Before Cantor, mathematicians treated infinity as a single thing — vast, undifferentiated, beyond comparison. The question "which infinity is larger?" would have seemed meaningless, like asking which endless hallway is longer. Cantor showed, in 1874, that this was wrong. Infinities have sizes. Some are strictly, provably larger than others. The proof that established this is one of the most elegant in mathematics, and it works by constructing, from any supposed complete list, something that cannot be on the list.
Start with the counting numbers: 1, 2, 3, 4, and so on. Call this set countably infinite — it's the baseline infinity, the smallest one. Now ask: are there more real numbers than counting numbers? The real numbers include everything on the number line: all the fractions, all the irrationals like √2 and π, all the decimals that go on forever without pattern. Are there more of these than there are counting numbers?
One way to make the question precise: can you pair up every real number with exactly one counting number, using each counting number once? If yes, the infinities are the same size. If some real numbers are always left over, the reals are a larger infinity. Cantor proved that the reals are always left over, no matter how clever your pairing scheme.
Here is the proof, in its essential form. Suppose you claim to have a complete list of all real numbers between 0 and 1. Now construct a new number by going down the diagonal — taking the first digit of the first number, the second digit of the second number, and so on. Then change every digit. The resulting number differs from every entry on your list in at least one position — the diagonal position. It cannot be anywhere on the list. But we assumed the list was complete. Contradiction. The list can never be complete. No pairing can exhaust the real numbers. They are a larger infinity than the counting numbers.
The argument has a slightly vertiginous quality when you follow it closely. You're not finding a specific real number that's missing. You're constructing a missing number from the list itself, using the list as raw material for its own refutation. Whatever list you propose, the diagonal construction hands you back something you forgot. The incompleteness is guaranteed by the structure of the argument, not by any particular gap you found.
Cantor called the size of the counting numbers aleph-null (ℵ₀), and the size of the real numbers the continuum. Then he asked the natural next question: is there an infinity between aleph-null and the continuum? He could not prove there was, and he could not prove there wasn't. This question — the continuum hypothesis — troubled him for decades.
The resolution came long after Cantor's death, and it is one of the stranger results in mathematical logic. Kurt Gödel showed in 1940 that the continuum hypothesis cannot be disproved from the standard axioms of set theory. Paul Cohen showed in 1963 that it cannot be proved from those axioms either. The question is independent of the axioms — mathematics cannot decide it either way. Both a universe where an intermediate infinity exists and one where it doesn't are mathematically consistent. The question has no answer, not because we haven't been clever enough, but because the axioms genuinely leave it open.
What strikes me most about the diagonal argument is its generality. The same idea proves that no set can be put in correspondence with the set of all its subsets — that the power set is always strictly larger. Apply this repeatedly and you get an infinite tower of strictly increasing infinities, each larger than the last. Cantor's diagonal is not just a proof about the reals. It is a machine for generating infinities without end.
There's something almost paradoxical about this. You start with a concept — infinity — that seems to resist comparison. Cantor defines a precise notion of size, applies it carefully, and finds that not only can infinities be compared, they proliferate. There is no largest infinity. For every infinity you name, I can show you a larger one. Cantor called this tower the transfinite, and he believed he was describing the structure of actual mathematical reality. Some of his contemporaries thought he had lost his mind. He hadn't. He had found something real.