# Infinity and beyond

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* We look at the different conceptions of 'infinity' that exist in mathematics, science and philosophy. * We ask whether it is possible for physical infinities to arise in the universe and whether a machine could ever complete an infinite number of tasks in a finite time. * And if so, what would it mean? * The answers to these questions will take us from Zeno's paradox to the beginning of the universe and the centre of black holes.

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INFINITY AND BEYOND

Professor John D Barrow

This is the last talk in this academic year’s series. Some of you came to a talk I gave a few weeks ago, which was largely to promote my new book about infinity, the infinite book. Today I will overlap that a little, but I will tell you some other and different things about infinity as well. However, thinking about that overlap, I was reminded that there might be a moral in the old story about the economics undergraduate at Oxford who, feeling rather worried about upcoming final examinations, went to his tutor and asked if he had any advice and guidance about what might come up in the final examinations, and the tutor to his amazement said, “Oh, I can tell you completely what the questions will be. They’ll be exactly the same as they were last year.” The student was astonished and said, “You mean the questions are always the same?” and the tutor said, “Of course, but the answers are different”!

Well, infinity, or lazy eight as it seems to be called in America, as a piece of iconography has a relatively short history. You’ll recognise the famous eight on its side symbol. The first occasion that it was used in a mathematical sense was by John Wallis, the Oxford mathematician - I think he was the first Savilian Professor of Geometry in Oxford - became renowned for his work on infinite series and geometry, and also must have been fairly astute. He was one of the first serious code breakers and code makers in the UK, and made a living making and breaking the codes for both sides in the English Civil War, which is a nice job if you can get it!

Next to it is a mathematical figure that looks the same. This is called the lemniscate - it’s a graph, in effect, in polar co-ordinates that was first written down by Jakob Bernoulli, one of the famous Bernoulli family of mathematicians. But if we go back further in time, we find this type of never-ending symbol appearing with religious significance – the famous cross of St Boniface, that you still sometimes see used as a religious symbol. Even older and with no really detectable single origin, is the famous ouroboros symbol, which is a snake eating its own tail, and in Ancient Egypt this would have a little kink in its configuration to make an infinity-like shape.

If you’re a scientist or a mathematician or a philosopher, you will perhaps make progress thinking about infinity best by going back to the time of Aristotle, making a start by distinguishing different types of infinity that Aristotle distinguished in order to think more clearly about them. Aristotle realised that there were two types of infinity of which we could conceive. The first we might call potential. This might be something like numbering the stars in a universe that goes on for ever; it’s something where you never get to the infinite number in the count, it’s a so-called potential infinity. On the other hand, there might be actual infinities, the idea that you could do something on this table which would produce, for example, an infinite temperature, an infinite brightness, an infinite density. These sorts of infinities are much more dangerous; they can jump up and bite you.

Aristotle, like many of the people that followed after him, was rather content with the idea of potential infinities, but denied that there could ever be any actual infinities. This thinking about infinity was very much tied up with Aristotle’s thinking about the vacuum and the void, which I mentioned in my last lecture because, just as Aristotle denied there could ever be a perfect vacuum created within the world, he denied that there could be an actual infinity. The two ideas were closely linked, because he argued that if you had a region that was a perfect vacuum, then motion would have no resistance when it moved through it, and it would move to infinite velocity.

The other area where Aristotle was particularly persuasive about infinity and the finiteness of things was the idea of an infinite universe. This was outlawed, Aristotelian thinking, the universe of matter, which we have contact with, we see around us, the universe of stars, this needed to be finite. There were reasons for this which we appreciate rather simply. If you have a finite universe, then it can have a centre, and it was important in these early days of thinking about the world that there be a centre so that we could be located at the centre. If you decide to have an infinite universe, as Giordano Bruno argued rather heretically 400 years ago, then there is no need for there to be a centre any more. In one of Escher’s famous engravings, so-called cubic space division in English, you could imagine what’s going on is that you have an infinite network of crossing girders, which are going on for ever, in every direction, and if you walk around in that and then look around you, wherever you’re standing, you will see exactly the same situation, so there is no centre in this infinite network. Every point is equally good as a centre. This is the observation that got Giordano Bruno burnt at the stake in 1600, arguing that in an infinite universe, there was no need to have a centre; Aristotle’s arguments were incorrect, not only did we not have to be at the centre, but there wasn’t even a centre.

In the Far East, in Eastern philosophy of different sorts, the view was much more enlightened. The ideas about vacuum and zero likewise were much more readily accepted, and in many Eastern philosophies, religious traditions, the idea of infinity is one that people are quite content to manipulate and think about. There’s not a logical barrier or religious barrier to conceiving of things which are infinitely large.

We tend automatically to think of infinity as being big, but you have to remember that it can also be very small. If you look again up to a few hundred years ago, to Blaise Pascal, he focuses attention very much on what he called the double infinity, this phrase that was used at the time, the idea that there is a double manifestation of infinite potentially in the world around us. On the one hand, the potentially infinite universe of stars and galaxies, as we would now call them, but on the other hand, what Pascal regarded as much more subversive potentially, the infinitely small, because if the world was infinitely divisible, then within your hand in front of you, there was a manifestation of something which was physically infinite.

Infinity, rather like buses, come in threes; there are three types of infinity which we might attempt to distinguish, and I’ll have a fair bit to say about the first two in this talk. It’s important to appreciate that they are really quite distinct in practice as well as in concept. The first is the infinity of the mathematicians. The second is a physical infinity, this question about whether you could do something in the world which would produce an infinite answer, result in an infinity being measured – very alarming and controversial. And the third is the more religious idea of a transcendental infinity, a sort of “make me one with everything” type of idea.

To show you that these are really different ideas, I’ve made a little table of some notable thinkers, some of whom you will have heard, others maybe you haven’t, and whether they believed in these three different types of infinity. I’ve picked them because every possible permutation of the three, the two cube possible permutations can be found. So if you’re Thomas Aquinas, you don’t believe that there can be mathematical infinities or physical infinities, because you believe in a theological or a transcendental infinity and you believe really that only God can be infinite. If you’re Abraham Robinson, a mathematician of the 20th century, a so-called finitist, you don’t believe in any of these things. It seems rather odd that you could have a mathematician who doesn’t believe in infinity. If you go back to, Luitzen Brouwer, a Dutch mathematician of the early 20th century, he led a somewhat vitriolic campaign to outlaw infinity from mathematics. He believed that mathematics should be re-convened as being the set of theorems which could be proved only by a finite number of step by step arguments from statements about the ordinary numbers, so it’s anything that a computer can do in our world. This truncates mathematics very dramatically, and there was huge controversy about this, most mathematicians not wanting to take that step. And someone like Hilbert was one of the people who opposed this, he believes in mathematical infinity, he doesn’t believe there can be physical infinities, and he doesn’t believe in transcendental infinities. George Cantor, one of the founders of our picture of mathematical infinity, believed in all three.

So just to carry on talking about the mathematical type of infinity, the first thing to appreciate about mathematical infinities is that infinity is not a big number. You shouldn’t think of it as being rather like an extremely large number, only just a little bit bigger. It doesn’t behave like any number, no matter how big it is. This is something that’s been appreciated for a long time – I’ll give you several examples. Here’s one from about 1350 by Albert of Saxony. He was a famous scholar and religious thinker of the time, usually known in the historical literature as Little Albert or Albertutius, to contrast him with Albert the Great, a famous theological reformer. What he pointed out was that you could build an entire infinite universe of space out of an infinitely long log of wood. How does this work? It looks very odd – so he’s got a little log, let’s suppose his log has a square cross section, it’s six inches square or something, but it goes on forever, and then what he wants to do is to chop it up into pieces – a long piece of spaghetti being chopped up into pieces of equal size, little cubic blocks, and then he reassembles those blocks, so when he’s got 27 blocks, he makes a cube, and then he cuts off some more so that he’s got four by four by four, he’s got 64 blocks, and then he’s got a four by four by four cube, and then he cuts more off, he makes a five by five by five, and so on forever. You can see what the effect is. He’s got an infinite number of blocks he can cut up, and he can make a cube that gets larger and larger and larger and can fill any volume that you care to specify. This is very odd; it seems that this volume is bigger than this one, and you can make it step by step.

In the early 17 th Century, Galileo drew attention to what is really a related paradox of a similar sort. Galileo’s paradox boils down to the fact that suppose we make a list of all the ordinary numbers – 1, 2, 3, 4, 5, and you let that list go on without end, and then we list some other numbers, let’s take the even numbers. If you stop the average person in the street and ask them are there more even numbers than ordinary numbers, they think about it, they’ve done GCSE mathematics maybe, they’ll probably guess that there were half as many even numbers as ordinary numbers; it would be a reasonable guess. But Galileo points out that well if you took a piece of string and you attached one to two, and two to four, and three to six, and four to eight and so on for ever, then every one of these even numbers is attached to precisely one number in the first list, so there must be an equal number in both lists, even though this list is just a sub-set of that list, so this is very strange. If this list was finite, this would not be possible, but an infinite collection of things seems to contain itself as a sub-set. Galileo regarded the lesson of this example as being don’t worry about infinity; it’s a paradoxical idea, if you try to incorporate it into your thinking about mathematics or anything else, things are going to go badly wrong. This was the reason why finitist mathematicians, even in the 19 th century, were keen to outlaw arguments using infinite sets, infinite collections, from mathematics. They felt that it had the potential to introduce a fallacy into mathematics, and the whole logical structure would then collapse.

There’s a wonderful graphic example that builds on this paradox, which was created by David Hilbert, who we mentioned earlier, who was a great enthusiast for infinities, and it’s the idea of having an infinite hotel. A finite hotel behaves well, in the sense that if it’s full and you arrive, you can’t get a room and that’s that, but in an infinite hotel, you can always get a room, even if it’s full. Suppose all the rooms are taken, you arrive, all that happens is the person in room 1 is moved to 2, 2 to 3, 3 to 4, and so on for ever, leaving room 1 vacant for you. If you come along with an infinite number of friends, you can all be accommodated. The concierge just moves the person in 1 to 2, the person in 2 to 4, the person in 3 to 6, 4 to 8 and so on, and all the infinite number of odd numbered rooms are now empty. Room service can get a little slow near the higher end of the spectrum, but you can see this is a manifestation of this curiosity that in infinite collections a set contains itself as a sub-set.

Well, this was rather perplexing to people. There’s a little story about Hilbert that I told when I talked the other week; I can’t resist telling it again. He was a strange man, as some mathematicians are, although not the Gresham Professors of Geometry, of course! Hilbert apparently once gave a problem to a research student, who was somewhat unstable, and the student struggled and struggled to solve this problem and in the end killed himself because he was unable to solve the problem which Hilbert had given him. His family were very distraught, but they decided they would ask Hilbert to make a special oration at the graveside in honour of their son as he was the most famous mathematician in the world, and Hilbert in his rather matter of fact, otherworldly way began by saying that it was a terrible tragedy because it wasn’t even a difficult problem!

In the 19 th century, a great German mathematician, George Cantor, really cleaned up this whole area of thinking about mathematics in a very remarkable way. Rather than being perplexed by these paradoxes that we’ve looked at about counting things, he realised that the definition of an infinite set was that it was one that could be put in this one-to-one correspondence with a sub-set of itself. That’s not possible for any finite collection of things, but just like our 1 to 2, 2 to 4, 3 to 6, the one-to-one correspondence with a sub-set of itself, that’s the definition of an infinite set, but he went rather further. He realised that there are different types of infinity, and this simple type of infinity, where we can put things in a one-to-one correspondence with the ordinary numbers, we can count them. He called this a countable infinity, so the even numbers are a countable infinity, the odd numbers are a countable infinity, and if you add them together, you get another countable infinity. This is not such a simple idea as it sounds because there are things which are not obviously countable infinities, and the trick is to know how to count them. Of course by defining a countable infinity, his next step, sort of tour de force, is to prove to you that there are other infinities which cannot be counted, and these are called uncountable infinities. Although you might not have guessed it, all fractions, like one number divided by another, 1 over 2, 3 over 4, 7 over 4, what mathematicians called the rational numbers, these are countable, so they’re no bigger than the ordinary numbers. Even though you might have thought there’s an awful lot more of them, it’s the same little trick as with the even numbers and the ordinary numbers.

The next step in his argument is rather dramatic: a new type of proof in mathematics that had not been seen before, and that’s the type of thing that mathematicians are really interested in, a new type of argument. He wants to prove that if you write down all the never ending decimals, things like pie and so forth, there are an infinite number of them, but is it an uncountable or a countable infinity? What he does is to assume that it’s a countable infinity, and show that this leads to a contradiction, and therefore the original assumption was wrong and the infinity is uncountable. So he said let’s imagine that we can list all the never ending decimals, and these are the first five in the list. It goes on forever, but there’s a systematic counting, as is required for a countable infinity. Then what he notices is suppose you take the first number after the decimal point, the second, the third, the fourth, the fifth, in each entry, and just change it by one in every case, so that’s the recipe, so you then create a number where the first point is 3, 8, 2, 1, 4 and so on forever, and by construction, this number cannot appear in this infinite list, because it’s been designed to be different from every number in the list in at least one decimal point. So this is the contradiction: you cannot systematically count all the never ending decimals. They are a higher order of infinity. They are infinitely larger than the collection of all fraction or all whole numbers.

He then went on to show that it doesn’t stop there, and in fact it doesn’t stop anywhere. There’s an even bigger infinity which you can’t put in correspondence with the unending decimals, and that infinity itself is topped by another one, and it goes on in fact for ever. There is no largest infinity. How this works you can appreciate from a simple finite example. Suppose you have a collection of three things, Tom, Dick and Harry, and you ask how many sub-sets, how many sub-collections are there? Well there’s one that doesn’t contain any of them, by convention, the empty set, and then there are three sets that just has one of them, Tom, Dick and Harry, and then there are all the three combinations that just has pairs of them, and then there’s the set itself with all of them, and so there are eight, so two cubed, and it turns out that whenever you have a set, finite or infinite, this collection of all the possible sub-sets can never be put in a one to one correspondence with the set itself. It’s always infinitely bigger when you’re dealing with infinite sets. This is how he constructs his never ending tower of infinity, it’s a sort of staircase to heaven, if you like. Here are the countable infinities, like the ordinary numbers, here are the never ending decimals, and up the staircase it goes and so on to absolute infinity, which is something that you just formally define but it’s not constructed as a specific thing.

The whole idea of taking infinity and putting it in mathematics, in a quantity that you can manipulate, and prove theorems about was very controversial in Germany when these ideas were first presented. Cantor himself left mathematics for quite long periods as a result of depression and oppression through influential people who just didn’t like this type of thinking; they regarded it as a subversive part of mathematics. During some of those periods when Cantor left mathematics, he worked on the history of mathematics, but his ideas were taken up by prominent members of the Catholic Church in Germany rather enthusiastically, because Catholic theologians realised that Cantor provided a way to resolve all these ancient disputes about whether having infinities in mathematics was theologically challenging for God; that if you decided that there could be an infinity in the world, that this was somehow a challenge to theological doctrine about the uniqueness of God in their tradition. You can see that by delineating these different orders of infinity, as they were called, showing how they’re distinguished, and also revealing that there was no ultimate infinity of sort, this is a rather new and, to theologians at the time, really exciting collection of ideas, and so, ironically, he found himself taken up much more enthusiastically by theologians than by mathematicians. Mathematical enthusiasm would come rather later, and today, Cantor’s ideas are really the bedrock of mathematical understanding of infinite sets and infinite collections.

But what about physical infinities? This is much closer to home in a way, easier to understand. Could we have an infinity appearing in a laboratory in a physical experiment, or is it the case that whenever you do a calculation that says there’s going to be an infinite answer, all it’s telling you is that you’ve got to try a bit harder, that you haven’t got all the physics, all the description of the world, in the problem to hand? You’ve made an approximation, and if you make a better approximation, the infinity will go away. If you’re an aeronautical engineer, and you study air flows by air wings, you’ll occasionally make a prediction that there’ll be an infinitely rapid change in the air flow. All that’s a signal of is that you haven’t included in your description the friction of air as completely as you could have, and if you do, you’ll get a shockwave – a very sudden change, but not an infinitely quick change. So there would be a school of opinion in physical science, a dominant opinion certainly in subjects like engineering, that infinities really are consequences of incomplete modelling, incomplete description, and if you work harder, then they’ll go away.

The great area where infinity has been paramount in the discussion of what’s going on is the search for a fundamental theory of elementary particles, and until 1980, for nearly 40 years, the working theories of particle physics were bedevilled by the so-called problem of infinities, that you could do calculations to predict what you should see in experiments, but you got infinite answers from those predictions. But the infinities were rather controlled, in that you could imagine that your answer had two pieces in the calculation, one bit was infinite and the other bit was finite. The bit that was finite agreed with observation to fantastic precision, 16 decimal places, so there’s something deep and correct about what you’re doing. But people always felt that somehow they didn’t have the right framework to do these calculations, that they would just throw away the infinite part, the so-called process of re-normalisation, appreciating that this really was not the best way of proceeding, that there must be another way of doing these calculations in which the infinities don’t appear. So what happened in particle physics was that the quest to avoid infinities became a guiding principle in your search for the best or even the ultimate theory of elementary particles. In the early 1980s, string theory, which you’ll have read about in the newspapers and on the underground or wherever you pick up your fundamental physics information, why everybody became so excited about string theory, and it was on the television and there were popular books and articles everywhere, and it still continues to a great extent, was for the first time string theory was a theory of elementary particles in which no infinites appeared, so it was a finite theory, it solved this problem of the infinities. It’s very interesting, you can see, just as in engineering, now in fundamental physics, the desire to avoid infinities is a very powerful principle in keeping you going to find a better theory.

Another area where infinity is rather obviously looming on the horizon is cosmology, astronomy, our picture of the universe as a whole. Think back to the conception of the expanding universe that we have today, the separation between distant clusters of galaxies, different points in the universe, against time, a picture of the expanding universe. The universe can either keep on expanding forever or ultimately head back to a big crunch, a small claustrophobic scenario, or we have our sort of British compromise universe, as I call it, just in between. The best buy universe today is slightly different. It’s one of these ones that expands forever, but it has a sort of kink in it and has started to accelerate again in the recent past. Now if you look at these, you can see that they immediately imply for you all sorts of potential and actual infinities. What are the potential infinities? Well, if you’re in one of these universes which is going to expand for ever, there’s apparently a potential infinity of time to the future, and in fact in these sorts of universe, the volume of space is infinite as well. This is another potential infinity. You could keep on going in your spaceship and you would never reach the edge of such a universe, because it has no edge. But there also are suggestive actual infinities, so in the initial state and the final state for this universe, there would be an infinity of density, of temperature, and of just about everything else. There is much argument and debate about how seriously you should take that prediction. If you’re a particle physicist or an engineer, your immediate reaction is to regard it like the other infinities, that it’s just a signal that you’ve got to try harder, that you don’t have a theory that holds in that very, very high density environment, that your theory of gravity in the expanding universe fails to include everything it needs to include in these high density environments, and it may be that the universe bounces into another contracting phase, or even is static, doesn’t expand at all, before that moment and then expands. Then there are other people who would agree with that view, that description is incomplete, we don’t know everything we need to know to understand these apparent infinities, but there are interesting reasons why we would still expect an infinity to occur, even when we include all the physics. That would be the view of someone like Roger Penrose, whereas someone like Stephen Hawking, who works in the same corridor as I, would argue that this is not a real infinity, that this expansion, even if it starts, starts from a finite density and state. So again, you can have a beginning which has finite conditions or you can have a beginning which has infinite conditions.

This story becomes more interesting if we focus on this type of universe which starts expanding, reaches a maximum and then contracts to a big crunch in the future. The way I’ve represented it before and the way I’ve represented it here, the universe is rather pear-shaped as it were, to the past, pear-shaped to the future, as we might have guessed, but you notice everybody begins, at the same time and everybody hits a crunch at the same time in the future, and you could argue that such infinities would be allowed and would even be in the spirit of Aristotle’s thinking, because no one could see those infinities. They would happen to everybody at the moment when the universe ended, and it would end for everyone everywhere.

However, in reality, the universe is not exactly the same in every place. In some places there are galaxies, in some there are not, so if you were to measure the expansion of the universe from one place to the other with fantastic precision, you would find it’s very slightly different. The differences are small. They’re about one part in a hundred thousand today, but as time goes on, they will magnify and become much larger, and in some parts of the universe, there are very dense regions which are doomed to collapse, to form black holes and a state of infinite density seemingly, relatively soon in the future, and so the real picture of the universe is where different places hit an infinity at different times. This is much more alarming because it means that if you were moving towards your infinity, you could look back and you could receive signals. That’s the sort of thing that Aristotle certainly would want to exclude, and even Roger Penrose wanted to exclude it, and so he proposed an idea long ago, which people in cosmology and relativity have worked on for a long time, gradually chipping away at it, and what the proposal is is something called cosmic censorship, so it’s really the prejudice, the idea that if infinities appear in density or temperature, then they can never influence the outside world, they’re always trapped inside black holes. So what does that mean? Well, if material is coming towards other material, increasing density, eventually the gravitational field becomes so strong that nothing can escape from this region, even light, and we say a black hole is formed, and the material becomes encompassed by a so-called event horizon, or black hole horizon. So if you’re inside the horizon, things may look quite normal. We could be inside the horizon at the moment of a billion solar mass black hole, and things would seem just like they are in this room, but if you tried to leave and return to base, you would fail. The laws of physics would not allow you to generate enough energy; you would have to go faster than the speed of light to leave. If you’re on the outside and there’s an infinite density at the centre, nothing that happens there can influence you here, so the conservation laws of physics can’t be affected by what goes on in the central region of the black hole. So the cosmic censorship hypothesis is that all infinities, all singularities as they’re called, are clothed by event horizons. There are no naked infinities in the universe which you can see and which can interact with you by sending light signals or exerting gravitational forces.

This idea is not fully proven as a consequence of Einstein’s theory of general relativity, but attempts to prove it are always very interesting, and very revealing about the way nature works. You see, if you have a rotating black hole, it turns out that there is a critical speed of rotation for the black hole which, if you exceeded it, the horizon would disappear and you’ll be able to see the infinite density at the centre of the black hole. It would become naked. So people thought, well, we’ll set up a little thought experiment, we’ll have a mathematical black hole, just rotating a little bit more slowly than this danger value, and then we’ll have a spinning particle coming in to the black hole which gets captured by the black hole, and because angular momentum and spin is conserved, that extra spin will be added to the black hole and just tip it over the limit, and then we’ll be able to see the naked singularity. Well you can set up this little mathematical thought experiment, but when you do it, you learn that in general relativity, a new force of nature arises, a spin-spin repulsive force between spinning objects, and it is just such that when the particle that’s going to make the singularity naked is about to enter the black hole, the force becomes large enough to repel the spinning object and stop it coming in. So this is the sort of thing that you invariably found, that all these simple ideas that you had to make a singularity, an infinity, naked were always defeated by some unusual interplay between the forces of nature, and so that’s why people suspect that there is a grand and simple argument which says that in the world of general relativity in the universe, you can’t have naked infinities.

Those sorts of infinities are not the only ones that people worry about in cosmology. The one that’s made people worry the most over the last few hundred years is the idea of the so-called infinite replication paradox. This was pursued by Friedrich Nietzsche in the 19 th century, and the idea was that if you have a situation which is infinite, and in fact it has to be more than infinite, it has to be infinitely random, so you have to be able to have an infinity which explores all the possibilities, because suppose you had an infinite number of even numbers, it may be an infinite collection, but you won’t find a single odd number within it. So what Nietzsche pointed out is that if you have an infinite universe, then any event that has a finite probability of occurring here and now will occur infinitely often at this very moment elsewhere in the universe, so this is all very alarming. There are an infinite number of copies of this very gathering occurring elsewhere in the universe at this very moment. Some people regard this as so unsatisfactory that they think it’s a powerful argument for the universe being finite, but in a modern picture of the astronomical universe, it’s not such an alarming idea. The fact that the speed of light is finite means that even if there is an infinite universe around us, we could only see a finite distance. So light has only had a time to travel a distance equal to the speed of light times the age of expansion of the universe. That distance is large, it’s about 10 to the 26 metres, it’s about 15 billion light years, something like that, so even if the universe is infinite in size, for all practical purposes, doing astronomy, sending out signals, receiving signals, we can only have contact with a finite part of it.

Suppose we try and do some arithmetic on Nietzsche’s idea. I can make various calculations and ask how far you would have to go before with essentially 100% probability you would find duplication of ourselves or of the Earth or even of a region the size of the whole visible universe. The size of the Earth is about 10 to the 7 metres, so in old currency I think the radius is about 6,400 miles, and the distance to the Sun, astronomical unit, about 10 to the 11 metres, the nearest star, 10 to the 16, the edge of the galaxy, 10 to the 19, and our distance to the edge of the visible universe, 10 to the 26, or 10 to the 27 metres. How far would you have to go before you found with 100% probability a copy of you and me? We’re asking for just a state which has the same number of atoms in it in the same configuration compared with all the possible configurations that it could be, and it’s a staggeringly huge number – 10 to the 10 to the 28 metres, so this is a number you have no chance of even writing down as a decimal expansion. How far would you have to go for the first copy of the whole Earth? In some sense, not that much further – 10 to the 10 to the 50 metres, and if you want the first copy of the entire region inside our visible horizon today, you would have to go 10 to the 10 to the 119 metres. These are some of the largest numbers that you ever see, but nonetheless the principle itself is rather worrying. If you consider things from an ethical point of view, the fact that there are copies of you doing exactly what you’ve decided to do at the same moment means that there’s always a copy doing the opposite, and so whatever good decision you might make, there is always a doppelganger making a bad decision. But if you look on the bright side, if you always make a bad decision, there is another person who is always making the right one!

The last topic I want to talk about in respect to physical infinities is infinity machines. Can you build a machine that would do an infinite number of things in a finite time? And even if you found you couldn’t but you discovered that you could build a machine that could do nearly an infinite number of things in a finite time, that might be jolly useful. This idea of carrying out an infinite number of tasks in a finite time has a name; such tasks are called super-tasks. So if you’re involved in super-tasking, you are trying to do an infinite number of things before breakfast. Why this is particularly interesting is that suppose, given that you know there are all these unusual twin paradoxes of relativity in which you can send a twin off on a space flight and come back and find that the twin has aged quite differently from the identical twin that stayed at home, it’s not inconceivable that there might be some way in which you could send your laptop computer on a trip through space time so that when it returned, it had carried out an infinite number of computations, even though a similar computer back on Earth had just carried out a finite number.

There’s an honourable tradition of thinking about things like this, going back certainly to Zeno, who was probably the most original philosopher of ancient times, whose paradoxes were not really refuted in any sense until modern times. There is a famous series of mathematics, known for a very long time, called the geometric series. You start with a term that’s a half, and then you halve the previous term to give you the next one, so you go from a half to a quarter to an eighth to a sixteenth to a thirty-secondth, and so on, and this series has a sum. The sum is equal to the first term divided by one minus the ratio between each successive term, which is a half, and so the sum is one. If you go away and on your computer you keep totalling this series to ever-larger terms, you will find that the sum gets closer and closer and closer to one. The tempting question that mathematicians like Herman Va asked long ago was whether you could have a physical operation in which each step corresponded to one of the terms in the series. So it might be, for example, that you are stepping out on an expedition, and in the first step, you went half a metre, and the next you went a quarter, and so on, and then these numbers show you the distance, the total distance you’ve travelled since your first step, and after an infinite number of steps, you will have gone one metre. Now of course the real question is whether these end steps are physically distinct. Are they physically realisable steps, or is it just a notional sub-division of a line?

Another example of this sort that became very topical in the 1950s due to an Oxford philosopher called James Thompson, and became known as Thompson’s lamp. He imagines the same series really in action. Suppose you have a desk lamp and you have a switch programmed on it such that for the first half minute, it is on, for the next quarter minute, it’s off, for the next eighth minute, it’s on, off, on, and so on forever, or not quite forever, but an infinite number of switchings, and the question is, after one minute, is the lamp on or off? Because remember this series converges, these halvings, means that after an infinite number of switchings, just one minute will have gone, so will the light be on or off after an infinite number of switching? Well, this lamp I fear is more like Aladdin’s lamp, in a way. My way of looking at this is like this. Suppose we think about another series, and this is a series that just alternates, one minus one plus one minus one, so on for ever. Mathematicians call it the alternating series. You score one when you put it on, the light, minus one when it goes off, plus one when it comes on again, and so on forever, and you can see that if you sum the series after one turn, it’s one when it’s on, it’s zero when it’s off, and so on, so if the sum of the series is zero, to a certain number of terms, then the light is off, and if the sum is one, then the light is on. It seems reasonable. But if you now look at the series, you’ll see that your infinite switchings creates a very odd, ambiguous situation. Suppose I take the series and I bracket off the terms in pairs, each bracket is one minus one, which is zero, add all the zeros up and the sum must be zero. But if I wanted to, I could have left out the first term and then bracketed the next pairs, and each pair would be minus one plus one, zero again, so they are all zero and the sum is one. To make matters worse, I could look at it another way: I could say well it’s one minus one minus one plus one minus one, and so on, but the thing in the big brackets now is just the same as the series I started with, so S is one minus S, 2S is one, so the sum is a half – not even equal to one of the terms in the series. What this shows you I think is really that the question, whether the light is on or off at the end of the minute, is not a meaningful question. In some ways, this is sort of the best answer. It’s sort of half on and half off, but you notice how an infinite series could be conceived to have a sum that’s not actually equal to any of the sub-sums that it could possibly have after a finite number of terms. The lesson from this is that infinite series that diverge or alternate; they don’t have uniquely defined sums that you have to define the process by which you’re producing the sum as well.

In Newtonian physics, it turns out that super-tasks are possible. So back in the 1970s, a mathematician looking at something else discovered a very unusual property of Newtonian mechanics. Suppose you have two stars orbiting around a common centre, rather like the Earth and the Moon do, and then another pair orbiting. One’s overall rotation is in one sense, and one in the opposite sense, so the total rotation is zero, and then in between them – they all have the same mass – you have a lighter particle, which you throw backwards and forwards at first and under gravity it oscillates backwards and forwards between the two pairs. When it runs into one pair, it gets kicked out rather rapidly, kicked back to interact with the other pair, they evict it rather quickly, and this happens an infinite number of times. As it does, these pairs of particles move away from one another, and dramatically, the whole system goes to infinite size in finite time, undergoing an infinite number of oscillations as it does so. In Newtonian mechanics, super-tasks are possible. Of course, this has eventually to recede faster than the speed of light. This type of situation cannot occur in relativistic physics.

But in relativity, it turns out that super-tasks are also possible, in principle, but unfortunately, whenever they are possible, they seem to require the universe to have rather unusual supporting properties. You can have a space time where you can send your computer off on a path and it will do an infinite number of steps, but that possibility means that any radiation that latches on to the same path, as it was incoming, would have its frequency driven to infinity as it followed the computer around, so these types of universe are unstable. They self-destruct, or any structures in them would self-destruct, because the paths that allow an infinite number of things to occur, allow an infinite number of oscillations or amplifications of radiation signals, so there’s probably extremely good reason why we do not live in a universe where these relativistic super-tasks are possible. But curiously, they’re not ruled out by relativity or by conservation of energy or other laws of physics.

Lastly, I should say one or two things about living for ever. I wrote a whole chapter in my book about the strange paradoxes of living forever. Philosophers like to debate about whether it is rather good or rather bad to live for ever, whether death is something to be overcome or whether it’s something that really limits human opportunity. Biologists tell us that death exists in the animal world for very good reasons. It’s a way of removing bad genes from the pool. It’s a way of creating diversity. We can also see that sociologically if we all lived for ever, then life would really be rather peculiar. There would be no limit to the amount of advice that we would have to take from friends and family and mothers-in-law and fathers-in-law, who would be present in huge abundance in our household. The world would be full of unfinished projects because there would be no end to the amount of advice that would need to be taken before they could be completed. If you wanted to run a legal system, there would be all sorts of strange paradoxes. What would life imprisonment mean? If you cause someone’s death by accident, the loss would be infinite, so how could you produce any proportionate punishment or restitution for any wrong that you’d done? You might indeed find that people in the end were clamouring for a finite life if they were given eternal life under the same conditions that we know and appreciate today. New religions might spring up which promised finite life rather than eternal life, as an escape from the never ending boredom of the world around us.

As our closing thought Woody Allen, when asked about immortality, just remarked he didn’t want to achieve immortality through his work: “I want to achieve it through not dying. I want to not live on in the hearts of my countrymen; I want to live on in my apartment.” Very sound thinking.

© Professor John D Barrow, Gresham College, 8 March 2005

## Professor John D Barrow FRS

### Professor of Astronomy

Professor John D Barrow FRS has been a Professor of Mathematical Sciences at the University of Cambridge since 1999, carrying out research in mathematical physics, with special interest in cosmology, gravitation, particle physics and associated applied mathematics.

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