To Infinity and Beyond

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0:00:04 > 0:00:06Archive programmes chosen by experts.

0:00:06 > 0:00:09For this collection, Prof Alice Roberts has selected

0:00:09 > 0:00:13a range of programmes to celebrate Horizon's 50th anniversary.

0:00:13 > 0:00:16More Horizon programmes and other BBC Four Collections

0:00:16 > 0:00:18are available on BBC iPlayer.

0:00:22 > 0:00:27Zero, one, two, three, four, five...

0:00:27 > 0:00:31I've seen things you people wouldn't believe.

0:00:31 > 0:00:35Things that would change how you see this world.

0:00:35 > 0:00:39Enough to drive men to madness.

0:00:39 > 0:00:44DIFFERENT SPEAKERS COUNT

0:00:47 > 0:00:50What is the biggest number?

0:00:50 > 0:00:52Is the universe infinite?

0:00:52 > 0:00:56Might every event repeat again and again and again and again...

0:00:56 > 0:00:59Your intuition is no use here.

0:00:59 > 0:01:01Faith alone can't save you.

0:01:01 > 0:01:04How did the universe begin?

0:01:04 > 0:01:08Is the Earth just one of uncountable copies,

0:01:08 > 0:01:11tumbling through an unending void?

0:01:11 > 0:01:13On one you are rich,

0:01:13 > 0:01:16on another you have yet to be born.

0:01:18 > 0:01:22These are the deepest mysteries of the universe.

0:01:22 > 0:01:24Ladies and gentlemen,

0:01:24 > 0:01:27pray silence as I present to you...

0:01:28 > 0:01:31...infinity.

0:01:37 > 0:01:39And so on.

0:01:51 > 0:01:56Planet Earth is so beautiful and so complex,

0:01:56 > 0:01:58humans can barely comprehend it.

0:02:01 > 0:02:04And yet humanity has always asked,

0:02:04 > 0:02:08"What's over the horizon? What lies beyond the stars?

0:02:10 > 0:02:12"Is this it?"

0:02:16 > 0:02:19I think infinity is one of those things

0:02:19 > 0:02:22that is an essential mystery of the universe.

0:02:23 > 0:02:27What happens after we die, why are we here,

0:02:27 > 0:02:29why were we born.

0:02:31 > 0:02:33Infinity is in that class of questions

0:02:33 > 0:02:36and humans have been thinking about whether

0:02:36 > 0:02:40there's an end to the world or whether the world goes on for ever,

0:02:40 > 0:02:43probably since the beginning of human thought.

0:02:44 > 0:02:49It's a natural human impulse to want to go beyond any boundary.

0:02:49 > 0:02:53It is simultaneously scary and exciting to think about.

0:02:57 > 0:02:59If infinity is real,

0:02:59 > 0:03:03it has implications far beyond the world of science.

0:03:03 > 0:03:07It strikes at the very heart of what it means to be you.

0:03:12 > 0:03:15There is actually, far out in space,

0:03:15 > 0:03:20a planet that looks just like Earth with people just like us.

0:03:20 > 0:03:24Some will be doing exactly the same things as we do,

0:03:24 > 0:03:27even with the same names and memories as us.

0:03:30 > 0:03:34No matter how much I study the field of cosmology and think about this,

0:03:34 > 0:03:38it still makes no sense to me that the universe is infinite.

0:03:38 > 0:03:42I prefer a finite universe because I can get my mind around that.

0:03:42 > 0:03:46It's the only universe that makes intuitive sense to me.

0:03:49 > 0:03:52Infinity. Impossible to comprehend.

0:03:52 > 0:03:58And yet it comes from something so simple, a child can understand it.

0:03:58 > 0:04:03One, two, three...

0:04:03 > 0:04:05four, five, six...

0:04:05 > 0:04:08seven, eight, nine...

0:04:08 > 0:04:1210, 11, 12, 13, 14...

0:04:12 > 0:04:1415, 16, 17...

0:04:14 > 0:04:1918, 19, 21, 22, 23...

0:04:19 > 0:04:23I was very proud first to be able to count to five, then to ten, and then

0:04:23 > 0:04:27I realised that you can always keep counting and there's no end to it.

0:04:27 > 0:04:3137, 38, 39...

0:04:34 > 0:04:37So I had this obvious intuition that everybody had that

0:04:37 > 0:04:42there is no end to counting, hence there must be infinity.

0:04:42 > 0:04:4691, 92, 93, 94, 95, 96...

0:04:46 > 0:04:531,374, 1,375, 1,376, 1,377...

0:04:53 > 0:04:57'Numbers can get so vast, it's impossible to imagine.'

0:04:57 > 0:05:02..1,380, 1,381, 1,382...

0:05:02 > 0:05:0912,763, 12,764, 12,765, 12,766...

0:05:09 > 0:05:12To count to a billion, it would take you about 30 years and to count

0:05:12 > 0:05:17to a trillion is not something you could even do in human history.

0:05:17 > 0:05:22111,330, 111,331,

0:05:22 > 0:05:25111,332...

0:05:25 > 0:05:291,372,365,

0:05:29 > 0:05:311,372,366...

0:05:31 > 0:05:33Billion and nine, billion and ten,

0:05:33 > 0:05:36billion and eleven, billion and twelve...

0:05:36 > 0:05:38'For most people, I suppose the biggest number they'll likely meet'

0:05:38 > 0:05:43will be somewhere in the billions or maybe the trillions,

0:05:43 > 0:05:47which might be something like the budget deficit or

0:05:47 > 0:05:49military spending or something like that.

0:05:49 > 0:05:54Mathematicians tend to use bigger numbers than that.

0:05:54 > 0:05:58Googolplexplexplex three, googolplexplexplex four...

0:05:58 > 0:05:59You made me do this!

0:05:59 > 0:06:03Googolplexplexplex five, googolplexplexplex six...

0:06:03 > 0:06:08When I get to 199,

0:06:08 > 0:06:11then that would be too hard, I would have to stop.

0:06:14 > 0:06:17TYPEWRITER KEYS CLICK, MONKEYS HOOT

0:06:21 > 0:06:24One of the largest numbers we have a name for is a googol

0:06:24 > 0:06:26and it's one followed by 100 zeros.

0:06:28 > 0:06:29A hundred zeros is a lot

0:06:29 > 0:06:32because each zero represents another factor of ten.

0:06:34 > 0:06:36So, it's a big number.

0:06:40 > 0:06:45You might be thinking 100 zeros isn't that many.

0:06:45 > 0:06:50But a googol is far bigger than the number of atoms in a human being,

0:06:50 > 0:06:54more than the number of atoms that make up planet Earth.

0:06:55 > 0:06:59A hundred zeros is more, even, than all the atoms

0:06:59 > 0:07:02in the entire observable universe.

0:07:09 > 0:07:12Sets your imagination going, doesn't it?

0:07:12 > 0:07:15'A googol sets your imagination going.'

0:07:15 > 0:07:17It knocks you off of your chair.

0:07:17 > 0:07:21It's already a number that's probably bigger than anything

0:07:21 > 0:07:23that makes sense in our experience.

0:07:23 > 0:07:29But, it's a very, very tiny, tiny, tiny large number.

0:07:29 > 0:07:32A googol itself was only a stepping stone on the way to

0:07:32 > 0:07:36a much, much larger number called a googolplex.

0:07:36 > 0:07:41A googolplex is ten raised to the power of a googol,

0:07:41 > 0:07:44that is it's one followed by a googol of zeros.

0:07:44 > 0:07:47And of course it's just not possible to imagine

0:07:47 > 0:07:50the size of a number like that.

0:07:53 > 0:07:58A googol has 100 zeros, but a googolplex has so many zeros

0:07:58 > 0:08:03that there's not enough space in the entire observable universe

0:08:03 > 0:08:06just to write the number down,

0:08:06 > 0:08:09even if you could write each zero on a single atom.

0:08:15 > 0:08:20But from my perspective, these are all very, very small.

0:08:21 > 0:08:24One of the biggest numbers ever used in mathematics

0:08:24 > 0:08:26is many times the size of a googolplex.

0:08:26 > 0:08:30It makes normal numbers like a trillion or a billion

0:08:30 > 0:08:32disappear into practically nothing.

0:08:34 > 0:08:36It's a number called Graham.

0:08:39 > 0:08:42Graham's number is much, much bigger than a googolplex.

0:08:42 > 0:08:47In fact, it's as large relative to a googolplex

0:08:47 > 0:08:50as a googolplex is to the number ten.

0:08:50 > 0:08:54In fact, it's much, much bigger than that.

0:09:03 > 0:09:07Graham's number was discovered in the 1970s

0:09:07 > 0:09:11by mathematician and former circus performer, Ron Graham.

0:09:11 > 0:09:14I don't know too many other people who have a number.

0:09:14 > 0:09:17Er... It's...

0:09:17 > 0:09:20It's not bad. It's not bad.

0:09:23 > 0:09:26I mean... I recommend it!

0:09:26 > 0:09:28'Graham's number is so big,

0:09:28 > 0:09:31'it even made it into the Guinness Book Of Records.'

0:09:31 > 0:09:34Oh, yes.

0:09:36 > 0:09:39This is the 1980 edition of Guinness Book Of World Records

0:09:39 > 0:09:45and I think if we turn to page, er, what is it, 192?

0:09:45 > 0:09:48Numeration. Yes, and then numbers.

0:09:48 > 0:09:51Prime numbers, perfect, highest numbers. Here we go, highest numbers.

0:09:51 > 0:09:54"The highest number ever used in a mathematical proof

0:09:54 > 0:10:00"is a bounding value published in 1977. It is known as Graham's number.

0:10:00 > 0:10:02"It concerns bichromatic hypercubes

0:10:02 > 0:10:05"and is inexpressible without the special arrow notation."

0:10:05 > 0:10:09It's really gigantic, I mean, it's just so large,

0:10:09 > 0:10:11that you can't compare it with anything

0:10:11 > 0:10:16you would normally associate large numbers, like the number of atoms

0:10:16 > 0:10:20in the universe or how many inches to the furthest galaxy

0:10:20 > 0:10:23or something like this, it's just way bigger than that.

0:10:25 > 0:10:27So vast is Graham's number,

0:10:27 > 0:10:34nobody knows how many digits it even has, including Ron Graham himself.

0:10:34 > 0:10:37In spite of the fact that it's Graham's number, and I'm Graham,

0:10:37 > 0:10:40er, I know very little about it.

0:10:41 > 0:10:44I have no idea what the first digit is.

0:10:45 > 0:10:48It has one. I don't know what it is.

0:10:48 > 0:10:51Maybe no-one will ever know what that digit is.

0:10:53 > 0:10:56Are there more zeros than ones in the number?

0:10:56 > 0:10:57Who knows?

0:10:58 > 0:11:02If you look at three to the three to the three to the n plus one...

0:11:02 > 0:11:04'Ron didn't just make up his number.

0:11:04 > 0:11:09'It's the upper limit to the solution of a pure mathematics problem

0:11:09 > 0:11:11'concerning multi-dimensional cubes.'

0:11:11 > 0:11:14That's divisible by five,

0:11:14 > 0:11:18only if the exponent three to the three to the n plus one...

0:11:18 > 0:11:20'While the problem itself is abstract,

0:11:20 > 0:11:23'the methods Ron used to solve it are now used to keep track

0:11:23 > 0:11:26'of data sent across the internet.'

0:11:26 > 0:11:31N plus one, minus three to the three to the n, minus one. OK. Great. So...

0:11:32 > 0:11:34'Just working out the last digit

0:11:34 > 0:11:37'of Graham's number is a lengthy calculation.'

0:11:37 > 0:11:40RON GRAHAM'S WORDS ECHO

0:11:40 > 0:11:43You can't really comprehend how large it is.

0:11:43 > 0:11:44Very large.

0:11:44 > 0:11:47I don't think anybody can know that.

0:11:50 > 0:11:53'But like all finite numbers, it comes to an end...'

0:11:53 > 0:11:56In the last stage, we have three to a certain power...

0:11:56 > 0:11:57'Eventually.'

0:11:57 > 0:12:01RON GRAHAM'S WORDS ECHO

0:12:04 > 0:12:07So that means that the remainder,

0:12:07 > 0:12:10when you divide by ten, is always seven.

0:12:10 > 0:12:13In other words, you can finally conclude that the last digit

0:12:13 > 0:12:15of Graham's number is seven.

0:12:16 > 0:12:18End of the story!

0:12:24 > 0:12:27Graham's number is not really any closer to infinity

0:12:27 > 0:12:29than the number one.

0:12:29 > 0:12:32You didn't really get started yet, even though you took a lot of steps

0:12:32 > 0:12:34to get to Graham's number,

0:12:34 > 0:12:38it takes so many more, infinitely more, to get to infinity.

0:12:41 > 0:12:45Infinity is just out there, out there.

0:12:45 > 0:12:47It's just a different beast.

0:12:56 > 0:12:58What's the biggest number?

0:12:58 > 0:13:02Erm... 120?

0:13:03 > 0:13:06Ten.

0:13:06 > 0:13:09Is ten the biggest number?

0:13:09 > 0:13:12Sometimes people just say, um,

0:13:12 > 0:13:17seventy hundred eighty nine hundred, but that's not even a number.

0:13:20 > 0:13:22Oh, dear. Oh, that's difficult.

0:13:22 > 0:13:28Well, I suppose there isn't really one, they just go on and on and on.

0:13:28 > 0:13:31There is no biggest number,

0:13:31 > 0:13:33because if there were, you could always add one to it.

0:13:35 > 0:13:39TYPEWRITER KEYS CLICK

0:13:39 > 0:13:41MONKEYS HOOT

0:13:46 > 0:13:52'Unlike normal numbers, infinity never comes to an end.

0:13:52 > 0:13:56'And that gives it some very strange properties.'

0:13:58 > 0:14:00OK, so this is one of the first things that you have to think about

0:14:00 > 0:14:02if you're thinking about infinity.

0:14:02 > 0:14:06You've got all the numbers - one, two, three, four, five, six.

0:14:06 > 0:14:10Let's have just a few.

0:14:10 > 0:14:12And they go on for ever.

0:14:13 > 0:14:18Now, suppose I went through and picked out just the even numbers.

0:14:18 > 0:14:23Two and four, six, eight, ten.

0:14:23 > 0:14:25I'll write them down here.

0:14:26 > 0:14:29Two, four, six, eight, ten. Well, it's pretty obvious,

0:14:29 > 0:14:32that there are more of these than there are of these?

0:14:32 > 0:14:37Because we've only picked out half of these numbers here.

0:14:37 > 0:14:42Well, in fact it's not. These two lists are exactly the same size.

0:14:42 > 0:14:45And this is the first real paradox about infinity.

0:14:45 > 0:14:47There are as many even though there are half as many,

0:14:47 > 0:14:52so half of this list has as many things in it as the whole list does.

0:14:52 > 0:14:57These sets look so different, but they're actually the same size.

0:14:57 > 0:14:59- How's that possible? - Well, OK. How's that possible?

0:14:59 > 0:15:02Well, you see, what are we doing? We're counting.

0:15:02 > 0:15:05And when you count something you match it up with the numbers.

0:15:05 > 0:15:08So if I'm counting my sheep, for example,

0:15:08 > 0:15:10I count one, two, three, four,

0:15:10 > 0:15:12I match the sheep up with the numbers.

0:15:12 > 0:15:15But you see, I can match the even numbers up with the numbers.

0:15:15 > 0:15:16I've already started doing it.

0:15:16 > 0:15:19Here's one matched with two, two matched with four,

0:15:19 > 0:15:22three matched with six, four matched with eight, five matched with ten,

0:15:22 > 0:15:26and that goes on for ever, so I've matched all of the even numbers up

0:15:26 > 0:15:31with all of the numbers, no gaps, perfect matching between the two.

0:15:31 > 0:15:35So there are the same number of these as there are of these.

0:15:35 > 0:15:39So this really is a characteristic property of infinity

0:15:39 > 0:15:42and it seems puzzling, but there it is.

0:15:42 > 0:15:44This is why infinity is a bit of a hard thing to deal with.

0:15:44 > 0:15:49And it troubled people for quite some time.

0:15:49 > 0:15:53'Two infinite lists are exactly the same size,

0:15:53 > 0:15:57'even though one appears to contain twice as many numbers as the other.

0:15:57 > 0:16:00'Which is just the start.

0:16:00 > 0:16:03'The more mathematicians thought about infinity,

0:16:03 > 0:16:06'the weirder it became.'

0:16:06 > 0:16:08- How are you? - I'm fine, thank you.

0:16:08 > 0:16:09Can I check in, please?

0:16:09 > 0:16:11'Around 100 years ago,

0:16:11 > 0:16:15'one mathematician tried to explain some of infinity's

0:16:15 > 0:16:19'strange properties by imagining arriving at a hotel

0:16:19 > 0:16:21'with infinite rooms.

0:16:21 > 0:16:24'He wondered if there would be any space for him,

0:16:24 > 0:16:26'even if it was fully booked?'

0:16:26 > 0:16:30In an ordinary hotel, I would be told, "I'm sorry, we're full up.

0:16:30 > 0:16:33"You'll have to go somewhere else." But in an infinite hotel,

0:16:33 > 0:16:35things are rather different. There's no problem at all.

0:16:35 > 0:16:37RECEPTION BELL RINGS

0:16:37 > 0:16:41'The infinite hotel was dreamt up by David Hilbert, one of the most

0:16:41 > 0:16:44'influential mathematicians of the early 20th century.'

0:16:46 > 0:16:48Your room is upstairs. Have a lovely stay with us. Bye-bye.

0:16:51 > 0:16:53The manager can't just put me into the last room,

0:16:53 > 0:16:56because there is no last room. It's an infinite hotel.

0:16:56 > 0:16:58The rooms go on for ever. There's no last room.

0:17:00 > 0:17:04And all the rooms are full anyway so, even if there were a last room,

0:17:04 > 0:17:06it would have somebody in it.

0:17:08 > 0:17:14But it's exactly for that reason that it's possible to find room for me.

0:17:14 > 0:17:18All you have to do is shift the guest from room one into room two,

0:17:18 > 0:17:23the guest from room two into room three and so on down the line.

0:17:23 > 0:17:27Because there's no last room, every guest has a next room to go to

0:17:27 > 0:17:31and that frees up room one, so I can stay in room one.

0:17:31 > 0:17:34So everything's fine.

0:17:35 > 0:17:40'It turns out, even if the hotel was packed to the rafters,

0:17:40 > 0:17:42'a room can always be found.

0:17:42 > 0:17:46'Infinity plus one is infinity.'

0:17:48 > 0:17:49LIFT BELL RINGS

0:17:52 > 0:17:55- Good morning, sir. How are you? - I'm very well, thank you.

0:17:55 > 0:17:57Can I check in, please?

0:17:57 > 0:18:03You can see that if two guests came in, infinity plus two is infinity.

0:18:03 > 0:18:06But suppose that I came along to this hotel

0:18:06 > 0:18:08with infinitely many of my friends

0:18:08 > 0:18:12and we all wanted to stay and the hotel was full, how could we do it?

0:18:14 > 0:18:17What we could do is arrange that the guest in room one

0:18:17 > 0:18:18moved into room two

0:18:18 > 0:18:22and the guest in room two moved into room four,

0:18:22 > 0:18:26each guest moved into the room with double the number.

0:18:26 > 0:18:30So all the even-numbered rooms have now got people in them

0:18:30 > 0:18:33and all the odd-numbered rooms are now free,

0:18:33 > 0:18:37so me and my friends can all stay in the odd-numbered rooms.

0:18:37 > 0:18:41So this shows that infinity plus infinity is still infinity.

0:18:44 > 0:18:49And it would make sense that the same rules also apply to subtraction.

0:18:50 > 0:18:53But if you think infinity will stay the same

0:18:53 > 0:18:58whatever you add or subtract, then think again.

0:19:01 > 0:19:05Suppose that in the morning, all the guests left.

0:19:05 > 0:19:08The number of occupied rooms would be infinity minus infinity,

0:19:08 > 0:19:10but it would be zero.

0:19:10 > 0:19:12So infinity minus infinity could be zero.

0:19:14 > 0:19:15Or it could be one.

0:19:17 > 0:19:20If I stayed on and all the other guests left,

0:19:20 > 0:19:23infinity minus infinity would be one in that case.

0:19:23 > 0:19:24So there's no definite answer.

0:19:24 > 0:19:28That's why you have to be very careful dealing with infinity.

0:19:28 > 0:19:30It's a very slippery concept.

0:19:32 > 0:19:37What if I and two other people stayed, say?

0:19:37 > 0:19:39Then infinity minus infinity would be three.

0:19:41 > 0:19:43So it could be anything you like.

0:19:51 > 0:19:52BELL RINGS

0:19:52 > 0:19:53- Good morning, sir. - Good morning.

0:19:53 > 0:19:54- How are you? - I'm fine, thank you.

0:19:54 > 0:19:57- Can I help you? - Yes. Can I check in, please?

0:19:57 > 0:19:59HE LAUGHS

0:20:00 > 0:20:01Oh, God!

0:20:01 > 0:20:02HE LAUGHS

0:20:05 > 0:20:09This is one of the reasons why you have to very careful

0:20:09 > 0:20:12dealing with a slippery character like infinity.

0:20:12 > 0:20:14Not to be trusted.

0:20:20 > 0:20:22Is infinity real?

0:20:22 > 0:20:24Well, not to most people,

0:20:24 > 0:20:29but you can't do mathematics without infinity.

0:20:29 > 0:20:31So if you don't feel comfortable with that

0:20:31 > 0:20:35then you're probably not going to become a mathematician.

0:20:35 > 0:20:36Get any mathematician

0:20:36 > 0:20:40to tell you about what he or she is doing at the moment,

0:20:40 > 0:20:48and their imagination will somehow be full of this idea of infinity.

0:20:51 > 0:20:54Infinity is like a landscape in which you work.

0:20:54 > 0:20:56A place in which you do mathematics.

0:20:58 > 0:20:59It's not a real place.

0:20:59 > 0:21:02You can't actually go there, except in your imagination.

0:21:02 > 0:21:07But to those who do mathematics, it seems very real indeed.

0:21:08 > 0:21:10But you see one of the problems with infinity

0:21:10 > 0:21:13is that it has some paradoxical properties

0:21:13 > 0:21:16and very basic questions about infinity that we can't answer.

0:21:16 > 0:21:20So you do have to be a little careful.

0:21:21 > 0:21:24It's like having a polar bear as a pet,

0:21:24 > 0:21:27you've grown up together,

0:21:27 > 0:21:32he's a wonderful pet, he's big, he's fast,

0:21:32 > 0:21:35he plays in the snow beautifully, but...

0:21:35 > 0:21:41there's always the chance that one day he'll get annoyed with you

0:21:41 > 0:21:44and bite off your head.

0:21:44 > 0:21:47So, we are playing with fire, I think.

0:21:51 > 0:21:53The paradoxes associated with infinity

0:21:53 > 0:21:56make some mathematicians uncomfortable.

0:21:56 > 0:22:00Not least Prof Doron Zeilberger.

0:22:05 > 0:22:08I first came across infinity like everybody else

0:22:08 > 0:22:10in a very early childhood when you start counting.

0:22:10 > 0:22:15First you count to three, then to four, then to five, then to ten, then

0:22:15 > 0:22:20to 100 and eventually you realise that you can keep counting for ever.

0:22:22 > 0:22:24Hence there is an infinity.

0:22:24 > 0:22:31341, 342, 343... 63,789, 63,790, 63...

0:22:31 > 0:22:36etc, etc, etc, ad infinitum.

0:22:36 > 0:22:40But, frankly, I don't think I ever liked it.

0:22:40 > 0:22:45I always found something repulsive about it.

0:22:45 > 0:22:49TYPEWRITER KEYS CLICK

0:22:49 > 0:22:55I prefer finite mathematics much more to infinite mathematics.

0:22:55 > 0:22:58I think that it's much more natural, much more appealing

0:22:58 > 0:23:04and the theory is much more beautiful.

0:23:06 > 0:23:09It is very concrete. It's something you can touch,

0:23:09 > 0:23:12something you can feel, something you can relate to.

0:23:15 > 0:23:18Infinite mathematics, to me, is meaningless

0:23:18 > 0:23:20because it's like abstract nonsense.

0:23:26 > 0:23:31In my opinion, infinity is only a fiction of the human mind.

0:23:34 > 0:23:39But not believing in infinity leaves Prof Zeilberger with a problem.

0:23:39 > 0:23:43If the numbers don't go on for ever, where do they end?

0:23:46 > 0:23:49When you start counting,

0:23:49 > 0:23:54you seemingly can go for ever,

0:23:54 > 0:23:58but eventually you will reach the biggest number

0:23:58 > 0:24:02and then when you add one to it you go back to zero.

0:24:02 > 0:24:04- You go back to zero? - Yeah.

0:24:04 > 0:24:06How is that possible?

0:24:06 > 0:24:10How is it not possible? Have you ever been there?

0:24:11 > 0:24:18The biggest number is much bigger than anybody can ever think of.

0:24:18 > 0:24:21It's bigger than a googol, bigger than a googolplex,

0:24:21 > 0:24:25bigger than a googolplex to the power of a googolplex.

0:24:25 > 0:24:28It's so big, we can never envision it.

0:24:29 > 0:24:32Nevertheless there is a biggest number

0:24:32 > 0:24:37and if you keep counting after that big number, we get back to zero.

0:24:37 > 0:24:39Like when we walk around our planet,

0:24:39 > 0:24:44if you keep walking, eventually we get back to the place we started.

0:24:45 > 0:24:48And if you think that this is ridiculous,

0:24:48 > 0:24:54look at the alternative. It's less ridiculous than infinity

0:24:54 > 0:24:57and all the paradoxes that go with it.

0:24:57 > 0:25:02Infinity may or may not exist, God may or may not exist

0:25:02 > 0:25:03but in mathematics

0:25:03 > 0:25:08there should not be any place for neither infinity nor God.

0:25:10 > 0:25:16Doron isn't the first person to feel that the infinite is an illusion.

0:25:16 > 0:25:21Until recently, infinity was too wild to be tamed by mathematics.

0:25:21 > 0:25:24Too unpredictable to be used in equations.

0:25:28 > 0:25:32Aristotle believed counting could go on for ever

0:25:32 > 0:25:35and that the universe was eternal.

0:25:35 > 0:25:39But he refused to accept that the universe was infinite in size.

0:25:40 > 0:25:44He believed the Earth was at the centre of the universe.

0:25:44 > 0:25:48However, without an end, there can be no middle

0:25:48 > 0:25:52so he banned infinity from his mathematics.

0:25:58 > 0:26:01Infinity was discussed by philosophers and priests,

0:26:01 > 0:26:04rather than mere mathematicians.

0:26:04 > 0:26:08The infinite was something closer to a god than a number.

0:26:12 > 0:26:18In 1600, philosopher Giordano Bruno claimed not only that the universe

0:26:18 > 0:26:23was infinite, but there would be many other Earths orbiting stars

0:26:23 > 0:26:25just like our own sun.

0:26:25 > 0:26:29His beliefs went down so poorly with the Catholic Church,

0:26:29 > 0:26:31they had him burnt at the stake.

0:26:31 > 0:26:34Only God himself could be truly infinite.

0:26:37 > 0:26:40HE SIGHS

0:26:41 > 0:26:45In the mid 19th century, one man, Gregor Cantor,

0:26:45 > 0:26:49made infinity truly part of mathematics.

0:26:49 > 0:26:54Something that could be used in equations as if it were a number.

0:26:54 > 0:26:59Cantor's idea was that we can gather together maybe even

0:26:59 > 0:27:02infinitely many things into a single set,

0:27:02 > 0:27:07like putting them into a bag and think of it as just a single object.

0:27:07 > 0:27:11And once it is a single object, we can then put it into mathematics,

0:27:11 > 0:27:14do calculations with it, because it's just one object.

0:27:14 > 0:27:18We don't have to know that there are infinitely many things in the bag,

0:27:18 > 0:27:20it's just a single object for us.

0:27:21 > 0:27:24But making infinity part of mathematics

0:27:24 > 0:27:27produced one surprising result.

0:27:27 > 0:27:30Some infinities are bigger than others.

0:27:32 > 0:27:35Cantor's great initial discovery

0:27:35 > 0:27:38was that the infinity of the decimal numbers

0:27:38 > 0:27:41was larger than the infinity of the counting numbers.

0:27:41 > 0:27:45And he did this by what's now called Cantor's diagonal argument.

0:27:45 > 0:27:51So what you have to show is that given any list of decimal numbers,

0:27:51 > 0:27:53that there's a decimal number not on that list.

0:27:53 > 0:27:57And what we can do is go down the diagonal of this list,

0:27:57 > 0:28:02so we might take the nine here,

0:28:02 > 0:28:05and the one here and the one here.

0:28:05 > 0:28:07We're going down the diagonal

0:28:07 > 0:28:09and we're generating another decimal number.

0:28:09 > 0:28:12Nine, one, one and so forth.

0:28:14 > 0:28:16Now we take this number and we change it.

0:28:16 > 0:28:19We change the nine to an eight, say,

0:28:19 > 0:28:22and we can change the ones to twos and so forth.

0:28:22 > 0:28:27And now we've generated a decimal number which can't be on this list.

0:28:27 > 0:28:31It can't be the first one because the nine has been changed to an eight.

0:28:31 > 0:28:35It can't be the second one because in the second position

0:28:35 > 0:28:37it's got a two instead of a one and so forth.

0:28:37 > 0:28:39And all the way down the list,

0:28:39 > 0:28:42this number will be different from the decimal number on the list.

0:28:42 > 0:28:44So what does that mean?

0:28:44 > 0:28:48It means there can be no matching of all the decimal numbers

0:28:48 > 0:28:50by the counting numbers

0:28:50 > 0:28:53and that tells us that the infinity of the decimal numbers

0:28:53 > 0:28:57is larger than the infinity of the counting numbers.

0:29:04 > 0:29:06There's no largest infinite number.

0:29:06 > 0:29:09For every infinite number, there's a bigger one.

0:29:11 > 0:29:16There are infinities beyond infinities and that's what we study.

0:29:18 > 0:29:22'In making infinity part of mathematics, Cantor had uncovered

0:29:22 > 0:29:27'a whole universe of infinities, each infinitely bigger than the last.

0:29:27 > 0:29:31'And that wasn't easy for many to accept.'

0:29:31 > 0:29:34At the time, I think that was a big surprise.

0:29:34 > 0:29:38No-one had really thought carefully about whether infinities could come

0:29:38 > 0:29:40in different sizes, after all, infinity is infinity.

0:29:40 > 0:29:43How can you have different sizes of infinity?

0:29:44 > 0:29:48'Cantor's breakthrough came at a price.

0:29:48 > 0:29:51'Cantor ended his days in an asylum.'

0:29:51 > 0:29:56Was it infinity that drove him there?

0:29:56 > 0:29:59Who knows? Who can tell?

0:30:03 > 0:30:07He faced a lot of opposition from his colleagues and

0:30:07 > 0:30:10it was possibly that, more than thinking about infinity itself

0:30:10 > 0:30:11that was the trouble.

0:30:11 > 0:30:17It was only later that mathematicians accepted and welcomed his theories

0:30:17 > 0:30:19into the body of mathematics.

0:30:22 > 0:30:26Cantor's work is absolutely fundamental to everything we do.

0:30:30 > 0:30:34'Today, Cantor's infinities are part of mainstream mathematics.

0:30:36 > 0:30:41'And the truth is, even those who would rather infinity didn't exist

0:30:41 > 0:30:44'use it in their equations every day.

0:30:45 > 0:30:52'Infinity is simpler and quicker to manipulate than large finite numbers.

0:30:53 > 0:30:57'Most mathematicians have made an uneasy peace with infinity

0:30:57 > 0:31:01'and accepted it as part of their universe.

0:31:04 > 0:31:08'And some have devoted their careers to studying it.'

0:31:11 > 0:31:14To the person who wants to deny infinity and say it doesn't exist,

0:31:14 > 0:31:17I don't see how that view enriches their world.

0:31:17 > 0:31:19I feel sorry for them.

0:31:19 > 0:31:25I mean, infinity, maybe it doesn't exist, but it is a beautiful subject.

0:31:25 > 0:31:29I could say the stars don't exist and stay inside, or always look down,

0:31:29 > 0:31:32but then I don't see the beauty of the stars.

0:31:37 > 0:31:40And until one has a real reason to doubt

0:31:40 > 0:31:45the existence of mathematical infinity, I just don't see the point.

0:31:45 > 0:31:49There's a whole world of infinities out there.

0:31:51 > 0:31:54Maybe they're real, maybe they're not.

0:31:57 > 0:32:01But could infinity be part of the world you call real?

0:32:01 > 0:32:07As yet unseen through any microscope and undetected by any telescope,

0:32:07 > 0:32:10might the heavens genuinely be unbounded

0:32:10 > 0:32:15and the depths of space deeper than love itself?

0:32:15 > 0:32:18Is space infinitely big

0:32:18 > 0:32:22or simply unimaginably big?

0:32:22 > 0:32:23TYPEWRITER KEYS CLICK

0:32:34 > 0:32:39The sky never ends, so I think space never ends, because there isn't like

0:32:39 > 0:32:45a wall all around our... all around our whole planet.

0:32:47 > 0:32:50I actually think the universe IS infinite

0:32:50 > 0:32:52and if I had to put odds on it,

0:32:52 > 0:32:57I would say I think there's a 95% chance that it is in fact infinite.

0:32:59 > 0:33:04I think space is very, very, very, very, very, very big.

0:33:06 > 0:33:10I would say 1,000 metres.

0:33:10 > 0:33:12That big.

0:33:15 > 0:33:20I think the universe is infinite on Mondays, Wednesdays and Fridays

0:33:20 > 0:33:23and I think it's finite the rest of the week.

0:33:23 > 0:33:27I'm having a very, very hard time making my mind up.

0:33:27 > 0:33:32Space. It hasn't got laws, or no electricity.

0:33:32 > 0:33:35It's just like a sky.

0:33:37 > 0:33:38It won't finish.

0:33:38 > 0:33:40It will keep on going.

0:33:44 > 0:33:49And going and it, like, never stops.

0:33:51 > 0:33:54The fundamental issue that most people come up with when you say

0:33:54 > 0:33:59the universe may be finite, is simply, what's outside of it?

0:33:59 > 0:34:01What happens when you come to the edge?

0:34:01 > 0:34:03Can't you go beyond it?

0:34:03 > 0:34:06And so this leads some people to the conclusion that

0:34:06 > 0:34:08the universe has to be infinite.

0:34:08 > 0:34:12'It might seem obvious that space goes on for ever.

0:34:12 > 0:34:14'Innocent even.'

0:34:14 > 0:34:19But unleash infinity into the universe and all bets are off.

0:34:19 > 0:34:25He makes the extraordinary mundane and the unbelievable inevitable.

0:34:25 > 0:34:28He makes the extraordinary mundane

0:34:28 > 0:34:30and the unbelievable inevitable.

0:34:33 > 0:34:38In an infinite universe, anything that's possible has to happen.

0:34:38 > 0:34:41Even something as unlikely as

0:34:41 > 0:34:45a monkey typing the complete works of Shakespeare.

0:34:45 > 0:34:48TYPEWRITER KEYS CLICK

0:34:48 > 0:34:52"My bounty is as boundless as the sea.

0:34:52 > 0:34:57"My love is deep. The more I give to thee, the more I have.

0:34:57 > 0:35:00"For both are infinite."

0:35:02 > 0:35:04If we imagine this monkey,

0:35:04 > 0:35:09all it's doing is thumping away at the keys completely at random.

0:35:11 > 0:35:12The monkeys don't have to evolve,

0:35:12 > 0:35:15they don't have to be able to read Shakespeare, they do have

0:35:15 > 0:35:20to be able to carry on typing, but that's all, just typing at random.

0:35:20 > 0:35:22"I could be bounded in a nutshell

0:35:22 > 0:35:25"and count myself king of infinite space,

0:35:25 > 0:35:27"were it not that I have bad dreams."

0:35:29 > 0:35:33To test the infinite monkey theorem, a computer was placed

0:35:33 > 0:35:37in the enclosure of a Cambridge University professor.

0:35:37 > 0:35:41Typing out the complete works of Shakespeare at random

0:35:41 > 0:35:43is really going to be a big job.

0:35:43 > 0:35:48It's a thick book. There's 37 plays in here, all the poems and sonnets.

0:35:48 > 0:35:54There's 884,429 words, every word has to be in exactly

0:35:54 > 0:35:58the right place, every character has to be exactly in sequence,

0:35:58 > 0:36:00including the spaces in between.

0:36:00 > 0:36:03And so doing that at random, bashing away on a keyboard,

0:36:03 > 0:36:05is a difficult thing to do.

0:36:08 > 0:36:09For the last week,

0:36:09 > 0:36:14David Spiegelhalter's computer has been randomly generating letters.

0:36:14 > 0:36:17We're only generating lower case at the moment. We're not using capitals.

0:36:17 > 0:36:20So we're giving it a bit of a chance like that.

0:36:20 > 0:36:23And it's generating them at the rate of 50 characters a second.

0:36:23 > 0:36:27And, as you can see, it's keeping on finding matches all the time.

0:36:27 > 0:36:29If it finds a match of four letters,

0:36:29 > 0:36:31four characters, it adds another character on, a random character,

0:36:31 > 0:36:35and sees if it's found a match for five characters and so on and so on.

0:36:39 > 0:36:42The programme's been running for more than a week now and in that time,

0:36:42 > 0:36:46it's managed to generate more than 34 million characters.

0:36:49 > 0:36:53If we assume that this monkey can type one character a second, it would

0:36:53 > 0:36:54have taken 34 million seconds,

0:36:54 > 0:36:56which is just over a year's typing for our monkey,

0:36:56 > 0:36:59but we've got to be kind and give it some breaks,

0:36:59 > 0:37:02so I would say just over two years probably typing.

0:37:09 > 0:37:15The longest match so far is eight letters and here's the string,

0:37:15 > 0:37:17"We space lover."

0:37:17 > 0:37:20This occurs once in the complete works.

0:37:20 > 0:37:25It occurs in Love's Labour's Lost, act two, scene one.

0:37:25 > 0:37:28So, it's in here...

0:37:28 > 0:37:30Somewhere...

0:37:30 > 0:37:33I've no idea where!

0:37:33 > 0:37:38It's not even in alphabetical order.

0:37:40 > 0:37:44Ah, Love's Labour's Lost, here we are. Act two, scene one.

0:38:08 > 0:38:10Yep, I've got it.

0:38:12 > 0:38:13Yeah, there we are,

0:38:13 > 0:38:17Boyet says, "With that which we lovers entitle affected."

0:38:17 > 0:38:19So the actual word is "we lovers"

0:38:19 > 0:38:22and that's where "we lover" fits into it.

0:38:25 > 0:38:27But eight letters doesn't seem like very much?

0:38:27 > 0:38:29No, it doesn't, but you have to think of just

0:38:29 > 0:38:33how unlikely it is to generate the exact works of Shakespeare.

0:38:33 > 0:38:35So we did some calculations

0:38:35 > 0:38:37that worked out that if we wanted 17 characters,

0:38:37 > 0:38:41which is, "To be or not to b..." Not even the whole phrase.

0:38:41 > 0:38:46We'd have had to set this going at about the time of the Big Bang

0:38:46 > 0:38:48around 14 billion years ago.

0:38:48 > 0:38:51And that's to get something just twice that length.

0:38:53 > 0:38:56But remember, we've got to get five million characters,

0:38:56 > 0:38:58all in the right order.

0:39:00 > 0:39:03We can calculate the chance of this happening

0:39:03 > 0:39:05as one in a very large number.

0:39:05 > 0:39:09It's ten with about nine million zeros written after it.

0:39:09 > 0:39:13So that's an incredibly tiny probability, very, very small.

0:39:13 > 0:39:16It's an unbelievably unlikely thing to occur.

0:39:19 > 0:39:21So if you imagine the current National Lottery,

0:39:21 > 0:39:26it would be like someone winning every single time, time and again,

0:39:26 > 0:39:32every single week for year after year, for 29,000 years.

0:39:32 > 0:39:35Same person.

0:39:35 > 0:39:37The same person buying their ticket

0:39:37 > 0:39:39and winning every single week for 29,000 years.

0:39:41 > 0:39:43But if we have an infinite amount of time,

0:39:43 > 0:39:45we can be certain that it will happen.

0:39:45 > 0:39:48And not just once, it's going to happen again and again and again.

0:39:52 > 0:39:54Because infinity is such a long time

0:39:54 > 0:39:56that everything, no matter how unlikely,

0:39:56 > 0:39:59as long as it's possible, will occur.

0:40:02 > 0:40:06'Infinity is so vast, one monkey, randomly typing for ever

0:40:06 > 0:40:08'could easily get the job done.'

0:40:08 > 0:40:10TYPEWRITER KEYS CLICK

0:40:20 > 0:40:21If he had an infinite amount of time,

0:40:21 > 0:40:24the monkey would produce far more than Shakespeare.

0:40:24 > 0:40:27He'd produce every book that's ever been written.

0:40:27 > 0:40:30Everything from the telephone directory

0:40:30 > 0:40:33to the latest celebrity autobiography.

0:40:35 > 0:40:40TYPEWRITER KEYS CLICK

0:40:40 > 0:40:46But in an infinite universe, there will be infinite number of monkeys.

0:40:46 > 0:40:51And that means, somewhere, one of them is typing Shakespeare right now.

0:40:51 > 0:40:55TYPEWRITER KEYS CLICK

0:41:04 > 0:41:08If the universe were infinite, it seems fairly simple and benign,

0:41:08 > 0:41:11but it has some really strange consequences.

0:41:13 > 0:41:15If we look far enough away,

0:41:15 > 0:41:19there would be regions like the one that we're in.

0:41:20 > 0:41:23There would be a room out there like the one that we're sitting in now.

0:41:23 > 0:41:26There would be Earths out there just like ours,

0:41:26 > 0:41:29except maybe the Roman Empire would still exist

0:41:29 > 0:41:34or Germany would have won the war. And on a personal level

0:41:34 > 0:41:39out there right now would be copies of Anthony giving interviews

0:41:39 > 0:41:44in a pink jumpsuit or rotting in jail or filthy rich.

0:41:55 > 0:41:59There would be every possible combination.

0:41:59 > 0:42:01Every sort of way your life could have gone,

0:42:01 > 0:42:04there is somebody else just like you leading that life.

0:42:07 > 0:42:09In fact, anything that we can really conceive of,

0:42:09 > 0:42:12anything that's physically possible will happen.

0:42:12 > 0:42:15Not only that, it will happen an infinite number of times.

0:42:15 > 0:42:18They're all out there in this infinite universe.

0:42:19 > 0:42:21HE CONTINUES: 'So whether you look at this

0:42:21 > 0:42:23'as a good thing or a bad thing

0:42:23 > 0:42:25'depends on how good your life is right now,

0:42:25 > 0:42:26'but they're out there.'

0:42:26 > 0:42:31Which is crazy. It means that there is actually far out in space,

0:42:31 > 0:42:33a planet that looks just like Earth,

0:42:33 > 0:42:35with people just like us,

0:42:35 > 0:42:40some will be doing exactly the same things as we do,

0:42:40 > 0:42:44even with the same names and memories as us.

0:42:45 > 0:42:47It feels a little bit spooky

0:42:47 > 0:42:50to know that there are all these other copies of me...

0:42:52 > 0:42:54...but I think it takes a little bit of the pressure off

0:42:54 > 0:42:58of getting things right all the time, knowing that when I screw up,

0:42:58 > 0:43:01one of the other Maxes perhaps fared better.

0:43:07 > 0:43:12Infinite space has consequences impossible to comprehend.

0:43:12 > 0:43:18Infinitely many copies of you, identical in every possible way.

0:43:18 > 0:43:23Every molecule, every heartbeat every atom, every breath,

0:43:23 > 0:43:25every thought the same.

0:43:25 > 0:43:29Each one convinced that they're the real you.

0:43:29 > 0:43:31In an infinite universe, you're not unique,

0:43:31 > 0:43:34you're insignificant, you're nothing.

0:43:38 > 0:43:43And it turns out it's a relatively simple calculation to work out

0:43:43 > 0:43:48how far you would need to travel to meet your nearest doppelganger.

0:43:55 > 0:43:58Imagine a ridiculously simple universe which only has space for

0:43:58 > 0:44:04four particles, and only two kinds of particles, purple and yellow.

0:44:04 > 0:44:09Then there are only 16 ways this universe can be arranged.

0:44:09 > 0:44:14Two times two times two times two, 16 possible arrangements.

0:44:14 > 0:44:15Yellow, purple...

0:44:21 > 0:44:23Purple, purple...

0:44:23 > 0:44:24Almost done.

0:44:30 > 0:44:33The top ones are all purple.

0:44:33 > 0:44:37This means that if we arrange a 17th universe in some random way,

0:44:37 > 0:44:41like yellow, purple, purple, yellow,

0:44:41 > 0:44:46it has to be a copy of one of the existing universes. Let's see...

0:44:46 > 0:44:48This one.

0:44:48 > 0:44:52And this is true no matter how we arrange these. If we do this,

0:44:52 > 0:44:56then it's a copy of...this.

0:44:56 > 0:44:58In other words,

0:44:58 > 0:45:02this guarantees that the new universe here is a duplicate.

0:45:02 > 0:45:05And it's also easy to see that

0:45:05 > 0:45:08the distance from any one to its nearest copy

0:45:08 > 0:45:10would be about the size of this square.

0:45:10 > 0:45:12In our observable universe,

0:45:12 > 0:45:16there's obviously more than 16 ways to arrange all the particles, but it

0:45:16 > 0:45:18still comes out to be a finite number,

0:45:18 > 0:45:21so we can use basically the same calculation to figure out

0:45:21 > 0:45:23how far away we have to go

0:45:23 > 0:45:27to find an exact copy of Earth and an exact copy of me.

0:45:30 > 0:45:33'All you need to do is work out how many

0:45:33 > 0:45:38'subatomic particles it's possible to cram into the observable universe.

0:45:38 > 0:45:42'Calculate the number of possible configurations of those particles

0:45:42 > 0:45:46'and multiply that by the diameter of the observable universe.'

0:45:46 > 0:45:49On the pool table there were four balls,

0:45:49 > 0:45:56so there are two to the power four equals 16 possible arrangements.

0:45:56 > 0:45:59In our actual observable universe,

0:45:59 > 0:46:03we can put in up to ten to the 118 particles.

0:46:03 > 0:46:07That's a huge number, but to get the number of ways in which they can

0:46:07 > 0:46:12be arranged in our universe, we have to take about two to the power that.

0:46:12 > 0:46:18So two to the power ten to the power 118. A honking big number.

0:46:18 > 0:46:21Then to get to the nearest copy of our universe, we multiply by

0:46:21 > 0:46:24the size of our universe, which is ten to the power

0:46:24 > 0:46:2726 metres.

0:46:27 > 0:46:2910 to the 26 is a tiny number

0:46:29 > 0:46:31compared to this huge number here.

0:46:31 > 0:46:34So the bottom line is that if we go

0:46:34 > 0:46:39two to the power 10 to the power 118 metres away or so,

0:46:39 > 0:46:42we're going to find a perfect copy

0:46:42 > 0:46:45of our entire universe, of Earth and of me.

0:46:49 > 0:46:52I find this quite dizzying, frankly.

0:46:55 > 0:47:01'Two to the power ten to the power 118 metres is further than any human

0:47:01 > 0:47:04'could ever travel, but if the universe is truly infinite,

0:47:04 > 0:47:10'these exact replicas of your universe have to exist.'

0:47:20 > 0:47:24'While no-one likes the idea of space coming to an end,

0:47:24 > 0:47:29'the consequences of an infinite universe are even more bewildering.'

0:47:34 > 0:47:37No matter how much I study the field of cosmology and think about this,

0:47:37 > 0:47:39it still makes no sense to me

0:47:39 > 0:47:42that the universe is infinite and always has been infinite.

0:47:42 > 0:47:45I don't understand that. I don't pretend to understand that.

0:47:46 > 0:47:50The idea that there may be an infinite number of Earths,

0:47:50 > 0:47:51an infinite number of people

0:47:51 > 0:47:54sitting here having this exact talk that I'm having right now,

0:47:54 > 0:47:57that just, that doesn't compute in my brain.

0:47:59 > 0:48:03I prefer a finite universe because I can get my mind around that.

0:48:03 > 0:48:06It's the only universe that makes intuitive sense to me.

0:48:08 > 0:48:12'Many physicists believe space could be curved

0:48:12 > 0:48:15'or even folded back on itself.

0:48:15 > 0:48:18'In the same way, you could sail round the Earth for ever

0:48:18 > 0:48:22'if you kept on going in a straight line through space,

0:48:22 > 0:48:25'and could travel long and fast enough,

0:48:25 > 0:48:29'then you will arrive back where you started.

0:48:29 > 0:48:34'You don't need infinity to produce a universe that has no edge.'

0:48:36 > 0:48:38We're probably never going to know

0:48:38 > 0:48:42whether our universe is infinite or actually finite in size.

0:48:42 > 0:48:45It's something I would really like to know.

0:48:45 > 0:48:47People have wondered about it for millennia.

0:48:48 > 0:48:50The best we can say right now

0:48:50 > 0:48:53is the universe is extraordinarily large.

0:48:56 > 0:48:59'But cosmologists might finally be

0:48:59 > 0:49:01'on the verge of an answer to the question of

0:49:01 > 0:49:03'whether the universe IS infinite.

0:49:03 > 0:49:08'And the clue that's led them there comes from something that has been

0:49:08 > 0:49:13'part of your lives since the first moment you opened your eyes.

0:49:13 > 0:49:15'Light.'

0:49:17 > 0:49:21Light travels extraordinarily fast, but not infinitely fast.

0:49:21 > 0:49:25It travels about 300,000 kilometres every second.

0:49:27 > 0:49:31And the moon happens to be about 300,000 kilometres away.

0:49:31 > 0:49:36So it takes light about one second to make it from the moon to the Earth.

0:49:36 > 0:49:40You might say the moon is one light second distant from the Earth.

0:49:43 > 0:49:47'As you gaze out into space, you are looking back in time.

0:49:47 > 0:49:50'You see the moon as it was a second ago,

0:49:50 > 0:49:53'Jupiter as it was an hour ago

0:49:53 > 0:49:59'and your nearest galaxy 2.5 million years in the past.

0:49:59 > 0:50:02'And some things are just so far away, their light would take

0:50:02 > 0:50:08'longer than the age of the universe to reach the Earth.

0:50:08 > 0:50:11'The most distant light ever detected

0:50:11 > 0:50:14'is also the oldest.

0:50:14 > 0:50:21'It began its journey just 400,000 years after the Big Bang.'

0:50:26 > 0:50:32'13.7 billion years ago, the universe was born.'

0:50:37 > 0:50:4013.7 billion years.

0:50:40 > 0:50:44To infinity, it's nothing.

0:50:44 > 0:50:48Seems like yesterday.

0:50:48 > 0:50:52'According to Big Bang theory, after a second the universe

0:50:52 > 0:50:55'is ten thousand million degrees,

0:50:55 > 0:50:59'and the first atomic nuclei condense out of the fireball.'

0:50:59 > 0:51:02And darkness was on the face of the deep...

0:51:02 > 0:51:08'Around 400,000 years later, the first atoms form...

0:51:08 > 0:51:11'and light is released into the universe.'

0:51:11 > 0:51:14And God called the light day.

0:51:14 > 0:51:17And darkness he called night.

0:51:17 > 0:51:18In the evening and the morning...

0:51:21 > 0:51:24'That light is still with you today.

0:51:24 > 0:51:28'It's called the cosmic microwave background.'

0:51:34 > 0:51:36Eventually this will become Shakespeare,

0:51:36 > 0:51:40monkeys, and even you.

0:51:42 > 0:51:44..we may know the traitors and the truth...

0:51:46 > 0:51:48..passing through nature...

0:51:53 > 0:51:56'The cosmic microwave background

0:51:56 > 0:51:59'is a snapshot of the universe when it was just a baby.

0:51:59 > 0:52:01'Over billions of years,

0:52:01 > 0:52:04'the cooler, denser blue regions in this image

0:52:04 > 0:52:09'will collapse to produce stars and galaxies.'

0:52:09 > 0:52:12It is astonishing that just this little picture

0:52:12 > 0:52:16can tell us something potentially about infinity.

0:52:16 > 0:52:19To think that we might get that sort of insight into the universe

0:52:19 > 0:52:20is pretty spectacular.

0:52:23 > 0:52:27'This image of the early cosmos might now reveal whether

0:52:27 > 0:52:33'the universe is infinite, because hidden within it lies a mystery,

0:52:33 > 0:52:37'something the Big Bang model alone couldn't answer.'

0:52:37 > 0:52:40What we actually see when we're looking at this

0:52:40 > 0:52:42map of the microwave background radiation

0:52:42 > 0:52:44is that it's showing us what temperature

0:52:44 > 0:52:47this radiation has all over the sky.

0:52:47 > 0:52:49So this is an image of the whole sky.

0:52:49 > 0:52:52And if you look at this red splotch versus that blue splotch,

0:52:52 > 0:52:56there's only a difference of about a ten thousandth of a degree

0:52:56 > 0:52:59so, in fact, this microwave background radiation

0:52:59 > 0:53:01is incredibly uniform on the sky.

0:53:03 > 0:53:04Now, for those two to be so similar,

0:53:04 > 0:53:07it seems that some sort of physical process,

0:53:07 > 0:53:09some sort of agreement should have taken place,

0:53:09 > 0:53:11and this is a rather baffling mystery.

0:53:11 > 0:53:12How did they do it,

0:53:12 > 0:53:14how did they come to this agreement in their temperature?

0:53:14 > 0:53:19This was a real enigma for the standard Big Bang cosmology.

0:53:19 > 0:53:21It was a real puzzle.

0:53:23 > 0:53:26'In the Big Bang model,

0:53:26 > 0:53:29'there was no way to explain how distant parts of the universe

0:53:29 > 0:53:32'could have such similar temperatures.

0:53:34 > 0:53:38'To make sense of it, physicists needed something else.

0:53:38 > 0:53:41'A new theory of the early universe.

0:53:41 > 0:53:44'A theory they called inflation.'

0:53:53 > 0:53:57'Inflation says the early universe expanded much faster and further

0:53:57 > 0:53:59'than previously thought,

0:53:59 > 0:54:03'a million, million, million, million times, in less than

0:54:03 > 0:54:07'a billionth of a billionth of a second.'

0:54:07 > 0:54:10If I faint here, I'm going to sue you!

0:54:10 > 0:54:12'This explains the uniform temperature

0:54:12 > 0:54:17'of the cosmic microwave background because everything you see

0:54:17 > 0:54:23'was stretched out from a small and uniform part of the whole universe.'

0:54:26 > 0:54:32Now, just imagine I did all this in ten to the power minus 32 seconds.

0:54:32 > 0:54:34That would be inflation.

0:54:34 > 0:54:36Inflation was devised as

0:54:36 > 0:54:40a way to explain the finite observable universe,

0:54:40 > 0:54:43and it does a very nice job of doing that, but it has a sort of

0:54:43 > 0:54:45side effect or a very interesting property

0:54:45 > 0:54:48that once you get inflation started, it just keeps going.

0:54:48 > 0:54:53It takes on a life of its own, like a genie you've let out of the bottle.

0:54:55 > 0:54:59The theory of cosmological inflation

0:54:59 > 0:55:02actually produces an infinite universe.

0:55:05 > 0:55:09'Inflation predicts your universe never stops expanding

0:55:09 > 0:55:14'and may, in fact, be infinite.

0:55:14 > 0:55:18'But following the mathematics to its logical conclusion

0:55:18 > 0:55:21'predicts an even more disturbing outcome.

0:55:21 > 0:55:25'Your infinite universe might not be the only one.'

0:55:27 > 0:55:30We used to think that inflation only gave us one Big Bang

0:55:30 > 0:55:33and one infinite space, but now

0:55:33 > 0:55:35it's becoming clear that actually it never stops

0:55:35 > 0:55:39and instead gives us an infinite number of infinite spaces.

0:55:41 > 0:55:46There are multiple universes, infinitely many multiple universes,

0:55:46 > 0:55:49infinitely many infinite universes even.

0:55:49 > 0:55:54'Inflation was devised to explain the finite observable universe,

0:55:54 > 0:55:57'and it does a very nice job of that...'

0:55:57 > 0:56:01WORDS ECHO

0:56:06 > 0:56:12'In an infinite universe, anything possible happens all the time.

0:56:12 > 0:56:18'But with infinite universes, the impossible is happening right now.

0:56:18 > 0:56:21'Because in some of those universes,

0:56:21 > 0:56:25'the laws of physics that govern your world simply don't apply.'

0:56:29 > 0:56:33What isn't appreciated by many even in the physics community

0:56:33 > 0:56:37is that this model of these infinitely many infinite universes

0:56:37 > 0:56:40is actually probably our current best bet

0:56:40 > 0:56:43as to what the real universe looks like.

0:56:45 > 0:56:48It's baffling and it's mind bending,

0:56:48 > 0:56:52but that's where our road of cosmology has taken us,

0:56:52 > 0:56:56to this confrontation with real infinity.

0:56:58 > 0:57:01I think we should expect us not to be able to intuitively grasp

0:57:01 > 0:57:05the ultimate nature of space and everything because we have intuition

0:57:05 > 0:57:09only for the things which were useful for our ancestors to understand.

0:57:09 > 0:57:13And we shouldn't expect our intuition to work when we ask

0:57:13 > 0:57:16really deep questions about the ultimate nature of reality.

0:57:16 > 0:57:18If one of our ancestors spent too much time thinking about

0:57:18 > 0:57:20what's outside of space, you know,

0:57:20 > 0:57:23they wouldn't have noticed that there was a tiger sneaking up

0:57:23 > 0:57:27from behind and they would have been cleaned right out of the gene pool.

0:57:28 > 0:57:31So it's very important for us scientists

0:57:31 > 0:57:35to not diss ideas just because they feel weird.

0:57:35 > 0:57:37Fortunately, our math doesn't have any inhibitions

0:57:37 > 0:57:39and we could still calculate all these things,

0:57:39 > 0:57:42even if they seem completely counter-intuitive and it's only

0:57:42 > 0:57:46through the math that we're able to actually deal with all these ideas.

0:57:49 > 0:57:54'Counting has led you to an infinite mathematical world of infinities,

0:57:54 > 0:57:58'each infinitely larger than the last.

0:57:59 > 0:58:03'And gazing out into the furthest depths of space,

0:58:03 > 0:58:10'some see an infinite universe, itself just one of infinitely many.'

0:58:16 > 0:58:20Infinity is a big topic. I don't think it's going to be

0:58:20 > 0:58:23understood fully in any finite period of time.

0:58:24 > 0:58:27We have a hint of just how rich that realm is,

0:58:27 > 0:58:31but we haven't understood the smallest fraction of it.

0:58:34 > 0:58:37This, of course, is because, by its very nature,

0:58:37 > 0:58:41the subject of infinity is a vast and infinite subject.

0:58:43 > 0:58:47CLOCK TICKS

0:58:55 > 0:58:57MONKEY SCREECHES