# Horizon (1964) s46e12 Episode Script

### To Infinity and Beyond...

Zero, one, two, three, four, five
I've seen things you
people wouldn't believe.

Things that would change how you see this world.

Enough to drive men to madness.

DIFFERENT SPEAKERS COUN What is the biggest number? Is the universe infinite? Might every event repeat again and again and again and again Your intuition is no use here.

Faith alone can't save you.

How did the universe begin? Is the Earth just one of uncountable copies, tumbling through an unending void? On one you are rich, on another you have yet to be born.

These are the deepest mysteries of the universe.

Ladies and gentlemen, pray silence as I present to you .

.

infinity.

And so on.

Planet Earth is so beautiful and so complex, humans can barely comprehend it.

And yet humanity has always asked, "What's over the horizon? What lies beyond the stars? "Is this it?" I think infinity is one of those things that is an essential mystery of the universe.

What happens after we die, why are we here, why were we born.

Infinity is in that class of questions and humans have been thinking about whether there's an end to the world or whether the world goes on forever, probably since the beginning of human thought.

It's a natural human impulse to want to go beyond any boundary.

It is simultaneously scary and exciting to think about.

If infinity is real, it has implications far beyond the world of science.

It strikes at the very heart of what it means to be you.

There is actually, far out in space, a planet that looks just like Earth with people just like us.

Some will be doing exactly the same things as we do, even with the same names and memories as us.

No matter how much I study the field of cosmology and think about this, it still makes no sense to me that the universe is infinite.

I prefer a finite universe because I can get my mind around that.

It's the only universe that makes intuitive sense to me.

Infinity.

Impossible to comprehend.

And yet it comes from something so simple, a child can understand it.

One, two, three four, five, six seven, eight, nine 10, 11, 12, 13, 14 15, 16, 17 18, 19, 21, 22, 23 I was very proud first to be able count to five, then to ten, and then I realised that you can always keep counting and there's no end to it.

37, 38, 39 So I had this obvious intuition that everybody had that there is no end to counting hence there must be infinity.

91, 92, 93, 94, 95, 96 1,374, 1,375, 1,376, 1,377 Numbers can get so vast, it's impossible to imagine.

1,380, 1,381, 1,382 1,2763, 1,2764, 1,2765, 1,2766 To count to a billion, it would take you about 30 years and to count to a trillion is not something you could even do in human history.

111,330, 111,331, 111,332 1,372,365, 1,372,366 Billion and nine, billion and ten, billion and eleven, billion and twelve, For most people, I suppose the biggest number they're likely to meet will be somewhere in the billions or maybe the trillions, which might be something like the budget deficit or military spending or something like that.

Mathematicians tend to use bigger numbers than that.

Googolplexplexplex three, googolplexplexplexplex four You made me do this! Googolplexplexplex five, googolplexplexplex six When I get to 199, then that would be too hard and I would have to stop.

TYPEWRITER KEYS CLICK One of the largest numbers we have a name for is a googol and it's one followed by 100 zeros.

A hundred zeros is a lot because each zero represents another factor of ten.

So, it's a big number.

You might be thinking 100 zeros, isn't that many.

But a googol is far bigger than the number of atoms in a human being, more than the number of atoms that make up planet Earth.

A hundred zeros is more even than all the atoms in the entire observable universe.

Sets your imagination going, doesn't it? A googol sets your imagination going.

It knocks you off of your chair.

It's already a number that's probably bigger than anything, that makes sense in our experience.

But, it's a very, very tiny, tiny, tiny large number.

A Googol itself was only a stepping stone on the way to a much much larger number called a googolplex.

A googolplex is ten raised to the power of a googol, that is it's one followed by a googol of zeros.

And of course it's just not possible to imagine the size of a number like that.

A googol has 100 zeros, but a googolplex has so many zeros that there's not enough space in the entire observable universe just to write the number down, even if you could write each zero on a single atom.

But from my perspective, these are all very, very small.

One of the biggest numbers ever used in mathematics is many times the size of a googolplex.

It makes normal numbers like a trillion or a billion disappear into practically nothing.

It's a number called Graham.

Graham's number is much much bigger than a googolplex.

In fact, it's as large relatively to a googolplex as a googolplex is to the number ten.

In fact, it's much much bigger than that.

Graham's number was discovered in the 1970s by mathematician and former circus performer, Ron Graham.

I don't know too many other people who have a number.

Er It's It's not bad.

It's not bad.

I mean I recommend it! Graham's number is so big, it even made it into the Guinness Book Of Records.

Oh, yes.

This is the 1980 edition of Guinness Book Of World Records and I think if we turn to page, er, what is it, 192? Numeration.

Yes, and then numbers.

Prime numbers, perfect, highest numbers.

Here we go, highest numbers.

"The highest number ever used in a mathematical proof "is a bounding value published in 1977.

It is known as Graham's number.

"It concerns bichromatic hypercubes and is inexpressible without the special arrow notation.

" It's really gigantic, I mean, it's just so large you can't compare it with anything you would normally associate large numbers, like the number of atoms in the universe or how many inches to the furthest galaxy or something like this, it's just way bigger than that.

So vast is Graham's number, nobody knows how many digits it even has, including Ron Graham himself.

In spite of the fact that it's Graham's number, and I'm Graham, er, I know very little about it.

I have no idea what the first digit is.

It has one.

I have no idea what it is.

Maybe no-one will ever know what that digit is.

Are there more zeros than ones in the number? Who knows? If you look at three to the three to the three to the n plus one Ron didn't just make up his number.

It's the upper limit to the solution of a pure mathematics problem concerning multi-dimensional cubes.

That's divisible by five, only if the exponent three to the three to the N plus one While the problem itself is abstract, the methods Ron used to solve it are now used to keep track of data sent across the internet.

N plus one, minus three to the three to the N, minus one.

OK.

Great.

So Just working out the last digit of Graham's number is a lengthy calculation.

RON GRAHAM'S WORDS ECHO You can't really comprehend how large it is.

.

.

Very large.

I don't think anybody can know that But like all finite numbers, it comes to an end In the last stage, we have three to a certain power Eventually.

RON GRAHAM'S WORDS ECHO So that means that the remainder, when you divide by ten, is always seven.

In other words, you can finally conclude that the last digit of Graham's number is seven.

End of the story! Graham's number is not really any closer to infinity than the number one.

You didn't really get started yet, even though you took a lot of steps to get to Graham's number, it takes so many more, infinitely more to get to infinity.

Infinity is just out there.

It's just a different beast.

What's the biggest number? Erm 120? Ten.

Is ten the biggest number? Sometimes people just say, um, seventy hundred eighty nine hundred, but that's not even a number.

Oh, dear.

Oh, that's difficult.

Well, I suppose there isn't really one, they just go on and on and on.

There is no biggest number, because if there were, you could always add one to it.

TYPEWRITER KEYS CLICK Unlike normal numbers, infinity never comes to an end.

And that gives it some very strange properties.

OK, so this is one of the first things that you have to think about if you're thinking about infinity.

You've got all the numbers - one, two, three, four, five, six.

Let's have just a few.

And they go on forever.

Now suppose I went through and picked out just the even numbers.

Two and four, six, eight, ten.

I'll write them down here.

Two, four, six, eight, ten.

Well, it's pretty obvious, that there are more of these than there are of these? We've picked out half of these numbers here.

Well, in fact it's not.

These two lists are exactly the same size.

And this is the first real paradox about infinity.

There are as many even, though there are half as many, so half of this list has as many things in it as the whole list does.

These sets look so different, but they're actually the same size.

How's that possible? Well, OK.

How's that possible? Well, you see, what are we doing, we're counting and when you count something you match it up with the numbers.

So if I'm counting my sheep, for example, I count one, two, three, four, I match the sheep up with the numbers.

But you see, I can match the even numbers up with the numbers.

I've already started doing it.

Here's one matched with two, two matched with four, three matched with six, four matched with eight, five matched with ten, and that goes on forever, so I've matched all of the even numbers up with all of the numbers, no gaps, perfect matching between the two.

So there are the same number of these as there are of these.

So this really is a characteristic property of infinity and it seems puzzling, but there it is.

This is why infinity is a bit of a hard thing to deal with.

And it troubled people for quite some time.

Two infinite lists are exactly the same size, even though one appears to contain twice as many numbers as the other.

Which is just the start.

The more mathematicians thought about infinity, the weirder it became.

How are you? I'm fine, thank you.

Can I check in, please? Around 100 years ago, one mathematician tried to explain some of infinity's strange properties by imagining arriving at a hotel with infinite rooms.

He wondered if there would be any space for him, even if it was fully booked? Now in an ordinary hotel, I'd be told, "I'm sorry we're full up.

You'll have to go somewhere else.

" But in an infinite hotel, things are rather different.

There's no problem at all.

RECEPTION BELL RINGS The infinite hotel was dreamt up by David Hilbert, one of the most influential mathematicians of the early 20th century.

Your room is upstairs.

Have a lovely stay with us.

Bye-bye.

The manager can't just put me into the last room, because there is no last room.

It's an infinite hotel.

The rooms go on forever.

There is no last room.

And all the rooms are full anyway so, even if there were a last room, it would have somebody in it.

But it's exactly for that reason that it's possible to find room for me.

All you have to do is shift the guest in room one to room two, the guest from room two to room three and so on down the line.

Because there's no last room, every guest has a next room to go to and that frees up room one, so I can stay in room one.

So everything's fine.

It turns out, even if the hotel was packed to the rafters, a room can always be found.

Infinity plus one is infinity.

RECEPTION BELL RINGS Good morning, sir.

How are you? I'm very well, thank you.

Can I check in, please? You can see that if two guests came in, infinity plus two is infinity.

But suppose I came along to this hotel with infinitely many of my friends and we all wanted to stay and the hotel was full, how could we do it? What we could do is arrange that the guests in room one moved into room two and the guest in room two moved into room four, each guest moved into the room with double the number.

So all the even-numbered rooms have now got people in them and all the odd numbered rooms are now free, so me and my friends can all stay in the odd-numbered rooms.

So this shows that infinity plus infinity is still infinity.

And it would make sense that the same rules also apply to subtraction.

But if you think infinity will stay the same whatever you add or subtract, then think again.

Suppose that in the morning, all the guests left.

The number of occupied rooms would be infinity minus infinity, but it would be zero.

So infinity minus infinity could be zero.

Or it could be one.

If I stayed on and all the other guests left, infinity minus infinity would be one in that case.

So there's no definite answer.

That's why you have to be very careful dealing with infinity.

It's a very slippery concept.

What if I and two other people stayed, say? Then infinity minus infinity would be three.

So it could be anything you like.

RECEPTION BELL RINGS Good morning, sir.

Good morning.

How are you? I'm fine, thank you.

Can I help you? Yes.

Can I check in, please? HE LAUGHS Oh, God! HE LAUGHS This is one of the reasons why you have to very careful dealing with a slippery character like infinity.

Not to be trusted.

Is infinity real? Well, not to most people, but you can't do mathematics without infinity.

So if you don't feel comfortable with that then you're probably not going to become a mathematician.

Get any mathematician to tell you about what he or she is doing at the moment, and their imagination will somehow be full of this idea of infinity.

Infinity is like a landscape in which you work.

a place in which you do mathematics.

It's not a real place.

You can't actually go there, except in your imagination.

But to those who do mathematics, it seems very real indeed.

But you see one of the problems with infinity is that it has some paradoxical properties and very basic questions about infinity that we can't answer.

So you do have to be a little careful.

It's like having a polar bear as a pet, you've grown up together, he's a wonderful pet, he's big, he's fast, he plays in the snow beautifully, but there's always the chance that one day he'll get annoyed with you and bite off your head.

So, we are playing with fire, I think.

The paradoxes associated with infinity make some mathematicians uncomfortable.

Not least Professor Doron Zeilberger.

I first came across infinity like everybody else in a very early childhood when you start counting.

First you count to three, then to four, then to five then to ten, then to 100 and eventually you realise that you can keep counting forever.

Hence there is an infinity.

341, 342, 343 63,789, 63,790, 63 etc, etc, ad infinitum.

But I don't think I ever liked it.

I always found something repulsive about it.

TYPEWRITER KEYS CLICK I prefer finite mathematics much more to infinite mathematics.

I think that it's much more natural much more appealing and the theory is much more beautiful.

It is very concrete.

It's something you can touch, something you can feel, something you can relate to.

Infinite mathematics, to me, is meaningless because it's like abstract nonsense.

In my opinion, infinity is only a fiction of the human mind.

But not believing in infinity leaves Professor Zeilberger with a problem.

If the numbers don't go on forever, where do they end? When you start counting, you seemingly can go forever, but eventually you will reach the biggest number and then when you add one to it you go back to zero.

You go back to zero? Yeah.

How is that possible? How is it not possible? Have you ever been there? The biggest number is much bigger than anybody can ever think of.

It's bigger than a googol, bigger than a googolplex, bigger than a googolplex to the power of a googolplex.

It's so big, we can never envision it.

Nevertheless there is a biggest number and if you keep counting after that big number, we get back to zero.

Like when we walk around our planet, if you keep walking, eventually we get back to the place we started.

And if you think that this is ridiculous.

Look at the alternative.

It's less ridiculous than infinity and all the paradoxes that go with it.

Infinity may or may not exist, God may or may not exist but in mathematics there should not be any place for neither infinity nor God.

Doron isn't the first person to feel that the infinite is an illusion.

Until recently, infinity was too wild to be tamed by mathematics.

Too unpredictable to be used in equations.

Aristotle believed counting could go on forever and that the universe was eternal.

But he refused to accept that the universe was infinite in size.

He believed the Earth was at the centre of the universe.

However, without an end, there can be no middle so he banned infinity from his mathematics.

Infinity was discussed by philosophers and priests rather than mere mathematicians.

The infinite was something closer to a god than a number.

In 1600, philosopher Giordano Bruno claimed not only that the universe was infinite, but there would be many other Earths orbiting stars just like our own sun.

His beliefs went down so poorly with the Catholic Church, they had him burnt at the stake.

Only God himself, could be truly infinite.

HE SIGHS In the mid 19th century, one man, Gregor Cantor, made infinity truly part of mathematics.

Something that could be used in equations as if it were a number.

Cantor's idea was that we can gather together maybe even infinitely many things into a single set, like putting them into a bag and think of it as just a single object.

And once it is a single object, we can then put it into mathematics, do calculations with it, because it's just one object.

We don't have to know that there are infinitely many things in the bag, it's just a single object for us.

But making infinity part of mathematics produced one surprising result.

Some infinities are bigger than others.

Cantor's great initial discovery was that the infinity of the decimal numbers was larger than the infinity of the counting numbers.

And he did this by what's now called Cantor's diagonal argument.

So what you have to show is that in any list of decimal numbers, there's a decimal number that's not on that list.

And what we can do is go down the diagonal of this list, so we might take the nine here, and the one here and the one here.

We're going down the diagonal and we're generating another decimal number.

Nine, one, one and so forth.

Now we take this number and we change it.

We change the nine to an eight, say, and we can change the ones to twos and so forth.

And now we've generated a decimal number, which can't be on this list.

It can't be the first one, because the nine has been changed to an eight.

It can't be the second one, because in the second position, it's got a two instead of a one and so forth.

And all the way down the list, this number will be different from the decimal number on the list.

So what does that mean? It means there can be no matching of all the decimal numbers by the counting numbers and that tells us that the infinity of the decimal numbers is larger than the infinity of the counting numbers.

There's no largest infinite number.

For every infinite number, there's a bigger one.

There are infinities beyond infinities and that's what we study.

In making infinity part of mathematics, Cantor had uncovered a whole universe of infinities, each infinitely bigger than the last.

And that wasn't easy for many to accept.

At the time, I think that was a big surprise.

No-one had really thought carefully about whether infinities could come in different sizes, after all infinity is infinity.

How can you have different sizes of infinity? Cantor's breakthrough came at a price.

Cantor ended his days in an asylum.

Was it infinity that drove him there? Who knows? Who can tell? He faced a lot of opposition from his colleagues and it was possibly that, more than thinking about infinity itself that was the trouble.

It was only later that mathematicians accepted and welcomed his theories into the body of mathematics.

Cantor's work is absolutely fundamental to everything we do.

Today, Cantor's infinities are part of mainstream mathematics.

And the truth is, even those who would rather infinity didn't exist, use it in their equations everyday.

Infinity is simpler and quicker to manipulate than large finite numbers.

Most mathematicians have made an uneasy peace with infinity and accepted it as part of their universe.

Some have devoted their careers to studying it.

To the person who wants to deny infinity and say it doesn't exist, I don't see how that view enriches their world.

I feel sorry for them.

I mean, infinity, maybe it doesn't exist, but it is a beautiful subject.

I could say the stars don't exist and stay inside, or always look down, but then I don't see the beauty of the stars.

And until one has a real reason to doubt the existence of mathematical infinity, I just don't see the point.

There's whole world of infinities out there.

Maybe they're real, maybe they're not.

But could infinity be part of the world you call real? As yet unseen through any microscope and undetected by any telescope, might the heavens genuinely be unbounded and the depths of space deeper than love itself? Is space infinitely big or simply unimaginably big? TYPEWRITER KEYS CLICK The sky never ends, so I think space never ends, because there isn't like a wall all around our all around our whole planet.

I actually think the universe IS infinite and if I had to put odds on it, I would say there's a 95% chance that it is infinite.

I think space is very, very, very, very, very, very big.

I would say 1,000 metres.

That big.

I think the universe is infinite on Mondays, Wednesdays and Fridays and it's finite the rest of the week.

I'm having a very, very hard time making my mind up.

Space.

It hasn't got laws or electricity.

It's just like a sky.

It won't finish.

It will keep on going.

And going and it likenever stops.

The fundamental issue that most people come up when you say the universe may be finite, is simply what's outside of it? What happens when you come to the edge? Can't you go beyond it? And so this leads some people to the conclusion that the universe has to be infinite.

It might seem obvious that space goes on forever.

Innocent even.

But unleash infinity into the universe and all bets are off.

He makes the extraordinary mundane and the unbelievable, inevitable.

He makes the extraordinary mundane and the unbelievable, inevitable.

In an infinite universe, anything that's possible has to happen.

Even something as unlikely as a monkey typing the complete works of Shakespeare.

TYPEWRITER KEYS CLICK "My bounty is as boundless as the sea.

"My love is deep.

The more I give to thee, the more I have.

"For both are infinite.

" If we imagine this monkey, all it's doing is thumping away at the keys completely at random.

The monkeys don't have to evolve, they don't have to be able to read Shakespeare, they do have to be able to carry on typing, but that's all, just typing at random.

"I could be bounded in a nut shell and count myself a king of infinite space, "were it not that I have bad dreams.

" To test the infinite monkey theorem, a computer was placed in the enclosure of a Cambridge University professor.

Typing out the complete works of Shakespeare at random is really going to be a big job.

It's a thick book.

There's 37 plays in here, all the poems and sonnets, there's 884,429 words, every word has to be in exactly the right place, every character has to be exactly in sequence including the spaces in between.

And so doing that at random, bashing away on a keyboard is a difficult thing to do.

For the last week, David Spiegelhalter's computer has been randomly generating letters.

We're only generating lower case at the moment we're not using capitals so we're giving it a bit of a chance like that.

And it's generating them at the rate of 50 characters a second.

And, as you can see, it's keeping on finding matches all the time.

If it finds a match of four letters, four characters, it adds another character on, a random character, and sees if it's found a match for five characters and so on and so on.

The programme's been running for more than a week now and in that time, it's managed to generate more than 34 million characters.

If we assume that this monkey can type one character a second, it would have taken 34 million seconds, which is just over a year's typing for our monkey, but we've got to be kind and give it some breaks, so I would say just over two years probably typing.

The longest match so far is eight letters and here's the string, "We space lover.

" This occurs once in the complete works.

It occurs in Love's Labour's Lost, Act Two, Scene One.

So, it's in here Somewhere I've no idea where! It's not even in alphabetical order.

Ah, Love's Labour's Lost, here we are.

Act Two, Scene One.

Yep, I've got it.

Yeah, there we are, Boyet says, "With that which we lovers entitle affected.

" So the actual word is "we lovers" and that's where "we lover" fits into it.

But eight letters doesn't seem like very much.

No, it doesn't, but you have to think of just how unlikely it is to generate the exact works of Shakespeare.

So we did some calculations and worked out if you wanted 17 characters, which is, "To be or not to b" Not even the whole, the whole phrase.

We'd have had to set this going at about the time of the big bang around 14 billion years ago.

And that's to get something just twice that length.

But remember, we've got to get five million characters, all in the right order.

We can calculate the chance of this happening as one in a very large number.

It's ten with about nine million zeros written after it.

So that's an incredibly tiny probability very, very small.

It's an unbelievably unlikely thing to occur.

So if you imagine the current National Lottery, it would be like someone winning every single time, time and again, every single week for year after year, for 29,000 years.

Same person.

The same person buying their ticket and winning every week for 29,000 years.

But if we have an infinite amount of time, we can be certain that it will happen.

And not just once, it's going to happen again and again and again.

Because infinity is such a long time that everything, no matter how unlikely, as long as it's possible, will occur.

Infinity is so vast, one monkey, randomly typing forever could easily get the job done.

TYPEWRITER KEYS CLICK If he had an infinite amount of time, the monkey would produce far more than Shakespeare.

He'd produce every book that's ever been written.

Everything from the telephone directory to the latest celebrity autobiography.

TYPEWRITER KEYS CLICK But in an infinite universe, there will be infinite number of monkeys.

And that means, somewhere, one of them is typing Shakespeare right now.

TYPEWRITER KEYS CLICK If the universe were infinite, it seems fairly simple and benign, but it has some really strange consequences.

if we look far enough away there would be regions like the one that we're in, there would be a room out there like the one that we're sitting in now.

There would be Earths out there just like ours, except maybe the Roman empire would still exist or Germany would have won the war and on a personal level out there right now would be copies of Anthony giving interviews in a pink jumpsuit or rotting in jail or filthy rich.

There would be every possible combination.

Every way your life could have gone, there is somebody else just like you leading that life.

In fact, anything that we can really conceive of anything that's physically possible will happen, not only that it will happen an infinite number of times.

They're all out there in this infinite universe.

So whether you look at this as a good thing or a bad thing depends a little bit on how good you life is right now, but there really out there.

Which is crazy.

It means that there is actually far out in space, a planet that looks just like Earth, with people just like us, some will be doing exactly the same things as we do, even with the same names and memories as us.

It feels a little bit spooky to know that there are all these other copies of me .

.

but it takes a little bit of the pressure off getting things right all the time, knowing that when I screw up, one of the other Maxes perhaps fared better.

Infinite space has consequences it's impossible to comprehend.

Infinitely many copies of you, identical in every possible way.

Every molecule, every heartbeat every atom, every breath, every thought, the same.

Each one convinced that they're the real you.

In an infinite universe you're not unique, you're insignificant, you're nothing.

And, it turns out it's a relatively simple calculation to work out how far you would need to travel to meet your nearest doppelganger.

Imagine a ridiculously simple universe which only has space for four particles and only two kinds of particles, purple and yellow.

Then there are only 16 ways this universe can be arranged two times two times two times two, 16 possible arrangements.

Yellow, purple Purple, purple Almost done.

The top ones are all purple.

This means that if we arrange a 17th universe in some random way, like yellow, purple, purple, yellow, it has to be a copy of one of the existing universes.

Let's see This one.

And this is true no matter how we arrange these.

If we do this, then it's a copy ofthis.

In other words, this guarantees that the new universe here is a duplicate.

And it's also easy to see that the distance from any one to it's nearest copy would be about the size of this square.

In our observable universe, there's obviously more than 16 ways to arrange all the particles, but it still comes out to be a finite number, so we can use basically the same calculation to figure out how far away we have to go to find an exact copy of Earth and an exact copy of me.

All you need to do is work out how many subatomic particles it's possible to cram into the observable universe.

Calculate the number of possible configurations of those particles and multiply that by the diameter of that observable universe.

On the pool table there were four balls, so there are two to the power four equals 16 possible arrangements.

In our actual observable universe, we can put in up to ten to the 118 particles.

That's a huge number, but to get the number of ways in which they can be arranged in our universe, we have to take about two to the power that.

So two to the power ten to the power 118.

A honking big number.

Then to get to the nearest copy of our universe, we multiply by the size of our universe, which is ten to the power 26 metres.

10 to the 26 is tiny compared to this number here.

so the bottom line is that if we go two to the power 10 to the power 118 metres away or so, we're going to find a perfect copy of our entire universe, of Earth and of me.

I find this quite dizzying, frankly.

Two to the power ten to the power 118 metres is further than any human could ever travel, but if the universe is truly infinite, these exact replicas of your universe have to exist.

While no-one likes the idea of space coming to an end, the consequences of an infinite universe are even more bewildering.

No matter how much I study the field of cosmology and think about this, it still makes no sense to me that the universe is infinite and always has been infinite.

I don't understand that.

I don't pretend to understand that.

The idea that there may be an infinite number of Earths, an infinite number of people having this exact talk that I'm having right now, that just, that doesn't compute in my brain.

I prefer a finite universe because I can get my mind around that.

It's the only universe that makes intuitive sense to me.

Many physicists believe space could be curved or even folded back on itself.

In the same way, you could sail round the Earth forever, if you kept on going in a straight line through space, and could travel long and fast enough, then you will arrive back where you started.

You don't need infinity to produce a universe that has no edge.

We're probably never going to know whether our universe is infinite or finite.

It's something I'd really like to know.

People have wondered about it for millennia.

The best we can say right now is the universe is extraordinarily large.

But cosmologists might finally be on the verge of an answer to the question of whether the universe IS infinite.

And the clue that's led them there comes from something that has been part of your lives since the first moment you opened your eyes.

Light.

Light travels extraordinarily fast, but not infinitely fast.

It travels about 300,000 kilometres every second.

And the Moon happens to be about 300,000 kilometres away.

So it takes light about one second to make it from the Moon to the Earth.

You might say the Moon is one light second distant from the Earth.

As you gaze out into space, you are looking back in time.

You see the Moon as it was a second ago, Jupiter as it was an hour ago and your nearest galaxy, 2.

5 million years in the past.

And some things are just so far away, their light would take longer than the age of the universe to reach the Earth.

The most distant light ever detected is also the oldest.

It began its journey just 400,000 years after the big bang.

13.

7 billion years ago, the universe was born.

13.

7 billion years.

To infinity, it's nothing.

Seems like yesterday.

According to Big Bang theory, after a second the universe is ten thousand million degrees and the first atomic nuclei condense out of the fireball.

And darkness was on the face of the deep Around 400,000 years later, the first atoms form and light is released into the universe.

And God called the light day.

And darkness he called night.

In the evening and the morning That light is still with you today.

It's called the cosmic microwave background.

Eventually this will become Shakespeare, Monkeys and even you.

.

.

we may know the traitors and the truth .

.

passing through nature The cosmic microwave background is a snapshot of the universe when it was just a baby.

Over billions of years, the cooler, denser blue regions in this image will collapse to produce stars and galaxies.

It is astonishing that just this little picture can tell us something potentially about infinity.

To think that we might get that sort of insight into the universe is pretty spectacular.

This image of the early cosmos might now reveal whether the universe is infinite, because hidden within it lies a mystery, something the big bang model alone couldn't answer.

What we see when we're looking at this map of the microwave background radiation is that it's showing us what temperature this radiation has all over the sky.

So this is an image of the whole sky.

If you look at this red splotch versus that blue splotch there's only a difference of about a ten thousandth of a degree so in fact, this microwave background radiation is incredibly uniform.

Now for those two to be so similar, it seems that some sort of physical process, some sort of agreement should have taken place and this is a rather baffling mystery.

How did they do it, how did they come to this agreement in their temperature? This was a real enigma for the standard big bang cosmology.

It was a real puzzle.

In the big bang model, there was no way to explain how distant parts of the universe could have such similar temperatures.

To make sense of it, physicists needed something else.

A new theory of the early universe.

A theory they called inflation.

HE INFLATES GLOBE Inflation says the early universe expanded much faster and further than previously thought, a million, million, million times, in less than a billionth of a billionth of a second.

If I faint, I'll sue you! This explains the uniform temperature of the cosmic microwave background because everything you see was stretched out from a small and uniform part of the whole universe.

Now just imagine I did all this in ten to the power minus 32 seconds.

That would be inflation.

Inflation was devised to explain the finite observable universe and it does a very nice job of doing that, but it has a sort of side effect or a very interesting property that once you get inflation started, it just keeps going.

It takes on a life of its own, like a genie you've let out of the bottle.

The theory of cosmological inflation actually produces an infinite universe.

Inflation predicts your universe never stops expanding and may, in fact, be infinite.

But following the mathematics to its logical conclusion predicts an even more disturbing outcome.

Your infinite universe might not be the only one.

We used to think that inflation only gave us one big bang and one infinite space, but now it's becoming clear that actually it never stops and instead gives us an infinite number of infinite spaces.

There are multiple universes, infinitely many multiple universes, infinitely many infinite universes even.

Inflation was devised to explain the finite observable universe, and it does a very nice job of that WORDS ECHO In an infinite universe, anything possible happens all the time.

But with infinite universes, the impossible is happening right now.

Because in some of those universes, the laws of physics that govern your world simply don't apply.

What isn't appreciated by many even in the physics community is that this model of these infinitely many infinite universes is actually probably our current best bet as to what the real universe looks like.

It's baffling and it's mind bending but that's where our road of cosmology has taken us, to this confrontation with real infinity.

I think we should expect us not to be able to intuitively grasp the ultimate nature of space and everything because we have intuition only for the things which were useful for our ancestors.

And we shouldn't expect our intuition to work for really big questions about the ultimate nature of reality.

If one of our ancestors spent too much time thinking about what's outside space, you know, they wouldn't have noticed that there was a tiger sneaking up from behind and they would have been cleaned right out of the gene pool.

So it's very important for us scientists to not diss ideas just because they feel weird.

Fortunately, our math doesn't have any inhibitions and we could still calculate all these things even if they seem completely counter intuitive and it's only through the math that we're able to actually deal with all these ideas.

Counting has led you to an infinite mathematical world of infinities, each infinitely larger than the last.

And gazing out into the furthest depths of space, some see an infinite universe, itself just one of infinitely many.

Infinity is a big topic I don't think it's gonna be understood fully in any finite period of time.

We have a hint of just how rich that realm is, but we haven't understood the smallest fraction of it.

This, of course, is because, by its very nature, the subject of infinity is a vast and infinite subject.

CLOCK TICKS MONKEY SCREECHES

Things that would change how you see this world.

Enough to drive men to madness.

DIFFERENT SPEAKERS COUN What is the biggest number? Is the universe infinite? Might every event repeat again and again and again and again Your intuition is no use here.

Faith alone can't save you.

How did the universe begin? Is the Earth just one of uncountable copies, tumbling through an unending void? On one you are rich, on another you have yet to be born.

These are the deepest mysteries of the universe.

Ladies and gentlemen, pray silence as I present to you .

.

infinity.

And so on.

Planet Earth is so beautiful and so complex, humans can barely comprehend it.

And yet humanity has always asked, "What's over the horizon? What lies beyond the stars? "Is this it?" I think infinity is one of those things that is an essential mystery of the universe.

What happens after we die, why are we here, why were we born.

Infinity is in that class of questions and humans have been thinking about whether there's an end to the world or whether the world goes on forever, probably since the beginning of human thought.

It's a natural human impulse to want to go beyond any boundary.

It is simultaneously scary and exciting to think about.

If infinity is real, it has implications far beyond the world of science.

It strikes at the very heart of what it means to be you.

There is actually, far out in space, a planet that looks just like Earth with people just like us.

Some will be doing exactly the same things as we do, even with the same names and memories as us.

No matter how much I study the field of cosmology and think about this, it still makes no sense to me that the universe is infinite.

I prefer a finite universe because I can get my mind around that.

It's the only universe that makes intuitive sense to me.

Infinity.

Impossible to comprehend.

And yet it comes from something so simple, a child can understand it.

One, two, three four, five, six seven, eight, nine 10, 11, 12, 13, 14 15, 16, 17 18, 19, 21, 22, 23 I was very proud first to be able count to five, then to ten, and then I realised that you can always keep counting and there's no end to it.

37, 38, 39 So I had this obvious intuition that everybody had that there is no end to counting hence there must be infinity.

91, 92, 93, 94, 95, 96 1,374, 1,375, 1,376, 1,377 Numbers can get so vast, it's impossible to imagine.

1,380, 1,381, 1,382 1,2763, 1,2764, 1,2765, 1,2766 To count to a billion, it would take you about 30 years and to count to a trillion is not something you could even do in human history.

111,330, 111,331, 111,332 1,372,365, 1,372,366 Billion and nine, billion and ten, billion and eleven, billion and twelve, For most people, I suppose the biggest number they're likely to meet will be somewhere in the billions or maybe the trillions, which might be something like the budget deficit or military spending or something like that.

Mathematicians tend to use bigger numbers than that.

Googolplexplexplex three, googolplexplexplexplex four You made me do this! Googolplexplexplex five, googolplexplexplex six When I get to 199, then that would be too hard and I would have to stop.

TYPEWRITER KEYS CLICK One of the largest numbers we have a name for is a googol and it's one followed by 100 zeros.

A hundred zeros is a lot because each zero represents another factor of ten.

So, it's a big number.

You might be thinking 100 zeros, isn't that many.

But a googol is far bigger than the number of atoms in a human being, more than the number of atoms that make up planet Earth.

A hundred zeros is more even than all the atoms in the entire observable universe.

Sets your imagination going, doesn't it? A googol sets your imagination going.

It knocks you off of your chair.

It's already a number that's probably bigger than anything, that makes sense in our experience.

But, it's a very, very tiny, tiny, tiny large number.

A Googol itself was only a stepping stone on the way to a much much larger number called a googolplex.

A googolplex is ten raised to the power of a googol, that is it's one followed by a googol of zeros.

And of course it's just not possible to imagine the size of a number like that.

A googol has 100 zeros, but a googolplex has so many zeros that there's not enough space in the entire observable universe just to write the number down, even if you could write each zero on a single atom.

But from my perspective, these are all very, very small.

One of the biggest numbers ever used in mathematics is many times the size of a googolplex.

It makes normal numbers like a trillion or a billion disappear into practically nothing.

It's a number called Graham.

Graham's number is much much bigger than a googolplex.

In fact, it's as large relatively to a googolplex as a googolplex is to the number ten.

In fact, it's much much bigger than that.

Graham's number was discovered in the 1970s by mathematician and former circus performer, Ron Graham.

I don't know too many other people who have a number.

Er It's It's not bad.

It's not bad.

I mean I recommend it! Graham's number is so big, it even made it into the Guinness Book Of Records.

Oh, yes.

This is the 1980 edition of Guinness Book Of World Records and I think if we turn to page, er, what is it, 192? Numeration.

Yes, and then numbers.

Prime numbers, perfect, highest numbers.

Here we go, highest numbers.

"The highest number ever used in a mathematical proof "is a bounding value published in 1977.

It is known as Graham's number.

"It concerns bichromatic hypercubes and is inexpressible without the special arrow notation.

" It's really gigantic, I mean, it's just so large you can't compare it with anything you would normally associate large numbers, like the number of atoms in the universe or how many inches to the furthest galaxy or something like this, it's just way bigger than that.

So vast is Graham's number, nobody knows how many digits it even has, including Ron Graham himself.

In spite of the fact that it's Graham's number, and I'm Graham, er, I know very little about it.

I have no idea what the first digit is.

It has one.

I have no idea what it is.

Maybe no-one will ever know what that digit is.

Are there more zeros than ones in the number? Who knows? If you look at three to the three to the three to the n plus one Ron didn't just make up his number.

It's the upper limit to the solution of a pure mathematics problem concerning multi-dimensional cubes.

That's divisible by five, only if the exponent three to the three to the N plus one While the problem itself is abstract, the methods Ron used to solve it are now used to keep track of data sent across the internet.

N plus one, minus three to the three to the N, minus one.

OK.

Great.

So Just working out the last digit of Graham's number is a lengthy calculation.

RON GRAHAM'S WORDS ECHO You can't really comprehend how large it is.

.

.

Very large.

I don't think anybody can know that But like all finite numbers, it comes to an end In the last stage, we have three to a certain power Eventually.

RON GRAHAM'S WORDS ECHO So that means that the remainder, when you divide by ten, is always seven.

In other words, you can finally conclude that the last digit of Graham's number is seven.

End of the story! Graham's number is not really any closer to infinity than the number one.

You didn't really get started yet, even though you took a lot of steps to get to Graham's number, it takes so many more, infinitely more to get to infinity.

Infinity is just out there.

It's just a different beast.

What's the biggest number? Erm 120? Ten.

Is ten the biggest number? Sometimes people just say, um, seventy hundred eighty nine hundred, but that's not even a number.

Oh, dear.

Oh, that's difficult.

Well, I suppose there isn't really one, they just go on and on and on.

There is no biggest number, because if there were, you could always add one to it.

TYPEWRITER KEYS CLICK Unlike normal numbers, infinity never comes to an end.

And that gives it some very strange properties.

OK, so this is one of the first things that you have to think about if you're thinking about infinity.

You've got all the numbers - one, two, three, four, five, six.

Let's have just a few.

And they go on forever.

Now suppose I went through and picked out just the even numbers.

Two and four, six, eight, ten.

I'll write them down here.

Two, four, six, eight, ten.

Well, it's pretty obvious, that there are more of these than there are of these? We've picked out half of these numbers here.

Well, in fact it's not.

These two lists are exactly the same size.

And this is the first real paradox about infinity.

There are as many even, though there are half as many, so half of this list has as many things in it as the whole list does.

These sets look so different, but they're actually the same size.

How's that possible? Well, OK.

How's that possible? Well, you see, what are we doing, we're counting and when you count something you match it up with the numbers.

So if I'm counting my sheep, for example, I count one, two, three, four, I match the sheep up with the numbers.

But you see, I can match the even numbers up with the numbers.

I've already started doing it.

Here's one matched with two, two matched with four, three matched with six, four matched with eight, five matched with ten, and that goes on forever, so I've matched all of the even numbers up with all of the numbers, no gaps, perfect matching between the two.

So there are the same number of these as there are of these.

So this really is a characteristic property of infinity and it seems puzzling, but there it is.

This is why infinity is a bit of a hard thing to deal with.

And it troubled people for quite some time.

Two infinite lists are exactly the same size, even though one appears to contain twice as many numbers as the other.

Which is just the start.

The more mathematicians thought about infinity, the weirder it became.

How are you? I'm fine, thank you.

Can I check in, please? Around 100 years ago, one mathematician tried to explain some of infinity's strange properties by imagining arriving at a hotel with infinite rooms.

He wondered if there would be any space for him, even if it was fully booked? Now in an ordinary hotel, I'd be told, "I'm sorry we're full up.

You'll have to go somewhere else.

" But in an infinite hotel, things are rather different.

There's no problem at all.

RECEPTION BELL RINGS The infinite hotel was dreamt up by David Hilbert, one of the most influential mathematicians of the early 20th century.

Your room is upstairs.

Have a lovely stay with us.

Bye-bye.

The manager can't just put me into the last room, because there is no last room.

It's an infinite hotel.

The rooms go on forever.

There is no last room.

And all the rooms are full anyway so, even if there were a last room, it would have somebody in it.

But it's exactly for that reason that it's possible to find room for me.

All you have to do is shift the guest in room one to room two, the guest from room two to room three and so on down the line.

Because there's no last room, every guest has a next room to go to and that frees up room one, so I can stay in room one.

So everything's fine.

It turns out, even if the hotel was packed to the rafters, a room can always be found.

Infinity plus one is infinity.

RECEPTION BELL RINGS Good morning, sir.

How are you? I'm very well, thank you.

Can I check in, please? You can see that if two guests came in, infinity plus two is infinity.

But suppose I came along to this hotel with infinitely many of my friends and we all wanted to stay and the hotel was full, how could we do it? What we could do is arrange that the guests in room one moved into room two and the guest in room two moved into room four, each guest moved into the room with double the number.

So all the even-numbered rooms have now got people in them and all the odd numbered rooms are now free, so me and my friends can all stay in the odd-numbered rooms.

So this shows that infinity plus infinity is still infinity.

And it would make sense that the same rules also apply to subtraction.

But if you think infinity will stay the same whatever you add or subtract, then think again.

Suppose that in the morning, all the guests left.

The number of occupied rooms would be infinity minus infinity, but it would be zero.

So infinity minus infinity could be zero.

Or it could be one.

If I stayed on and all the other guests left, infinity minus infinity would be one in that case.

So there's no definite answer.

That's why you have to be very careful dealing with infinity.

It's a very slippery concept.

What if I and two other people stayed, say? Then infinity minus infinity would be three.

So it could be anything you like.

RECEPTION BELL RINGS Good morning, sir.

Good morning.

How are you? I'm fine, thank you.

Can I help you? Yes.

Can I check in, please? HE LAUGHS Oh, God! HE LAUGHS This is one of the reasons why you have to very careful dealing with a slippery character like infinity.

Not to be trusted.

Is infinity real? Well, not to most people, but you can't do mathematics without infinity.

So if you don't feel comfortable with that then you're probably not going to become a mathematician.

Get any mathematician to tell you about what he or she is doing at the moment, and their imagination will somehow be full of this idea of infinity.

Infinity is like a landscape in which you work.

a place in which you do mathematics.

It's not a real place.

You can't actually go there, except in your imagination.

But to those who do mathematics, it seems very real indeed.

But you see one of the problems with infinity is that it has some paradoxical properties and very basic questions about infinity that we can't answer.

So you do have to be a little careful.

It's like having a polar bear as a pet, you've grown up together, he's a wonderful pet, he's big, he's fast, he plays in the snow beautifully, but there's always the chance that one day he'll get annoyed with you and bite off your head.

So, we are playing with fire, I think.

The paradoxes associated with infinity make some mathematicians uncomfortable.

Not least Professor Doron Zeilberger.

I first came across infinity like everybody else in a very early childhood when you start counting.

First you count to three, then to four, then to five then to ten, then to 100 and eventually you realise that you can keep counting forever.

Hence there is an infinity.

341, 342, 343 63,789, 63,790, 63 etc, etc, ad infinitum.

But I don't think I ever liked it.

I always found something repulsive about it.

TYPEWRITER KEYS CLICK I prefer finite mathematics much more to infinite mathematics.

I think that it's much more natural much more appealing and the theory is much more beautiful.

It is very concrete.

It's something you can touch, something you can feel, something you can relate to.

Infinite mathematics, to me, is meaningless because it's like abstract nonsense.

In my opinion, infinity is only a fiction of the human mind.

But not believing in infinity leaves Professor Zeilberger with a problem.

If the numbers don't go on forever, where do they end? When you start counting, you seemingly can go forever, but eventually you will reach the biggest number and then when you add one to it you go back to zero.

You go back to zero? Yeah.

How is that possible? How is it not possible? Have you ever been there? The biggest number is much bigger than anybody can ever think of.

It's bigger than a googol, bigger than a googolplex, bigger than a googolplex to the power of a googolplex.

It's so big, we can never envision it.

Nevertheless there is a biggest number and if you keep counting after that big number, we get back to zero.

Like when we walk around our planet, if you keep walking, eventually we get back to the place we started.

And if you think that this is ridiculous.

Look at the alternative.

It's less ridiculous than infinity and all the paradoxes that go with it.

Infinity may or may not exist, God may or may not exist but in mathematics there should not be any place for neither infinity nor God.

Doron isn't the first person to feel that the infinite is an illusion.

Until recently, infinity was too wild to be tamed by mathematics.

Too unpredictable to be used in equations.

Aristotle believed counting could go on forever and that the universe was eternal.

But he refused to accept that the universe was infinite in size.

He believed the Earth was at the centre of the universe.

However, without an end, there can be no middle so he banned infinity from his mathematics.

Infinity was discussed by philosophers and priests rather than mere mathematicians.

The infinite was something closer to a god than a number.

In 1600, philosopher Giordano Bruno claimed not only that the universe was infinite, but there would be many other Earths orbiting stars just like our own sun.

His beliefs went down so poorly with the Catholic Church, they had him burnt at the stake.

Only God himself, could be truly infinite.

HE SIGHS In the mid 19th century, one man, Gregor Cantor, made infinity truly part of mathematics.

Something that could be used in equations as if it were a number.

Cantor's idea was that we can gather together maybe even infinitely many things into a single set, like putting them into a bag and think of it as just a single object.

And once it is a single object, we can then put it into mathematics, do calculations with it, because it's just one object.

We don't have to know that there are infinitely many things in the bag, it's just a single object for us.

But making infinity part of mathematics produced one surprising result.

Some infinities are bigger than others.

Cantor's great initial discovery was that the infinity of the decimal numbers was larger than the infinity of the counting numbers.

And he did this by what's now called Cantor's diagonal argument.

So what you have to show is that in any list of decimal numbers, there's a decimal number that's not on that list.

And what we can do is go down the diagonal of this list, so we might take the nine here, and the one here and the one here.

We're going down the diagonal and we're generating another decimal number.

Nine, one, one and so forth.

Now we take this number and we change it.

We change the nine to an eight, say, and we can change the ones to twos and so forth.

And now we've generated a decimal number, which can't be on this list.

It can't be the first one, because the nine has been changed to an eight.

It can't be the second one, because in the second position, it's got a two instead of a one and so forth.

And all the way down the list, this number will be different from the decimal number on the list.

So what does that mean? It means there can be no matching of all the decimal numbers by the counting numbers and that tells us that the infinity of the decimal numbers is larger than the infinity of the counting numbers.

There's no largest infinite number.

For every infinite number, there's a bigger one.

There are infinities beyond infinities and that's what we study.

In making infinity part of mathematics, Cantor had uncovered a whole universe of infinities, each infinitely bigger than the last.

And that wasn't easy for many to accept.

At the time, I think that was a big surprise.

No-one had really thought carefully about whether infinities could come in different sizes, after all infinity is infinity.

How can you have different sizes of infinity? Cantor's breakthrough came at a price.

Cantor ended his days in an asylum.

Was it infinity that drove him there? Who knows? Who can tell? He faced a lot of opposition from his colleagues and it was possibly that, more than thinking about infinity itself that was the trouble.

It was only later that mathematicians accepted and welcomed his theories into the body of mathematics.

Cantor's work is absolutely fundamental to everything we do.

Today, Cantor's infinities are part of mainstream mathematics.

And the truth is, even those who would rather infinity didn't exist, use it in their equations everyday.

Infinity is simpler and quicker to manipulate than large finite numbers.

Most mathematicians have made an uneasy peace with infinity and accepted it as part of their universe.

Some have devoted their careers to studying it.

To the person who wants to deny infinity and say it doesn't exist, I don't see how that view enriches their world.

I feel sorry for them.

I mean, infinity, maybe it doesn't exist, but it is a beautiful subject.

I could say the stars don't exist and stay inside, or always look down, but then I don't see the beauty of the stars.

And until one has a real reason to doubt the existence of mathematical infinity, I just don't see the point.

There's whole world of infinities out there.

Maybe they're real, maybe they're not.

But could infinity be part of the world you call real? As yet unseen through any microscope and undetected by any telescope, might the heavens genuinely be unbounded and the depths of space deeper than love itself? Is space infinitely big or simply unimaginably big? TYPEWRITER KEYS CLICK The sky never ends, so I think space never ends, because there isn't like a wall all around our all around our whole planet.

I actually think the universe IS infinite and if I had to put odds on it, I would say there's a 95% chance that it is infinite.

I think space is very, very, very, very, very, very big.

I would say 1,000 metres.

That big.

I think the universe is infinite on Mondays, Wednesdays and Fridays and it's finite the rest of the week.

I'm having a very, very hard time making my mind up.

Space.

It hasn't got laws or electricity.

It's just like a sky.

It won't finish.

It will keep on going.

And going and it likenever stops.

The fundamental issue that most people come up when you say the universe may be finite, is simply what's outside of it? What happens when you come to the edge? Can't you go beyond it? And so this leads some people to the conclusion that the universe has to be infinite.

It might seem obvious that space goes on forever.

Innocent even.

But unleash infinity into the universe and all bets are off.

He makes the extraordinary mundane and the unbelievable, inevitable.

He makes the extraordinary mundane and the unbelievable, inevitable.

In an infinite universe, anything that's possible has to happen.

Even something as unlikely as a monkey typing the complete works of Shakespeare.

TYPEWRITER KEYS CLICK "My bounty is as boundless as the sea.

"My love is deep.

The more I give to thee, the more I have.

"For both are infinite.

" If we imagine this monkey, all it's doing is thumping away at the keys completely at random.

The monkeys don't have to evolve, they don't have to be able to read Shakespeare, they do have to be able to carry on typing, but that's all, just typing at random.

"I could be bounded in a nut shell and count myself a king of infinite space, "were it not that I have bad dreams.

" To test the infinite monkey theorem, a computer was placed in the enclosure of a Cambridge University professor.

Typing out the complete works of Shakespeare at random is really going to be a big job.

It's a thick book.

There's 37 plays in here, all the poems and sonnets, there's 884,429 words, every word has to be in exactly the right place, every character has to be exactly in sequence including the spaces in between.

And so doing that at random, bashing away on a keyboard is a difficult thing to do.

For the last week, David Spiegelhalter's computer has been randomly generating letters.

We're only generating lower case at the moment we're not using capitals so we're giving it a bit of a chance like that.

And it's generating them at the rate of 50 characters a second.

And, as you can see, it's keeping on finding matches all the time.

If it finds a match of four letters, four characters, it adds another character on, a random character, and sees if it's found a match for five characters and so on and so on.

The programme's been running for more than a week now and in that time, it's managed to generate more than 34 million characters.

If we assume that this monkey can type one character a second, it would have taken 34 million seconds, which is just over a year's typing for our monkey, but we've got to be kind and give it some breaks, so I would say just over two years probably typing.

The longest match so far is eight letters and here's the string, "We space lover.

" This occurs once in the complete works.

It occurs in Love's Labour's Lost, Act Two, Scene One.

So, it's in here Somewhere I've no idea where! It's not even in alphabetical order.

Ah, Love's Labour's Lost, here we are.

Act Two, Scene One.

Yep, I've got it.

Yeah, there we are, Boyet says, "With that which we lovers entitle affected.

" So the actual word is "we lovers" and that's where "we lover" fits into it.

But eight letters doesn't seem like very much.

No, it doesn't, but you have to think of just how unlikely it is to generate the exact works of Shakespeare.

So we did some calculations and worked out if you wanted 17 characters, which is, "To be or not to b" Not even the whole, the whole phrase.

We'd have had to set this going at about the time of the big bang around 14 billion years ago.

And that's to get something just twice that length.

But remember, we've got to get five million characters, all in the right order.

We can calculate the chance of this happening as one in a very large number.

It's ten with about nine million zeros written after it.

So that's an incredibly tiny probability very, very small.

It's an unbelievably unlikely thing to occur.

So if you imagine the current National Lottery, it would be like someone winning every single time, time and again, every single week for year after year, for 29,000 years.

Same person.

The same person buying their ticket and winning every week for 29,000 years.

But if we have an infinite amount of time, we can be certain that it will happen.

And not just once, it's going to happen again and again and again.

Because infinity is such a long time that everything, no matter how unlikely, as long as it's possible, will occur.

Infinity is so vast, one monkey, randomly typing forever could easily get the job done.

TYPEWRITER KEYS CLICK If he had an infinite amount of time, the monkey would produce far more than Shakespeare.

He'd produce every book that's ever been written.

Everything from the telephone directory to the latest celebrity autobiography.

TYPEWRITER KEYS CLICK But in an infinite universe, there will be infinite number of monkeys.

And that means, somewhere, one of them is typing Shakespeare right now.

TYPEWRITER KEYS CLICK If the universe were infinite, it seems fairly simple and benign, but it has some really strange consequences.

if we look far enough away there would be regions like the one that we're in, there would be a room out there like the one that we're sitting in now.

There would be Earths out there just like ours, except maybe the Roman empire would still exist or Germany would have won the war and on a personal level out there right now would be copies of Anthony giving interviews in a pink jumpsuit or rotting in jail or filthy rich.

There would be every possible combination.

Every way your life could have gone, there is somebody else just like you leading that life.

In fact, anything that we can really conceive of anything that's physically possible will happen, not only that it will happen an infinite number of times.

They're all out there in this infinite universe.

So whether you look at this as a good thing or a bad thing depends a little bit on how good you life is right now, but there really out there.

Which is crazy.

It means that there is actually far out in space, a planet that looks just like Earth, with people just like us, some will be doing exactly the same things as we do, even with the same names and memories as us.

It feels a little bit spooky to know that there are all these other copies of me .

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but it takes a little bit of the pressure off getting things right all the time, knowing that when I screw up, one of the other Maxes perhaps fared better.

Infinite space has consequences it's impossible to comprehend.

Infinitely many copies of you, identical in every possible way.

Every molecule, every heartbeat every atom, every breath, every thought, the same.

Each one convinced that they're the real you.

In an infinite universe you're not unique, you're insignificant, you're nothing.

And, it turns out it's a relatively simple calculation to work out how far you would need to travel to meet your nearest doppelganger.

Imagine a ridiculously simple universe which only has space for four particles and only two kinds of particles, purple and yellow.

Then there are only 16 ways this universe can be arranged two times two times two times two, 16 possible arrangements.

Yellow, purple Purple, purple Almost done.

The top ones are all purple.

This means that if we arrange a 17th universe in some random way, like yellow, purple, purple, yellow, it has to be a copy of one of the existing universes.

Let's see This one.

And this is true no matter how we arrange these.

If we do this, then it's a copy ofthis.

In other words, this guarantees that the new universe here is a duplicate.

And it's also easy to see that the distance from any one to it's nearest copy would be about the size of this square.

In our observable universe, there's obviously more than 16 ways to arrange all the particles, but it still comes out to be a finite number, so we can use basically the same calculation to figure out how far away we have to go to find an exact copy of Earth and an exact copy of me.

All you need to do is work out how many subatomic particles it's possible to cram into the observable universe.

Calculate the number of possible configurations of those particles and multiply that by the diameter of that observable universe.

On the pool table there were four balls, so there are two to the power four equals 16 possible arrangements.

In our actual observable universe, we can put in up to ten to the 118 particles.

That's a huge number, but to get the number of ways in which they can be arranged in our universe, we have to take about two to the power that.

So two to the power ten to the power 118.

A honking big number.

Then to get to the nearest copy of our universe, we multiply by the size of our universe, which is ten to the power 26 metres.

10 to the 26 is tiny compared to this number here.

so the bottom line is that if we go two to the power 10 to the power 118 metres away or so, we're going to find a perfect copy of our entire universe, of Earth and of me.

I find this quite dizzying, frankly.

Two to the power ten to the power 118 metres is further than any human could ever travel, but if the universe is truly infinite, these exact replicas of your universe have to exist.

While no-one likes the idea of space coming to an end, the consequences of an infinite universe are even more bewildering.

No matter how much I study the field of cosmology and think about this, it still makes no sense to me that the universe is infinite and always has been infinite.

I don't understand that.

I don't pretend to understand that.

The idea that there may be an infinite number of Earths, an infinite number of people having this exact talk that I'm having right now, that just, that doesn't compute in my brain.

I prefer a finite universe because I can get my mind around that.

It's the only universe that makes intuitive sense to me.

Many physicists believe space could be curved or even folded back on itself.

In the same way, you could sail round the Earth forever, if you kept on going in a straight line through space, and could travel long and fast enough, then you will arrive back where you started.

You don't need infinity to produce a universe that has no edge.

We're probably never going to know whether our universe is infinite or finite.

It's something I'd really like to know.

People have wondered about it for millennia.

The best we can say right now is the universe is extraordinarily large.

But cosmologists might finally be on the verge of an answer to the question of whether the universe IS infinite.

And the clue that's led them there comes from something that has been part of your lives since the first moment you opened your eyes.

Light.

Light travels extraordinarily fast, but not infinitely fast.

It travels about 300,000 kilometres every second.

And the Moon happens to be about 300,000 kilometres away.

So it takes light about one second to make it from the Moon to the Earth.

You might say the Moon is one light second distant from the Earth.

As you gaze out into space, you are looking back in time.

You see the Moon as it was a second ago, Jupiter as it was an hour ago and your nearest galaxy, 2.

5 million years in the past.

And some things are just so far away, their light would take longer than the age of the universe to reach the Earth.

The most distant light ever detected is also the oldest.

It began its journey just 400,000 years after the big bang.

13.

7 billion years ago, the universe was born.

13.

7 billion years.

To infinity, it's nothing.

Seems like yesterday.

According to Big Bang theory, after a second the universe is ten thousand million degrees and the first atomic nuclei condense out of the fireball.

And darkness was on the face of the deep Around 400,000 years later, the first atoms form and light is released into the universe.

And God called the light day.

And darkness he called night.

In the evening and the morning That light is still with you today.

It's called the cosmic microwave background.

Eventually this will become Shakespeare, Monkeys and even you.

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we may know the traitors and the truth .

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passing through nature The cosmic microwave background is a snapshot of the universe when it was just a baby.

Over billions of years, the cooler, denser blue regions in this image will collapse to produce stars and galaxies.

It is astonishing that just this little picture can tell us something potentially about infinity.

To think that we might get that sort of insight into the universe is pretty spectacular.

This image of the early cosmos might now reveal whether the universe is infinite, because hidden within it lies a mystery, something the big bang model alone couldn't answer.

What we see when we're looking at this map of the microwave background radiation is that it's showing us what temperature this radiation has all over the sky.

So this is an image of the whole sky.

If you look at this red splotch versus that blue splotch there's only a difference of about a ten thousandth of a degree so in fact, this microwave background radiation is incredibly uniform.

Now for those two to be so similar, it seems that some sort of physical process, some sort of agreement should have taken place and this is a rather baffling mystery.

How did they do it, how did they come to this agreement in their temperature? This was a real enigma for the standard big bang cosmology.

It was a real puzzle.

In the big bang model, there was no way to explain how distant parts of the universe could have such similar temperatures.

To make sense of it, physicists needed something else.

A new theory of the early universe.

A theory they called inflation.

HE INFLATES GLOBE Inflation says the early universe expanded much faster and further than previously thought, a million, million, million times, in less than a billionth of a billionth of a second.

If I faint, I'll sue you! This explains the uniform temperature of the cosmic microwave background because everything you see was stretched out from a small and uniform part of the whole universe.

Now just imagine I did all this in ten to the power minus 32 seconds.

That would be inflation.

Inflation was devised to explain the finite observable universe and it does a very nice job of doing that, but it has a sort of side effect or a very interesting property that once you get inflation started, it just keeps going.

It takes on a life of its own, like a genie you've let out of the bottle.

The theory of cosmological inflation actually produces an infinite universe.

Inflation predicts your universe never stops expanding and may, in fact, be infinite.

But following the mathematics to its logical conclusion predicts an even more disturbing outcome.

Your infinite universe might not be the only one.

We used to think that inflation only gave us one big bang and one infinite space, but now it's becoming clear that actually it never stops and instead gives us an infinite number of infinite spaces.

There are multiple universes, infinitely many multiple universes, infinitely many infinite universes even.

Inflation was devised to explain the finite observable universe, and it does a very nice job of that WORDS ECHO In an infinite universe, anything possible happens all the time.

But with infinite universes, the impossible is happening right now.

Because in some of those universes, the laws of physics that govern your world simply don't apply.

What isn't appreciated by many even in the physics community is that this model of these infinitely many infinite universes is actually probably our current best bet as to what the real universe looks like.

It's baffling and it's mind bending but that's where our road of cosmology has taken us, to this confrontation with real infinity.

I think we should expect us not to be able to intuitively grasp the ultimate nature of space and everything because we have intuition only for the things which were useful for our ancestors.

And we shouldn't expect our intuition to work for really big questions about the ultimate nature of reality.

If one of our ancestors spent too much time thinking about what's outside space, you know, they wouldn't have noticed that there was a tiger sneaking up from behind and they would have been cleaned right out of the gene pool.

So it's very important for us scientists to not diss ideas just because they feel weird.

Fortunately, our math doesn't have any inhibitions and we could still calculate all these things even if they seem completely counter intuitive and it's only through the math that we're able to actually deal with all these ideas.

Counting has led you to an infinite mathematical world of infinities, each infinitely larger than the last.

And gazing out into the furthest depths of space, some see an infinite universe, itself just one of infinitely many.

Infinity is a big topic I don't think it's gonna be understood fully in any finite period of time.

We have a hint of just how rich that realm is, but we haven't understood the smallest fraction of it.

This, of course, is because, by its very nature, the subject of infinity is a vast and infinite subject.

CLOCK TICKS MONKEY SCREECHES