Cantor's First Uncountability Proof

Cantor's First Uncountability Proof

Georg Cantor's first uncountability proof demonstrates that the set of all real numbers is uncountable. This proof differs from the more familiar proof that uses his diagonal argument. Cantor's first uncountability proof was published in 1874, in an article that also contains a proof that the set of real algebraic numbers is countable, and a proof of the existence of transcendental numbers.

Two controversies have developed about that article:

• Is Cantor's proof of the existence of transcendental numbers constructive or non-constructive?
• Why did Cantor emphasize the countability of the real algebraic numbers rather than the uncountability of the real numbers?

In 1891 Cantor published his diagonal argument, which produces an uncountability proof that is generally considered simpler and more elegant than his first proof. Both uncountability proofs contain ideas that can be used elsewhere. The diagonal argument is a general technique that is useful in mathematical logic and theoretical computer science, while Cantor's first uncountability proof can be generalized to any infinite ordered set with the same order properties as the real numbers.