Dedekind's contributions to Cantor's article Since 1856, Dedekind had developed theories involving infinitely many infinite sets—for example: ideals, which he used in algebraic number theory, and Dedekind cuts, which he used to construct the real numbers. This work enabled him to understand and contribute to Cantor's work. Dedekind's first contribution concerns the theorem that the set of real algebraic numbers is countable. Cantor is usually given credit for this theorem, but the mathematical historian José Ferreirós calls it "Dedekind's theorem." Their correspondence reveals what each mathematician contributed to the theorem. In his letter introducing the concept of countability, Cantor stated without proof that the set of positive rational numbers is countable, as are sets of the form (an1, n2, ..., nν) where n1, n2, ..., nν, and ν are positive integers. Cantor's second result uses an indexed family of numbers: a set of the form (an1, n2, ..., nν) is the range of a function from the ν indices to the set of real numbers. His second result implies his first: let ν = 2 and an1, n2 = . The function can be quite general—for example, an1, n2, n3, n4, n5 = + Dedekind replied with a proof of the theorem that the set of all algebraic numbers is countable. In his reply to Dedekind, Cantor did not claim to have proved Dedekind's result. He did indicate how he proved his theorem about indexed families of numbers: "Your proof that (n) [the set of positive integers] can be correlated one-to-one with the field of all algebraic numbers is approximately the same as the way I prove my contention in the last letter. I take n12 + n22 + ··· + nν2 = and order the elements accordingly." However, Cantor's ordering is weaker than Dedekind's and cannot be extended to -tuples of integers that include zeros. Dedekind's second contribution is his proof of Cantor's second theorem. Dedekind sent this proof in reply to Cantor's letter that contained the uncountability theorem, which Cantor proved using infinitely many sequences. Cantor next wrote that he had found a simpler proof that did not use infinitely many sequences. So Cantor had a choice of proofs and chose to publish Dedekind's. Cantor thanked Dedekind privately for his help: "… your comments (which I value highly) and your manner of putting some of the points were of great assistance to me." However, he did not mention