of this proof and a proof that this intersection contains a single real number in [a, b] are given below. The development of Cantor's ideas The development leading to Cantor's 1874 article appears in the correspondence between Cantor and 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 individual of the other?" Cantor added that collections having such a correspondence include the collection of positive rational numbers, and collections of the form (an1, n2, . . . , nν) where n1, n2, . . . , nν, 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 responded 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 a proof by contradiction that the set of real numbers is uncountable. Cantor starts by assuming that the real numbers in can be written as a sequence. Then, he applies a construction to this sequence to produce a number in that is not in the sequence, thus contradicting his assumption. Together, the letters of December 2 and 7 provide a non-constructive proof of the existence of transcendental numbers. Also, the proof in Cantor's December 7th letter shows some of the reasoning that led to his discovery that the real numbers form an uncountable set. Dedekind received Cantor's proof on December 8. On that same day, Dedekind simplified the proof and mailed his proof to Cantor. Cantor used Dedekind's proof in his article. The letter containing Cantor's December 7th proof was not published until 1937. On December 9, Cantor announced the theorem that allowed him to construct transcendental numbers as well as prove the uncountability of the set of real numbers: This is the second theorem in Cantor's article. It comes from realizing that his construction can be applied to any sequence, not just to sequences that supposedly enumerate the