Cantor's Diagonal Argument (1891)
“Diagonalization seems to show that there is an inexhaustibility phenomenon for definability similar to that for provability” — Franzén (2004)
Georg Cantor (1845-1918)’s correspondence with mathematician Richard Dedekind (1831-1916) in the years 1873-74 has been narrated in a previous newsletter. Their letters detail the process by which Cantor arrived at the discovery that the set of the real numbers is uncountably infinite. That is, Cantor showed that although both the natural numbers and the real numbers are infinite in number and so go on forever, there “aren’t enough” natural numbers to create a one-to-one correspondence between them and the real numbers. Cantor’s brilliant discovery, in other words, showed rigorously and undeniably that infinity comes in different sizes, some of which are larger than others. One of his methods of proving this assertion, Cantor’s diagonal ar…