Infinity seems to be the simplest idea in mathematics. It is endless, boundless, beyond all counting. Surely there is only one infinity, since nothing can be larger than something that never ends. In the late nineteenth century, the German mathematician Georg Cantor proved this assumption wrong. Some infinities, he showed, are larger than others.
Cantor began with a simple question: what does it mean for two collections to be the same size? With finite sets, we can count. With infinite sets, counting fails, so Cantor used pairing instead. Two sets are the same size if every member of one can be matched with exactly one member of the other, with nothing left over.
This method leads to surprising results. The even numbers seem to be half as numerous as all the whole numbers. Yet we can pair them perfectly: 1 with 2, 2 with 4, 3 with 6, and so on forever. Neither set runs out first, so they are the same size. Cantor called any infinity that can be paired with the counting numbers countable.
David Hilbert later illustrated this with an imaginary hotel. The hotel has infinitely many rooms, all occupied. A new guest arrives. The manager simply asks every guest to move from room n to room n + 1, freeing room 1. Even infinitely many new guests can be accommodated: each existing guest moves to the room with double their number, leaving every odd-numbered room empty.
Cantor’s greatest discovery came when he examined the real numbers, all the decimals between 0 and 1. He imagined someone claiming to have a complete list of them. Using his famous diagonal argument, he built a new decimal by changing the first digit of the first number, the second digit of the second number, and so on. The new number differs from every number on the list in at least one place, so it cannot be on the list. Any such list must be incomplete. The real numbers are uncountable, a strictly larger infinity.
Cantor went further, showing that there is an endless ladder of ever-larger infinities. His ideas were fiercely attacked by some contemporaries, and he suffered greatly during his lifetime. Today, his work forms the foundation of modern mathematics.
Infinity, it turns out, is not one thing but many, and the endless comes in different sizes.