Cantor's Bombshell: Infinity Has Sizes

In 1891, Georg Cantor proved a result so radical it drew fierce hostility from leading mathematicians of his day: some infinities are bigger than others. The whole numbers (1, 2, 3, ...) are infinite. The real numbers between 0 and 1 are also infinite. But Cantor showed you cannot pair them one-to-one — the reals are a "larger" infinity.

His proof is strikingly simple. Suppose you list every real number between 0 and 1. Cantor builds a new number not on the list: change the first digit of the first number, the second digit of the second, and so on down the diagonal. The result differs from every listed number in at least one digit, so the "complete" list was never complete.