What Is Infinity? Why Some Infinities Are Bigger Than Others

The Greeks distrusted it. Aristotle allowed a potential infinity, a process that can always be continued, such as counting, and denied an actual infinity, a completed infinite thing, and that distinction held for two thousand years. Zeno's paradoxes showed why it was feared: Achilles can never catch the tortoise because he must first reach where it was, and by then it has moved, and so on for ever, an argument that seems to prove motion impossible. Galileo noticed in 1638 that the square numbers could be paired off one to one with all the whole numbers, 1 with 1, 2 with 4, 3 with 9, so that in one sense there were as many squares as numbers, and concluded that the words greater, less and equal simply did not apply to infinite quantities. He was nearly right, and the exact way in which he was wrong is the subject.

Georg Cantor, working in Halle in the 1870s, took Galileo's pairing seriously as a definition. Two collections are the same size if their members can be matched one to one with none left over on either side, whatever the collections are. By that test the evens are as many as the whole numbers, since 1 pairs with 2, 2 with 4, and so on without end; so are the fractions, which can be listed in a zigzag through a grid and numbered off; and so, it seems, is anything that can be put in a list. Cantor called that size aleph-null, after the first Hebrew letter, and a collection of that size is called countable. Then he asked whether the real numbers, all the points on a line, could be listed too, and showed they could not.

The proof is a page long and is called the diagonal argument. Suppose someone claims to have a complete list of all the decimals between 0 and 1. Build a new decimal by taking the first digit of the first number and changing it, the second digit of the second number and changing it, and so on down the diagonal. The new number differs from every number on the list in at least one place, so it is not on the list, so the list was not complete. No list can be. The points on a line are uncountable: a larger infinity than the whole numbers.

Cantor went on to show that the same trick applies at every level. For any collection, the collection of all its subsets is strictly larger, and so there is no biggest infinity but an endless ascending tower of them, which he called the transfinite numbers. Some of what follows from his definitions:

• •The whole numbers, the evens, the primes and the fractions are all the same size, aleph-null
• •The points on a line, in a square and in all of space are all the same size, the continuum, and it is bigger than aleph-null
• •Every infinite set has a subset the same size as itself, which is in fact the definition of infinite
• •There is no largest infinity: the subsets of any set outnumber its members
• •Whether there is a size between the whole numbers and the continuum, the continuum hypothesis, cannot be settled from the standard axioms of mathematics, as was proved in 1963

Cantor's work was attacked in his lifetime by the most influential mathematician in Germany, Leopold Kronecker, who held that only the whole numbers were real and blocked Cantor's appointment to Berlin, and Cantor spent his later years in and out of sanatoria, dying in 1918. The theory also broke things. Bertrand Russell found in 1901 that the collection of all collections that are not members of themselves cannot exist without contradiction, which meant that the naive idea of a collection had to be replaced by careful axioms, and the effort to found mathematics safely on those axioms led to Godel's proof in 1931 that no such foundation can prove everything true about even the whole numbers. Hilbert, defending the theory, said that no one would expel mathematicians from the paradise Cantor had created, and no one has.

Physics uses infinity cautiously. Calculus works with quantities that shrink towards zero without arriving, which is Aristotle's potential infinity made rigorous, and an infinity that appears in a physical equation, as at the centre of a black hole or in the early calculations of quantum theory, is generally taken as a sign that the theory has broken down rather than that something infinite exists. Whether the universe is infinite in extent is not known; it is at least 46 billion light years in radius, and the measurements are consistent with its going on for ever. Cantor's infinities are a different kind of thing, not sizes of objects but sizes of collections, and their existence is a matter of definition and proof rather than of observation, which is why they can be known with certainty about a world in which no one has ever seen one.

The mathematician David Hilbert popularised a thought experiment that makes countable infinity feel concrete. Imagine a hotel with infinitely many rooms, numbered 1, 2, 3 and onwards, every one of them occupied. A new guest arrives. The manager asks everyone to move from room n to room n plus 1, so the guest in room 1 goes to room 2, and so on for ever. Nobody is left without a room, and room 1 is now free. The hotel was full and yet it had space, which is exactly the property Cantor's definition of an infinite set relies on.

It gets stranger. If a coach arrives carrying infinitely many new guests, one for each whole number, the manager moves every current guest from room n to room 2n, which puts them all in the even rooms and frees every odd one. Even infinitely many coaches, each with infinitely many guests, can be fitted in with a cleverer scheme, since pairs of whole numbers can be listed like the fractions. What the hotel cannot do is take a coach with one passenger for every point on a line. The diagonal argument shows that no room plan could ever list them all.

A classic puzzle shows how infinite processes trip up intuition. Does 0.999 recurring, with nines going on for ever, equal 1, or fall just short of it? It equals 1 exactly. One quick argument starts from a third, which is 0.333 recurring; multiply both by three and you get 1 on one side and 0.999 recurring on the other. Another calls the number x, notes that ten times it is 9.999 recurring, and subtracts the first from the second: nine x equals 9, so x is 1. Both arguments lean on the ordinary rules of arithmetic, and a fully rigorous version agrees with them.

The objection most people feel is that the nines never finish, so there must always be a tiny gap left. That treats the decimal as a process still being written out, the potential infinity Aristotle accepted. But the notation names a single number, the limit of the sequence 0.9, 0.99, 0.999 and so on, and that limit is 1. If the two differed, the difference would have to be a positive number smaller than every positive number, and no such real number exists. The lesson is that some numbers have two decimal spellings, just as a half can be written as 0.5 or as 0.4999 recurring.

The familiar infinity sign, a figure of eight lying on its side, was introduced by the English mathematician John Wallis in 1655. Wallis did not explain why he chose it. One suggestion is that he adapted a form of the Roman numeral for one thousand, which was sometimes used to mean a very large number, and another is that it echoes the last letter of the Greek alphabet.

The curve itself is known as a lemniscate, from a Latin word for a ribbon, and mathematicians later studied lemniscates as curves in their own right. The symbol is useful precisely because it is not a number: in calculus it marks a value that grows without limit, and in set theory Cantor and his successors needed whole new symbols, such as the Hebrew letter aleph, to tell different sizes of infinity apart.

Infinity, since Cantor, is the size of a collection whose members can be paired off with a part of itself, and such collections come in different sizes: the whole numbers, fractions and evens are all countable, the points on a line are a strictly larger uncountable infinity by the diagonal argument, and above every infinity is a bigger one. The theory cost its inventor his career and his peace, forced mathematics to rebuild its foundations, and remains the paradise Hilbert refused to leave.