one kind of infinity; in fact, he claimed that there are an infinite number of different infinities, or transfinite numbers. (He also believed these results had been communicated to him by God.)

The aim of this book is to explain and investigate these claims of Cantor’s in depth (and question them, where appropriate). It’s not a textbook, though; instead, it’s a popular account—it tells a story—and the target audience is interested lay readers, not mathematicians or logicians. What little mathematics is needed to understand the story is explained in the book itself.