Some Infinities Are Bigger: Cantor and the Sizes of the Endless

The claim sounds like a category error — that one endlessness can exceed another. The proof needs no advanced mathematics, only the nerve to take counting seriously past the end of the numbers.

Ask most people whether one infinity can be larger than another and they will say the question makes no sense. Infinity is infinity. It was a reasonable position until the 1870s, when Georg Cantor proved it was wrong. There are different sizes of infinity, an unending ladder of them, and the proof is simple enough to follow without any advanced mathematics. It is one of the cleanest surprises in the subject.

The first thing Cantor needed was a way to compare the sizes of sets that never end. For finite collections this is easy: two groups are the same size if their members can be paired off with none left over, three cups matched to three saucers. Cantor took the same idea and applied it without flinching to infinite sets. Two sets have the same size, he said, if you can match their members one to one. No counting required, just pairing.

This leads immediately to results that feel wrong. Consider the whole numbers 1, 2, 3, and so on, and the even numbers 2, 4, 6. There are obviously fewer evens, since you have thrown half the numbers away. But you can pair each whole number with its double: 1 with 2, 2 with 4, 3 with 6, forever, with nothing left out on either side. By Cantor’s rule the two sets are the same size, even though one sits inside the other. The same trick works for the integers including negatives, and even, with a cleverer pairing, for all the fractions. The fractions seem far more numerous than the whole numbers, packed infinitely densely on the number line, yet they too can be arranged in a single list and matched one to one with 1, 2, 3. All of these sets share one size of infinity, the one Cantor labeled with the Hebrew letter aleph: aleph-null, the size of the counting numbers. A set that can be listed this way is called countable.

At this point you might expect every infinite set to be countable, that infinity comes in only one flavor after all. Cantor’s great result is that it does not. The real numbers, meaning all the points on a continuous line, including the endless nonrepeating decimals like pi, cannot be listed. There are strictly more of them than there are whole numbers, and here is the argument, which he published in 1891.

Suppose someone claims to have a complete list of all the real numbers between 0 and 1, written as decimals, one per row, going on forever down the page. Cantor shows the list must be incomplete by building a number it cannot contain. Look at the first digit of the first number, the second digit of the second number, the third digit of the third, and so on down the diagonal. Now write a new decimal whose first digit differs from that first digit, whose second digit differs from that second one, whose every digit differs from the corresponding diagonal digit. This new number cannot be the first on the list, because it disagrees in the first place. It cannot be the second, because it disagrees in the second place. It cannot be anywhere on the list, because for every row it differs in at least one digit. So no list, however cleverly built, can capture all the reals. They are uncountable. Their infinity is genuinely larger than the infinity of the whole numbers.

That single diagonal stroke cracks the subject open. Cantor went on to show, with a related argument, that the trick never stops. Take any set at all and form the set of all its subsets, and the new set is always strictly larger than the one you started with. Apply this to an infinite set and you get a still larger infinity, and you can apply it again, and again. There is no biggest infinity. They tower upward without end, each one provably larger than the last.

Cantor’s contemporaries did not all welcome this. Leopold Kronecker, an influential older mathematician, found the whole edifice of completed infinities offensive and blocked Cantor’s career where he could. Cantor, who suffered from depression that the hostility surely did not help, spent years defending ideas that much of the establishment treated as a kind of mathematical disease. The judgment did not last. David Hilbert, the leading mathematician of the next generation, called Cantor’s work a paradise from which no one would drive us out, and set theory became the foundation on which much of modern mathematics is built.

One question Cantor could not answer still hangs over the field. He believed there was no size of infinity strictly between the whole numbers and the reals, a guess called the continuum hypothesis. It turned out to be undecidable from the standard axioms of mathematics, a fact established in two stages by Kurt Gödel and Paul Cohen in the twentieth century. You can assume it is true or assume it is false and get consistent mathematics either way. So the ladder of infinities that Cantor revealed has, near its bottom rung, a gap whose size no one can pin down, and perhaps no one ever will.

Cantor handed mathematics a paradise and an unanswerable question in the same stroke: the ladder of infinities is real, and the size of its lowest gap may stay forever beyond reach.