“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…