Georg Cantor

By Lin Yue

Georg Cantor tamed infinity — proved there are different sizes of it — and the effort broke his mind and remade mathematics.

Born in 1845 in St. Petersburg to a German-Danish family, Cantor moved to Germany as a boy and became professor at Halle. In the 1870s he began the work that would define him: a rigorous theory of infinite sets. The first shock came in 1873 — the real numbers, the continuum, are uncountably infinite: there are strictly more of them than there are integers, though both are infinite. Infinity comes in sizes. He developed transfinite arithmetic — aleph-null, aleph-one, the hierarchy of the infinite — and proved the continuum could be mapped, paradoxically, onto higher-dimensional spaces, so that a line contains as many points as a cube.

The mathematical establishment revolted. Leopold Kronecker, the powerful Berlin professor, attacked Cantor's work as a "grave disease" infecting mathematics, blocked his publications, and sabotaged his career — Cantor never got the Berlin chair he deserved and remained at minor Halle. Poincaré called set theory a disease from which mathematics would recover. The opposition was partly mathematical, partly theological: Cantor's infinities trespassed on God's prerogative, and Cantor — who corresponded with theologians and believed his theory revealed the divine — felt every attack personally.

He suffered repeated breakdowns from 1884 onward, spending his later years in and out of sanatoriums, turning to Shakespeare authorship theories, the Baconian ciphers, and theology — a great mind seeking refuge anywhere but the battlefield that had wounded it. He died in a Halle clinic in 1918, impoverished by the war. But David Hilbert's defense held — "No one shall expel us from the paradise Cantor created" — and set theory became the foundation of all modern mathematics. Cantor's infinities, once heresy, are now the first thing every mathematics student learns.

Impact on civilization

Cantor's set theory became the foundation of modern mathematics — the language in which all of analysis, topology, and logic are now written. By proving infinities come in different sizes, he opened the study of the infinite as a rigorous science and forced mathematics to confront its own foundations, triggering the logicist and formalist programs of Russell and Hilbert. Every proof taught today stands on ground Cantor cleared while his contemporaries called him mad — and every computer program runs on his sets.