Most numbers can never be named, and we can prove it

We ran out of words for numbers a long time ago, and we can prove it.

Start with counting: 1, 2, 3, on forever. Now take all the numbers between 0 and 1, every endless decimal. Both sets are infinite, so it is tempting to call them the same size. In the 1870s Georg Cantor asked whether they really are, and proved they are not.

His argument is something you can almost do by hand. Suppose you hand me a complete list of every number between 0 and 1: line 1, line 2, line 3, all the way down. I build a new number like this. I make its first digit different from line 1’s first digit, its second digit different from line 2’s second digit, and so on down the diagonal. The number I end up with differs from every line on your list in at least one place, so it cannot be on the list. No list can hold them all. There are strictly more real numbers than counting numbers. One infinity is bigger than another.

Here is the part that unsettles me. Every name, formula, or computer program is a finite string of symbols, and you can line up all possible finite strings in a single list. So there are only as many descriptions as there are counting numbers, the smaller infinity. But the reals are the bigger one. There are not enough descriptions to go around. Almost every number that exists has no name, no formula, no rule, no finite description in any language at all. Not undiscovered. Undescribable. The numbers we ever actually talk about, like pi or the square root of two, are a vanishingly thin dust on a line that is almost entirely unspeakable.

And it does not stop at two sizes. Cantor showed that the set of all subsets of any set is always strictly bigger than the set itself. So above the reals is a bigger infinity, and a bigger one above that, with no top. An endless staircase of infinities.

Cantor saw this further than anyone alive, and it cost him. His old teacher called him a charlatan and a “corrupter of youth” and blocked his career. He spent his last years in and out of sanatoriums and died in one. Decades later David Hilbert defended what he had built:

“No one shall expel us from the paradise that Cantor has created for us.”

A paradise its own discoverer was hounded out of, built on one simple and devastating fact: almost every number that exists can never be named, and they are all sitting right there, between 0 and 1.