Contributions to the Founding of the Theory of Transfinite Numbers

As a teenager studying mathematics, I was fascinated by Cantor's notion of an infinity of infinities. Then I bought this book and read the arguments behind his theories, and realized that, although he had invented an interesting new mathematical game, it was not a part of the standard arithmetical game, for he made several errors. There are many I could list, but two are crucial. Firstly he makes a subtle change in the meaning of equinumerous sets, which should mean that all bijections between the members of two sets hold - Cantor reduces this to the sets being equinumerous as long as there is at least one bijection which holds, ie his sense of the notion is weaker than the standard interpretation. Secondly his diagonal proceedure show more requires the ability to complete an infinite process, which is impossible. He claims that at any stage he produces a number that differs from all those so far processed, which is true, but there always remain an infinite number of numbers which may contain the "different" number he has constructed, and since he must work his process an infinite number of times to produce a number different from all in an infinite set he has to shew that an infinite process can be completed, which is not possible. Nice try Georg, but yours is a game I can admire, but not one I can play! show less

Surprisingly accessible for a century old mathematical treatise. Cantor's demonstration of how the same infinite set of rational numbers could have different orderings remains one of the most mind-blowing ideas I've ever encountered.