Cantor's First Uncountability Proof - The Development of Cantor's Ideas

The Development of Cantor's Ideas

The development leading to Cantor's article appears in the correspondence between Cantor and his fellow mathematician Richard Dedekind. On November 29, 1873, Cantor asked Dedekind whether the collection of positive integers and the collection of positive real numbers "can be corresponded so that each individual of one collection corresponds to one and only one of the other?" Cantor added that collections having such a correspondence include the collection of positive rational numbers, and collections of the form (a_{n1, n2, …, nν}) where n₁, n₂,…, n_ν, and ν are positive integers.

Dedekind replied that he was unable to answer Cantor's question, and said that it "did not deserve too much effort because it has no particular practical interest." Dedekind also sent Cantor a proof that the set of algebraic numbers is countable.

On December 2, Cantor pointed out that his question does have interest: "It would be nice if it could be answered; for example, provided that it could be answered no, one would have a new proof of Liouville's theorem that there are transcendental numbers."

On December 7, Cantor sent Dedekind an intricate proof by contradiction that the set of real numbers is uncountable. This proof uses infinitely many sequences of real numbers while the published proof uses only two sequences. Taken together, the letters of December 2 and 7 provide a non-constructive proof of the existence of transcendental numbers.

On December 9, Cantor announced the theorem that allows him to construct transcendental numbers as well as prove the uncountability of the set of real numbers:

I show directly that if I start with a sequence

(I) ω₁, ω₂, …, ω_n, …

I can determine, in every given interval, a number η that is not included in (I).

This theorem is the second theorem in Cantor's article.