IN ANYONE’S book, infinity is pretty paradoxical. If you add one apple to an infinite pile of apples, you still have an infinite pile of apples – and it is exactly the same size as the original pile. If your bank has an infinite number of pounds in the vault, then you can pay in one pound, take out a million pounds, and the bank won’t have lost any money. And that’s just the beginning. There’s even a way you can take out an infinite number of pounds, and the bank still won’t have lost any money.
If you are confused already, don’t worry: that’s just an indication that your brain is working as it should. When we start thinking about infinity we are on dangerous ground. But it’s not just philosophically threatening – it’s also a problem to mathematics. Mathematicians would gladly banish the infinite from their minds, were it not for one thing: infinity is far too useful to do without. Even if it doesn’t really exist, mathematics reeks of infinity; in many ways, it is what makes mathematics tick.
What do we mean by “infinity”? On an informal, intuitive level, the main feature of infinity is that it’s big. Very big. No, a lot bigger than that. Bigger than you imagine. Bigger than you can imagine. When children learn to count, they often go through a stage where they become fascinated by really big numbers – a million, a billion, a trillion. Most of them reach a stage when they wonder what the biggest possible number is. Shortly afterwards, they realise that there can’t be a biggest number, because if there were, then you could add 1 to get a number that is bigger still. The counting numbers go on forever, they never stop. They are, in some sense, infinite. But, again, what does that really mean?
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What infinity does not mean is that when you keep counting forever you reach a number called infinity. Every counting number, however big, is finite. So whatever “infinity” means in this context, it’s not a number as such. It is a metaphor, and it captures the idea that the process of generating new numbers never ends.
The first serious work on the infinite in mathematics goes back to the ancient Greeks, and the work of the mathematician Euclid on prime numbers. In Euclid’s Elements (the first geometry text), he proves: “Prime numbers are more than any assigned multitude of prime numbers”. In the vernacular? There are infinitely many prime numbers.
Philosophers call this kind of concept “potential infinity” and generally consider it a relatively harmless kind of infinity because you never actually get to it. (There are other kinds that seem positively dangerous, but I’ll get to those later.) Potential infinity saved the day at a crucial turning point in the history of mathematics. When Gottfried Leibniz and Isaac Newton invented calculus they had to grapple with a close relative of the infinite, the infinitesimal.
If you loosely think of infinity as something bigger than any finite number, then the infinitesimal is something that is not zero but is smaller than any non-zero number. Mathematicians and philosophers got very confused about this concept at first, because they failed to recognise a basic point. Just as infinity can’t be a number like all the others, so an infinitesimal can’t be a number like all the others. The only number smaller than any non-zero number is zero – yet the main reason for wanting infinitesimals to exist was to avoid using zero.
Eventually mathematicians realised that “infinitesimal” is not a number but a process (èƵ, 26 January 2002, p 27). Just as the process “keep counting” generates an adequate substitute for “infinity”, so the process “keep shrinking” generates an adequate substitute for the infinitesimal.
All this lets infinity in by the back door, while keeping it respectable. It even gets its own symbol,. Infinity lets us do forbidden things, such as divide by zero. When a mathematician writes 1/0 = ∞, she doesn’t mean that 1 divided by 0 makes ∞. She means that if a number x keeps shrinking ever closer to 0, then 1/x keeps growing bigger and bigger, without limit. And she has to be a very senior mathematician to be allowed to be that sloppy, even then.
A vast swathe of today’s mathematics, physics and other science relies on Newton and Leibniz’s calculus, which underlines the importance of firming up our concepts of the infinite – be it infinitely big or small. Having sorted out potential infinity as a shorthand for a process, it became possible to make sense of a lot of fascinating but irritatingly vague hand-waving by great mathematicians of earlier years. In particular, you could add up infinitely many numbers and get a perfectly sensible result. What, for example, is the sum 1/2 + 1/4 + 1/8 +…and so on forever, where each denominator is twice the previous one? If you stop at any point along the way, it’s a bit messy. For instance, if you stop after “…+ 1/1024” then the sum is 1023/1024. But if you go on for ever the result is 1. Exactly 1.
Defining infinity by a process enables mathematicians to resolve the paradoxes raised by the Greek philosopher and mathematician Zeno. One of the best known is about a race between a hare and a tortoise. The tortoise starts half a mile ahead of the hare, and the hare runs twice as fast as the tortoise. By the time the hare reaches the half-mile mark, the tortoise has moved on a quarter of a mile. By the time the hare reaches that point, 3/4 of a mile out, the tortoise has moved on again by 1/8 of a mile. By the time the hare reaches that point, the 7/8 mile mark, the tortoise has moved on again, and so on. Zeno’s conclusion was that the hare has to do infinitely many things before catching the tortoise, which makes no sense.
Putting aside the deep issue here (can you do infinitely many things in a finite time?), there is a hole in the logic. What Zeno proves is that the hare has not caught the tortoise after travelling 1/2 mile, 3/4 mile, 7/8 mile, and so on. That is entirely correct, but somewhat beside the point. When you are chasing something, there are lots of places where you don’t catch it. The big question is: where do you catch it? The place where the hare catches the tortoise is exactly 1 mile along the road. When the hare has done 1 mile, the tortoise has done half a mile, so taking the head start into account, they are in the same place.
Another way to see this is to observe that in order to catch the tortoise, the hare must travel 1/2 + 1/4 + 1/8 +…miles. The mathematical process of forming the sum goes on forever (and has the answer 1). That doesn’t mean the hare takes forever to do this, because the time taken shrinks exactly the same way as the distances do.
This resolution is just one of the proofs that infinity is something that mathematicians can’t live without. But it’s more than a necessity. Infinity can also provide a fundamental insight into the relationship between mathematics and the real world.
Take this rope trick, for example. A stage magician’s assistant passes him a thick, soft rope, and he ties a knot in it. Then, further along the rope, he ties another completely separate knot. He grasps the rope by its two ends, gives it a flick and, lo and behold, the knots have cancelled each other out and vanished! Impressive stuff – because “anti-knots” don’t exist.
Suppose instead that he had wound the rope round a pole a few times, then wound it back the same number of times. You wouldn’t be surprised that he could then hold the two ends and pull the rope away from the pole. The anticlockwise turns just cancel out the clockwise ones. So what’s the difference with knots? In case you’re wondering, it’s not a pointless question: knots are an important part of modern physics, underlying many properties of the universe. But you could never establish by experiment that it’s not possible to tie an anti-knot. Mathematically, however, it is provable. And the simplest proof that anti-knots don’t exist depends on the infinite.
How does a slippery concept like infinity tell us anything sensible about something as mundane as a knot? Well, suppose 0 denotes an unknotted rope, and let K be a knot. For instance, K could be the usual overhand knot. Now suppose that there is an anti-knot L such that a rope with knots K and L in it can be deformed to an unknotted rope without manipulating the ends. Symbolically, K + L = 0. By making L very loose and K very tight, and sliding K along the rope through L to the other side, L + K = 0 as well. (This observation is crucial, but only a mathematician would notice.) Now comes the cunning use of infinity. Imagine tying an infinite knot K + L + K + L + K + L +…going on forever (don’t try this at home). Think of Zeno, and make each knot half as big as the previous one so that the entire infinite sequence fits into a finite length of (very thin) rope. If we think of this “sum” as (K + L) + (K + L) + (K + L) +…then it is clearly the same as 0 + 0 + 0 +…, which is lots of unknotted ropes glued end to end – that is, an unknotted rope 0. On the other hand, we can also think of this sum as K + (L + K) + (L + K) +…going on forever. This is K + 0 + 0 +…, which is K + 0, which is K. Since both calculations must give the same answer, we see that 0 = K. That is, K was unknotted all along.
So we can conclude that, if K is a genuine knot that can’t be untied except by sliding it off the end of the rope, then there does not exist an anti-knot L. Note that if K and L are numbers, this kind of calculation is fallacious because the mathematical infinite sum has no sensible meaning as the result of some sensible mathematical process, let alone in the real world. That is because it does not “converge”, or settle down towards a well-defined value.
But an infinite sum of knots does: it represents the process of tying more and more tinier and tinier knots, and this has a well-defined outcome. I should admit that this outcome is an infinitely intricate knot so complicated that you can’t make one from ordinary string: only a mathematician’s infinitely thin line will do. But the point remains: with real knots, the infinite sum has a meaning. So in this instance, infinity works better for knots than it does for numbers.
What do we conclude from all this? Not that mathematics simply mimics the “real world”. There are no infinite knots in the real world, but thinking about what they would do if they existed tells us something important about real knots: anti-knots don’t exist. The strength of mathematics is that while its raw materials parallel the real world in some respects, its thought-patterns can go outside of it. The consequences can tell us useful things about the real world, even when some of the steps involved have no obvious real counterpart. It’s the ultimate in thinking sideways.
Most of the uses of “infinity” in classical mathematics, then, are really potential infinities, and can be restated as processes that provide useful models of the real world that we can’t get using finite quantities. Whatever you might make of the potential infinities we have discussed so far – the ones that involve generating unending sequences – they aren’t the ones that should blow your mind. Save that for the ones that blew Georg Cantor’s mind: actual infinities.
In 1874 Cantor, a Russian-born German mathematician, discovered that some infinities are bigger than others. Within a few years he had a nervous breakdown – and most of his colleagues weren’t surprised, since they thought his work was becoming increasingly unhinged. But to give Cantor his due, the reason for the breakdown was his isolation and frustration at his colleagues’ failure to understand what he was telling them. It was only later generations of mathematicians that grasped his profound insight.
Cantor was developing what is now called set theory. A “set” is just a collection of mathematical objects. Finite sets can be counted; for example, the set whose members are the numbers 2, 3, 5 and 7 contains four members. Cantor started wondering what would happen if you tried to count an infinite set, like the set of all whole numbers. You get some kind of measure of infinity, he decided, and called it an “infinite cardinal”. Not being sure what that meant, he used the symbol N0 to denote this particular size of infinity, where the symbol is the Hebrew letter “aleph”, and the zero just meant it was the first infinite cardinal. Not only does the set of all whole numbers have aleph-0 members, so does any set whose members can be paired up one-to-one with the whole numbers. For instance, the set of even numbers can be matched like this:
1 2 3 4 5…
2 4 6 8 10…
Even though the even numbers do not exhaust the supply of whole numbers, both sets contain the same number of members.
Cantor then proved that various apparently larger sets – the positive and negative integers together, even the set of all possible fractions – also contain exactly aleph-0 members. Great: that surely meant that aleph-0 was just a fancy name for “infinity”. He could dump the zero and just call it N, or even simply ∞. But then he discovered something fascinating and unprecedented: some sets have more than aleph-0 members.
The set of all “real” numbers – which includes not just integers and fractions, but all the numbers in between them – is a case in point. Since it was the only other example he had found, Cantor’s first instinct was to call this new infinity “aleph-1”. But, he admitted, he couldn’t be sure that it was the next one up from aleph-0. Could there be an infinity in between? This problem was not solved until the 1960s. When I say “solved”, what I mean is that American mathematician Paul Cohen proved that the answer is “yes and no”: it depends on what properties you want mathematics to have.
That’s because maths is a human construct rather than a god-given absolute, and so there is flexibility in the logical foundations we can lay when setting up our mathematical processes – especially when it comes to infinity. And so either answer to Cantor’s question could be logically consistent.
One of Cantor’s greatest triumphs mirrored the discovery that every child makes about ordinary whole numbers – that there isn’t a biggest one – only Cantor went much further: there isn’t a biggest infinity. The list of “infinite cardinals”, starting with aleph-0 and increasing at every step, has no end.
Exotic though they may sound, Cantor’s ideas are fundamental and useful, underpinning many areas of mathematics (probability theory, for instance), physics (quantum mechanics and quantum field theory are just two examples), and even biology: population dynamics relies, through statistics, on understanding differing degrees of infinity. These higher infinities may be complex to deal with, but they produce some fascinating surprises (see “To infinity and beyond”).
And this leads us to a suitably paradoxical conclusion. Even infinity – whichever one you choose – is not “the biggest thing that can exist”. There’s always something bigger still. But it’s OK, you do learn to live with it eventually. Especially when you realise you can’t live without it.
To infinity and beyond
It may surprise you to learn that infinity is actually quite small. Oh yes, its staggering vastness may be impossible to imagine, but, mathematically speaking, the ordinary infinity – the one behind the idea of endless space and time, for example – is a trifle. It is so disappointingly small that mathematicians are driven to search for bigger infinities, such as what you get when you raise infinity to the power of infinity.
It was the mathematician Georg Cantor who first showed there was more than one kind of infinity. In the 1870s he proved that there are as many even numbers as integers. It sounds wrong: you’d think there should be twice as many integers. But he pointed out that you can pair up each integer with each even number – 1 with 2, 2 with 4, 3 with 6 and so on forever. The two infinities are paired off exactly, so they must be the same size. He also showed that infinity plus 1, infinity times 2, even infinity squared, are all the same size. So how do you get a bigger infinity?
Well, just take the set of all numbers between 0 and 1. Since you can specify each number with an infinite decimal (0.17902673838…, say), then it is clear that there are infinitely many of them. But more than that, Cantor showed that this infinity of “real numbers”, as they are known, is truly bigger than the infinity of the integers. In fact the reals are infinitely more numerous.
Cantor went on to show that there is an endless succession of infinities. He named the infinity of the integers aleph-0, with aleph-1, aleph-2 and so on for each successively larger infinity (they were just names, however; he had no idea what aleph-1, aleph-2 and so on actually represented).
Doing sums with them is surprisingly simple. In most cases, the answer is simply the biggest infinity that appears in the calculation. So aleph-0 + aleph-1 = aleph-1; aleph-0 × aleph-1 = aleph-1; aleph-346 × aleph-120 = aleph-346. Each bigger infinity is so much vaster than its predecessors that it is unperturbed by having them added, or even being multiplied by them. “Schoolchildren would love such arithmetic,” says Saharon Shelah of the Hebrew University of Jerusalem.
Shelah has been making it his business to take this extreme mathematics a little further out. In the 1980s, he began to play with more offbeat infinities, such as the infinity you get if you multiply together every infinity up to the infinitieth infinity. You might want to read that again. It’s aleph-0 × aleph-1 × aleph-2 ×…forever.
How big is this? Pretty big, you might think. But raising to powers of infinity is a tricky business. The infinity of real numbers, for instance, is equivalent to 2 raised to the power aleph-0 (2aleph-0). But is this the same as aleph-1, Cantor’s “next biggest infinity” after the number of integers? Or are there some other infinities between aleph-0 and 2aleph-0?
We don’t know. Worse, we know that we can never know – mathematicians proved last century that this question can never be resolved. We’ll never know whether the infinity of real numbers is aleph-1, aleph-57, or something much bigger.
This uncertainty makes raising to infinite powers a vague operation. For example, (aleph-7)aleph-3 may be aleph-7, or it may be just 2aleph-0, whichever is the bigger. So, disappointingly, raising a big infinity to another big infinity might simply give you just the number of numbers between 0 and 1; even infinity to the power of infinity might be relatively small.
Indeed, Shelah’s work suggests we are likely to be disappointed in our hunt for truly gargantuan numbers. When he used an abstruse branch of set theory to multiply together every infinity up to the infinitieth infinity, as described above, what he got was an infinity that is smaller than aleph-aleph-4. You might want to read that again too – aleph-aleph-4 is the first infinity below which there are aleph-4 smaller infinities. So Shelah has shown that, even after multiplying infinitely many infinities together, the result is smaller than another infinity. It seems a forlorn hope but, perhaps one day mathematicians will find the infinity that is truly worthy of its name…Stephen Battersby
- Everything and More: A compact history of infinity by David Foster Wallace, to be published next monthby Weidenfeld & Nicolson
- The Art of the Infinite: Our lost language of numbers by Robert Kaplan and Ellen Kaplan, published by Allen Lane Infinity by Brian Clegg, published by Robinson