Cantor's Set Theory and the Mathematics of Infinity

Before Cantor, infinity was a single mystical word. After him, it was a tower — an unending ladder of infinities, each one strictly larger than the last.

For most of its history, mathematics handled infinity the way a careful diner handles a hot dish: at arm's length, never directly. Aristotle had drawn the line. There were two kinds of infinity, he said. The potential infinity — you can always add one more number, always extend the line a little further — was respectable. The actual infinity — a completed totality with infinitely many things in it all at once — was the property of mystics. For more than two thousand years mathematicians worked with limits, with “as much as you like,” with arrows pointing into the distance, but never with the distance itself.

Then, in the 1870s, a German mathematician named Georg Cantor crossed the line. He treated infinite collections as ordinary objects you could compare, count, and stack on top of each other. The result was a tower of actual infinities of strictly different sizes, stretching upward without end. He was attacked viciously for it. Henri Poincaré called it “a grave disease.” Cantor's former teacher Leopold Kronecker called him a corrupter of youth and blocked his career. Cantor died in a sanatorium. But he had been right, and within a generation set theory had become the language in which essentially all of modern mathematics is written.

Two baskets of fruit

Cantor began with the simplest possible question: when are two collections of things the same size? Forget counting. If you have a basket of apples and a basket of oranges and you want to know which has more, you don't need numbers. You pair them off. One apple, one orange. One apple, one orange. If you run out of apples first, there were fewer apples. If they match up perfectly, with nothing left over on either side, the baskets are the same size. Mathematicians call this a one-to-one correspondence, or a bijection.

For finite collections this is obvious. Cantor's genius was to insist on using exactly the same test for infinite ones. And the moment you do, strange things happen. Consider the natural numbers 1, 2, 3, 4, … and the even numbers 2, 4, 6, 8, … Naively there are “twice as many” naturals, because the evens are half of them. But pair them off: 1 with 2, 2 with 4, 3 with 6, 4 with 8, and so on. Every natural number is paired with a unique even number; every even number is paired with a unique natural number. Nothing is left over. The two infinities are the same size.

The whole is no larger than the part. In finite mathematics that would be a contradiction. For Cantor it was the defining feature of infinite sets — the thing that finally distinguished them from finite ones.

He called the size of the natural numbers ℵ₀ (“aleph-null”). A set is countable if it can be paired off with the natural numbers. The evens are countable. The integers are countable. More surprisingly, the rational numbers — all fractions p/q, of which there seem to be wildly more than there are integers — are also countable. The trick is to lay them out in a grid and walk through it on a diagonal zig-zag.

The diagonal argument

If everything were countable, ℵ₀ would be the only infinity and Cantor's theory would have stopped there. It does not. In 1891 he proved that the real numbers — the points on the number line, including √2 and π and 0.10110001110… — cannot be paired off with the natural numbers. There are strictly more reals than naturals.

The proof, the famous diagonal argument, is one of the shortest profound proofs in mathematics. Suppose, for contradiction, that you could list all real numbers between 0 and 1 in a single column. Now construct a new number digit by digit by walking down the diagonal: take the nth digit of the nth number on the list, and change it. The number you have just built differs from the first number in its first digit, from the second in its second, from the third in its third — from every number on the supposed list. It cannot be on the list. So no such list exists.

The reals are uncountable. This was the door swinging open. There is more than one infinity. There is a hierarchy.

The ladder, and the gap in it

Cantor then showed something even stronger. Given any set S — finite, countable, or already uncountable — the set of all its subsets, the power set P(S), is always strictly larger than S itself. Apply it once: the natural numbers are countable, but their power set is uncountable. Apply it again: the power set of that is larger still. And again. And again.

There is no largest infinity. There is an unending tower, each level dwarfing the one below.

This is dizzying enough that it created its own famous gap. The real numbers have the same size as P(ℕ), which Cantor called the cardinality of the continuum, c. He knew c > ℵ₀. The natural question: is there an infinity between them? Some collection bigger than the naturals but smaller than the reals? Cantor believed not. The claim that there isn't became the continuum hypothesis, and it sat at the top of Hilbert's famous list of open problems in 1900.

It turned out to have the strangest possible answer. In 1940 Kurt Gödel showed you cannot disprove the continuum hypothesis using the standard axioms of set theory. In 1963 Paul Cohen showed you cannot prove it either. It is independent — neither true nor false on the basis of the axioms. You can consistently do mathematics either way. Deep down, the fabric of mathematics has a fork in it.

“The essence of mathematics lies precisely in its freedom.” — Georg Cantor, 1883.

Why this matters

Cantor's work did three things. The first was practical: set theory turned out to be the right language for every other branch of mathematics. A function is a particular kind of set. A number system is a set with operations. Topology, algebra, analysis, probability — all of it, in the twentieth century, was rebuilt on the bedrock of sets. When the Bourbaki group sat down in the 1930s to rewrite mathematics from scratch, set theory was page one.

The second was philosophical. Cantor showed that “size,” “infinity,” and even “exists” could be mathematical properties — not mystical ones. Whether a real number with a particular property exists is a question with a definite answer, decided by axioms and proof, not feeling. He took infinity away from the theologians.

The third — and this is the part that still feels uncanny — is that there are more mathematical objects than mathematics can describe. The real numbers form an uncountable set, but there are only countably many possible descriptions in any human language: each description is a finite string from a finite alphabet, and the countable union of finite things is countable. It follows that almost every real number is forever undescribable. No formula. No name. No algorithm. The vast majority of mathematics is dark matter. We can prove it is there. We cannot point at it.

That is Cantor's gift, and the price of it. A tower of infinities, beautifully ordered. A language strong enough to carry all of modern mathematics. And, looking down from the top, the quiet discovery that almost everything that exists in mathematics is permanently out of reach.