There are infinitely many positive integers: 1, 2, 3, 4, and so on. There are also infinitely many even integers: 2, 4, 6, 8, and so on. Are there twice as many integers as evens? Or the same number?

Georg Cantor, working in 1870s Germany, said the same. You can pair every integer with an even number — 1 pairs with 2, 2 pairs with 4, 3 with 6 — and you'll never run out of either. When two infinite sets can be paired one-to-one, Cantor said, they have the same size. He called this size cardinality. The integers and the even numbers have the same cardinality, written ℵ₀ — aleph-naught.

So far so weird. But then Cantor proved something remarkable: not every infinite set has cardinality ℵ₀.

Consider the real numbers between 0 and 1 — every possible decimal. Cantor showed that no matter how cleverly you try to pair them with the integers, you can always construct a decimal that you missed. He proved this with a now-famous diagonal argument. The conclusion: the real numbers are a strictly larger infinity than the integers. He called this size 𝑐.

There are not just two sizes of infinity. Cantor showed there are infinitely many — a hierarchy of larger and larger infinities, each one strictly bigger than the last.

Other mathematicians thought he was crazy. Leopold Kronecker called Cantor a 'scientific charlatan' and worked to block his career. Cantor suffered repeated breakdowns and spent time in psychiatric hospitals. He died in poverty in 1918.

His work is now considered one of the foundations of modern mathematics.