Georg Cantor (1845–1918) did something no one had dared: he counted the infinite — and discovered, to the fury of many of his contemporaries, that some infinities are bigger than others. Single-handedly he founded set theory, the language in which essentially all of modern mathematics is now written.

Taming infinity

Cantor's masterstroke was the diagonal argument, a single page of reasoning proving the real numbers cannot be listed — they are uncountable, a strictly larger infinity than the counting numbers. He went further: Cantor's theorem shows there is an endless tower of ever-bigger infinities, and the size of the number line, the continuum, sits partway up it. On finding that a line and a plane have the same number of points, he wrote to a friend: "I see it, but I don't believe it."

Ridiculed, then vindicated

Cantor's infinities were too much for the mathematical establishment. His former teacher Leopold Kronecker called him a "corrupter of youth" and a "charlatan", and blocked his work at every turn. The stress, and repeated bouts of depression, dogged the rest of his life. History took his side utterly: David Hilbert declared, "No one shall expel us from the paradise that Cantor has created for us."