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[crickets chirping]
[eerie music playing]
[dogs barking]
[car alarm blaring]
-[eerie music intensifies] -[girl] Four, five…
[man 1] Well, the first time that I thought about infinity,
I was looking up at the sky one night
when I was about ten years old.
[woman 1] I was really little, it was before I had started school,
and I was sitting under the dining room table
and counting.
And at a certain point,
I realize this goes on forever.
[man 2] Lying on a beach late at night and just wondering
the vastness of outer space.
Does it go on forever?
[tense music plays]
[music intensifies]
[tense music ends]
[jaunty jazz music plays]
[cat] Mmm…
[producer] I want to start with sort of the premise of the film,
which is that we're looking for infinity.
-So where should we go looking? -[chuckles]
Well, I don't think everybody's looking for infinity. [laughs]
[upbeat music plays]
[woman] So is infinity a number, a place,
an idea, a concept?
If you ask me, probably all of the above.
[man 1] Numbers never end. That's the basic idea of infinity.
[man 2] If I imagine continuing to count for as long as I can count,
infinity will definitely last longer than my lifetime.
[man 3] The funny thing is, all these numbers that you can imagine,
they're like zero, nothing.
Any number, no matter how big,
is just absolutely insignificant compared to infinity.
For me, infinity is not scary.
I find infinity beautiful, and haunting, and thrilling.
I love Infinity.
You know, I guess if I start thinking about, "I'll be dead forever…"
there's a side of me that's worried about it. [laughs]
But then again, I don't plan to be thinking about it.
I think to be conscious is to be wrangling with infinity.
I mean, not to be grandiose.
When our hearts are broken, are we going to be in pain forever?
And if you really fall in love with someone,
are you gonna be in love with them forever?
Love is certainly infinite, because when we are overwhelmed by love,
we have this constant sense of breaking the limits and the boundaries.
All of these abstractions, suddenly it seemed to me,
well, maybe they're just the product of our minds.
Who invented infinity?
I think, uh, to some extent,
the concept goes back earlier than anything that we know.
[woman] One, two, three…
[Kenny] In fact, counting is the very earliest writing that we have.
[girl] …seven…
[Kenny] Whoever invented writing already had the idea of counting,
and the idea that you can always count one more
must surely go back at least that far.
[girl] …fifteen, sixteen…
[Steven] We have all kinds of different visions of things that are infinite,
and they're usually terrifying.
There's the vision of the bottomless pit.
There's the idea of time that never ends.
[clock ticking]
If you believe in hell,
and if you believe about torture going on forever,
a nightmare that you could never wake up from…
So I understand that a lot of people find thinking about infinity terrifying,
or nauseating.
At least very upsetting.
[Rebecca] But for some of us, it's one of the deepest pleasures.
[girl] …2001…
[Rebecca] We're so small…
[girl] …2003, 2004…
[Rebecca] …and yet we can touch something so…
[girl] …1000000, 10000002…
[Rebecca] …explosively large.
[girl] …1000004…
[Rebecca] That feeling of, "I'm bigger because I know how small I am."
[girl] …399, 400, 401…
[Rebecca] I've been chasing that feeling
my entire life.
[girl continues counting]
[Janna] But does infinity exist outside of our minds?
[Moon] So does infinity exist?
Um…
You know, in math, I think-- [smacks lips]
[producer] Can we wait for this train to go by?
Oh yeah.
[train rattling]
[producer] Okay.
All right. Sorry about that.
Oh, yeah. It's good to collect thoughts.
Um, okay, so I think I was gonna make a comment
about what it means to exist mathematically, right?
Um, so, in math, I think we have an interesting relationship to stuff,
to things in existence, right?
If you can conceive of something,
if you can kind of create rules for handling it,
then it exists.
[train rattling]
So how does infinity fit into all that?
Well, okay, infinity is super counterintuitive,
and it leads to a lot of surprising paradoxes,
contradictions, intellectual quagmires, quicksand.
Many of these are summed up in a parable
that goes by the name of the Infinite Hotel.
[jaunty jazz music plays]
Imagine a hotel with infinitely many rooms.
It's a really popular hotel.
And in fact, it's always booked solid. Every room is occupied.
Nevertheless, there's always room at the Infinite Hotel.
[Steven laughs]
So, one night, a new guest shows up.
-[bell dinging] -Bang, bang, bang on the bell.
"I'd like a room."
And the manager says,
"Sure. Of course, we can accommodate you here at the Infinite Hotel."
"Just hold on for one minute."
The manager tells everyone over the loudspeaker--
[manager] Please gather your things.
Get ready to move to the next room down the hall.
[Steven] Person in room one moves into room two.
Person in room two moves into room three.
You might think there's gonna be trouble because all the rooms are occupied.
Nevertheless, as it's an Infinite Hotel, I mean, this can happen, right?
The numbers never stop.
One can move to two, two can move to three,
a million can move to a million and one.
So everybody moves down to the next room,
which means that now room one is open, and the new guest can be accommodated.
[producer] You could keep everyone in the same room
and just have the guest go to the end.
Yeah. So a tempting thought would be just have the guest take the last room.
[laughs] Why are we making everybody else move for this one person?
Well, because there is no last room.
[jaunty jazz music plays]
Infinity doesn't work like that.
You have to start at the beginning, not at the end. There is no end.
Now, a bigger challenge for the manager comes the next night
when, instead of just one guest showing up,
suddenly a whole busload full of cranky, sweaty, ill-tempered people
all pound on the bell at the same time, saying, "We all want rooms."
There's infinitely many new guests.
They all want to be accommodated at this hotel.
How can you do that?
Can you somehow find room for infinitely many new people,
given that infinitely many rooms are occupied?
[panting]
It turns out the manager has seen this problem before,
and so she calls out over the loudspeaker…
[manager] Everyone be prepared in a few minutes
to move to the room that is double your current room.
[Steven] So, person in room one, you're gonna be in room two.
Person in room two, you're now going to be moving into room four.
Person in room three, you're going to six.
Now, it's a big inconvenience for person in room one million.
They have to move to room two million, which is a big schlep down the hall.
[jaunty jazz music intensifies]
So all these people dutifully move to their new rooms.
You'll notice that what's happened is all the odd-numbered rooms
have been vacated.
And so all those new guests can just start filing in.
[jaunty jazz music ends]
There's one other custom at the Infinite Hotel,
which is that this manager is very conscientious
and always checks in on the rooms, infinitely many of them.
Fortunately, the manager is very speedy,
so she can get her whole job done in a minute.
-[jaunty jazz music plays] -[grunts]
She spends half a minute making sure everything's okay in room one,
and a quarter of a minute, so 15 seconds, on room two,
and then half of that, so 7.5 seconds, on room three,
and so on, going all the way down the hall in the Infinite Hotel.
A half, plus a quarter, plus an eighth, plus a sixteenth,
and if you add that up…
But if you just think about that, when you go out to infinity,
that's gonna add up to one.
She's basically gonna be done
checking infinitely many rooms in one minute.
[producer] How does she get back from infinity?
She get-- Oh, that's good. Well… [laughs]
Yeah, let's see. So, um… Hmm.
That's an interesting puzzle. Yeah. Where is she at the end? Uh…
Don't ask me.
I've been thinking about infinity my whole career,
and even I don't really see any way out for her.
[eerie music plays]
She's out at the end of this infinitely long hallway,
and I don't see how she's gonna come back from there.
I told you it was a strange hotel.
Well, so I think that, you know, the moral of this crazy parable
is that infinity doesn't behave like anything we're used to.
We're used to thinking about collections of finite numbers of things.
Fish, fish, fish. That's three fish, you know.
So we're used to small numbers, or even big numbers.
You hear nowadays about trillions of dollars
needing to be spent on the budget in the United States.
But, um, those are nothing like infinity.
Any finite number, no matter how big, is nothing compared to infinity.
[marching drum playing]
We just don't have good intuition about how infinite things work.
[tense music plays]
Even very small children can have a gut feeling about infinity.
It's the biggest possible thing.
It's bigger than anything you can ever think of.
Well, what happens if you add one?
So if infinity is the biggest thing there is,
then when you add one, it should still be infinity. Maybe?
What happens if I subtract infinity from both sides?
Then you get one equals zero.
And so immediately you've landed yourself in some kind of an issue,
and that's just the start.
It turns out you can write things down that don't have an answer.
I mean, that doesn't happen in finite arithmetic.
If I say one plus two, the answer is three, and I'm done with it.
But if I say one minus one, plus one, minus one…
[voice echoes]
One minus one, plus one, minus one, plus one, minus one, dot, dot, dot.
Keep doing that forever.
There is no answer to that. It's not zero and it's not one.
It's neither. It's both. What is it?
You can try adding infinity to infinity. Multiply it.
You get these weird paradoxes.
Well, what's infinity plus infinity?
That should also be infinity because it's the biggest thing.
Then you subtract infinity from both sides
and you get infinity equals zero.
That's even worse. So this is terrible.
[tense music continues]
But it's wonderful, because what is going on?
[waves crashing]
Is infinity beautiful? Of course, it's beautiful.
I mean, yeah, it's--
Like I say, it's beautiful because it is…
Uh…
Maybe…
a good way to think about it is with our circle.
Shall we talk about the circle?
[eerie music plays]
[Steven] There's a very beautiful shape
that often people think of as the most perfect shape, a circle.
Perfectly round, never-ending.
We see circles everywhere.
I look in your eyes, I see the circle of your iris.
I see the circle of your pupil.
I see the circle of the sun and the moon.
I've got my wedding ring in the shape of a circle.
[Eugenia] I love circles.
But if you sit down and think about what they are for a second,
how many sides do they have, for example? How many corners do they have?
[Steven] That immediately brings us into face-to-face confrontation
with the infinite,
because you can't think about a circle in terms of straight lines.
[Eugenia] If we think about pointy shapes, say a triangle,
if we tried to use it as a wheel,
it would be kind of painful going up the street.
If we had ten corners, then that wouldn't be so bad as a wheel.
And then, if you keep going, you can go, "Oh, wait,
if we then had infinity sides,
then we would be completely like a circle,
and we would have infinity corners, which is like having no corners."
And so, somehow, infinity has come round to being just like zero.
[bell rings]
One of the amazing things about infinity
is that, if you scale it, it's still infinity.
Whereas if you take any finite number, if you scale it,
which is like multiplying it, it gets bigger.
But infinity doesn't do that somehow,
and that's a very odd thing for us to get our heads around.
So let's try and picture this, right?
When it comes to a circle as a set of points…
'Cause that's what it is. It's an infinite collection of points.
I would like to try to convince you
that a small circle, like a circle of radius one,
um, has the same number of points as a huge circle,
a circle of radius billion.
Right? Okay.
So, to do that,
I need to get you to agree on what it means to be the same size.
Like, anytime you're making a high-stakes bet with a friend,
you wanna settle… This is good advice, by the way.
You want to settle in advance what qualifies as winning.
I propose that two sets will be said to have the same size
if I can put their elements in a perfect one-to-one correspondence.
Now, all I need to give you is a rule that puts the points of a small circle
in correspondence with the points of a larger circle.
You can't count the points on the circle,
but you can match them up with the points on a bigger circle.
I take my small circle
and I just center it somewhere at a point,
and then I take the huge circle and I center it at the same point.
I'm gonna-- From that center point, I'm gonna draw all the radii. Right?
I'll draw these kind of line segments
stretching out from the center to the outer circle.
Well, each of those line segments hits the outer circle on one point
and hits the inner circle on one point, and that's the correspondence.
I'm gonna say the point, um, on a particular radius in the small circle
is matched to its corresponding point from the large circle.
Well, by sweeping those radii all the way around,
I clearly hit all the points on each circle.
And so I'm done.
I've got a perfect one-to-one correspondence.
You now have to agree
that the two infinite sets have the same size.
[upbeat music plays]
So mathematicians don't literally sit there counting to infinity.
We just match things up in pairs.
And if you can find a way to match things up in pairs,
you can go, "Oh, that's the same then."
So, for example, we can match up all the numbers, the whole numbers,
with the even numbers by just multiplying each one by two.
You sit there and go, "Wait, the even numbers is only half of them."
And then you go, "But wait, I can match them up,
because if I pair up one with two,
and two with four,
and three with six,
then they pair up exactly."
So then you go, "Wait, there's the same number,
even though it's half of them."
And that's the amazing thing about infinity.
Once you've figured out this way of matching things up
to see how big an infinite set is,
it turns out that there might be some that you can never match up,
which means that there is some kind of bigger possible infinity.
[tense music plays]
[Moon] How can there be larger infinities?
Infinity is already as big as we can imagine, right?
No. No.
[hesitates] There's something beyond that.
-Let me see if this works for you. -[producer] Okay.
Um, so, provably the smallest infinity
is the one that goes one, two, three, dot, dot, dot.
If I was to take the numbers zero and one
and ask you how many numbers are there between that,
well, you could keep dividing and dividing.
No matter how many times you divide,
there's always a number
that you could add another zero behind the last decimal you had put.
[Eugenia] And I can make a one-to-one little map,
like a little dictionary between those numbers and the whole numbers.
One, two, three, four, five.
And so what we say is that those two infinite sets are the same size.
But then there's also other numbers.
Things like the square root of two,
and pi, and E.
And those are just a few that we know about and use.
These are numbers that aren't ordinary fractions,
and if we try and write them down as decimals,
they'll somehow go on forever without ever repeating themselves.
So how can we ever say what they are?
It took mathematicians hundreds of years to figure out how to do it rigorously,
and when they did, they realized that they're so complicated,
it's actually impossible to put them in a list.
No matter how we try to list them,
we're doomed to miss some out along the way.
We can't make a one-to-one correspondence
between those numbers and the counting numbers.
Now that we have two infinities,
would we really stop there?
[Moon] There are bigger and bigger and bigger ones.
This is a trail that just keeps on going.
[eerie music plays]
[Eugenia] An infinite hierarchy of infinities.
This is the kind of thing
that, hopefully, turns some people into mathematicians.
And no doubt sends other people running in the other direction. [laughs]
[Steven] My wife gets nauseous when I bring up infinity.
And my kids don't want to hear about it either.
[Eugenia] Some people love peering over cliff edges.
There are now those glass bridges
where you can step out over the Grand Canyon.
There's no way I would do that.
But when it's a mathematical cliff edge, I love it.
I think this gets back to this concept of the sublime.
When you're out by a waterfall…
[water running]
…you have this feeling of this thing that is so much larger than yourself.
But then you climb onto the mountain peak and you look out at the next valley,
and you see another waterfall that's even larger,
but it's in the distance, so it looks tiny.
[birds singing]
Infinity is some kind of monster that has to be tamed.
[violin music plays]
"INFINITY"
[loud thump]
[roars]
[roars, grunts]
[alarm blaring]
[in Japanese] It's infinity! It's attacking the city!
[Moon in English] Mathematicians needed to invent ways to deal with infinity.
Taking something totally weird and unintuitive
and taming it to the point where you can
walk around and study it from all sides.
[roars]
[grunts]
And the ways that they came up with led to the field of rigorous calculus.
There's one idea at the heart of calculus,
and it's an idea I like to call the infinity principle.
[upbeat music plays]
[Steven] You can make sense of any complicated motion or phenomenon,
or anything that's changing, or any curved shape,
by thinking of it as being made of an infinite number
of infinitesimally small simpler motions or shapes.
This is one of the greatest ideas in history.
[monster roars]
Everything to do with movement,
and everything to do with things that are continuously changing,
can only be studied rigorously using calculus.
[roars]
[woman] Hmm…
[in Japanese] Maybe it's trying to communicate with us!
We can use calculus to study its roar!
[Steven in English] So the infinity principle is
you can make sense of complicated things
by breaking them up into infinitely many simpler things.
[graphic beeps]
Solve the problem for the simpler things
and then add the results back together to get the original whole.
[upbeat music continues]
[in Japanese] A picture encoded in the roar!
Stop fighting infinity!
It's peaceful. Look!
[electrical sparking]
[Eugenia in English] The understanding of electricity
was made possible by calculus.
[loud thump]
[roars]
And electricity has enabled the entire modern world.
[phone dials]
Moshi moshi!
[talks indistinctly]
So does infinity exist?
Well, in one of the senses of math,
absolutely no question about it, because we have a symbol for it,
we know how to manipulate that symbol,
we can agree on the conclusions that we reach when doing so.
And by doing so, we can solve practical questions.
Right? So, from a pure math point of view,
that's your certificate of existence right there.
So then, does infinity exist out there?
[upbeat music ends]
That's above my pay grade. [laughs]
[loud thump]
If you mean does it physically exist,
who knows?
That's a question for the physicists to answer.
And so now they've got to go do their expensive things, right?
And figure out whether space-time is infinite or not. It's a real question.
[mysterious music plays]
Oh, this is a great question. So where can one look for real infinities?
[Brian] We've been struggling with the question
of whether the infinite infinity is a real thing
or something that is a human invention for a very long time.
[Stephon] Maybe we could approach the infinite.
As a physicist, that's the business I'm in.
[Anthony] How thrilling would that be, to do something in the physical world
that would tell us that something is physically infinite?
[Carlo] I think all of us have this experience
of, you know, going out in the night in the starry sky.
[Stephon] Lying on a beach late at night in Trinidad.
I mean, I probably was, like, eight years old
and just wondering the vastness of outer space.
Does it go on forever?
I question the notion of infinity
when I am using a particular yardstick
which is, can we measure it?
Can we access it? [voice echoes]
Can we, in any point
wrap our arms around it and say, "Here it is."
"There's the infinite."
And using that particular yardstick, the answer is no.
There's simply no way that we can measure the infinite.
[Janna] Nobody will ever give you back pi to all of its infinite digits
as the result of a measurement.
All I'll ever measure, um, in a laboratory,
or my friends will measure in laboratories,
are approximations of infinite numbers.
But then you start to wonder, "Maybe these numbers don't really exist."
"And nature doesn't actually make use of them."
And, um…
And I don't know the answer to that question.
That's why sometimes you think, "Oh, when the universe was created,
maybe the speed at which it came out expanding
is an irrational number,
like 0.500187923… [voice echoes]
…and the universe is going to compute that infinite list of digits
over the course of time."
So the universe itself is performing a computation,
and it's computing infinite numbers possibly.
[voices echoing] Five, zero, one, seven, eight…
[voices echoing indistinctly]
[voices stop]
I feel like I'm freaking you out,
but, you know, Steven Strogatz had to be pretty out there.
-[producer] Oh. -[laughs]
-Oh yeah. -[laughs]
You think of a piece of Jell-O.
You know, a blob, a block of Jell-O
jiggling on the table or on a plate.
You don't picture it as made of atoms.
You think it's a continuum of Jell-O stuff
that is infinitely subdivisible and has no gaps.
It hangs together perfectly.
It's what most of us think stuff really is like.
[eerie music plays]
The mathematicians are great in talking about the continuum.
And continuity is the idea that if I take a line, just a short line,
I can cut it in half,
and then in half again, and then in half again, and in half again.
And I always have a line, and I never stop.
I can go forever.
But a completely different question
is whether things are actually continuous in reality.
Take a rope, cut it in half,
and then again in half, and then again in half.
Can we go forever?
And this is a question that was debated since antiquity.
[food processor whirring]
Now, we have understood quite clearly that a piece of rope is not continuous.
That no piece of matter is continuous.
It's made by little individual pieces that we call molecules, atoms, particles.
My job,
as a physicist,
is to study another kind of continuity, which is a continuity--
It's not of matter. String, or a piece of wood, or a piece of metal.
But the continuity of space itself.
So just consider the space between my hands
and imagine to have it divide in half,
and half, and half, and half.
Can we go forever?
[Brian] Is space truly continuous?
Could it be infinitely divided?
My gut feeling is that it can't.
[Carlo] I think if we take what we best know about the world,
which is Einstein's General Relativity Theory and quantum mechanics,
and we bring them together,
the clear consequence of that
is that there's a minimal amount of space. It's very small. Incredibly small.
[Brian] Ten to the minus 33 centimeters, the so-called Planck length.
So incredibly tiny
that it certainly stretches the human imagination to…
think about.
So if you were to…
take an individual atom
and magnify it, expand it to be as large as the observable universe,
that's a huge scale of magnification.
The Planck length under that magnification
would grow to roughly be the size of an average tree.
So a tree is to the observable universe as the Planck length is to an atom.
So even on atomic scales,
the distances that we're talking about
where the notion of continuity may break down,
where discreteness may emerge, they are fantastically small.
[ethereal music plays]
So I have a sort of, uh,
pixelated vision of reality at this small scale.
It's like, uh, if God didn't draw the universe
with continuous lines but just little pixels.
It's many, many, many, many little things, but it's not continuity.
It's discreteness.
It's finite.
There is nothing infinitely small.
[tense music plays]
My favorite example where we can find real infinities
is when thinking about black holes.
[tense music continues]
Black holes are these massive objects
where all the matter is condensed so tightly
that we get this region around it called the event horizon,
from which we get no information.
[Janna] Now, there's nothing infinite about the event horizon.
It's actually empty. A neutral region of space.
You wouldn't even notice when you cross the event horizon.
You wouldn't feel anything weird happen to your body.
[Janna] What happens that's problematic is on the inside of the black hole.
[somber music plays]
[Delilah] In the very literal sense, we know nothing about
what goes on inside of the black hole.
But Einstein's theory of general relativity
says that, once you make it past the event horizon,
eventually if you keep just falling in…
you'll reach a point called the singularity.
[Janna] A region which is an infinite curvature in space-time.
[Delilah] Where all the mass of the black hole is concentrated.
[Janna] Infinite densities, and completely catastrophic.
What happens that's really, um, considered such a horror show
is that, in a finite time,
in microseconds, in fact,
you would hit this region of infinite curvature and infinite density
and simply fall out of existence.
[somber music ends]
As though you weren't part of the natural world anymore,
so you weren't-- Physics stopped.
It's really a violation
of the whole continuity of the program
of understanding nature.
It starts to say nature's fundamentally unknowable
in this one secret place inside this black hole.
And that just doesn't feel right.
[Stephon] What do you do when you have a theory
that works really well
except for when you have this infinity?
Do you throw the baby out with the bathwater,
or is the infinity trying to tell you something?
The singularity, this region of infinite curvature,
infinite density, I bet that doesn't really happen.
[Stephon] The infinity the equations predict is a hint.
It hints at us, "Hey, there's something new there."
[Janna] Sometimes, when I think about the infinity
at the center of a black hole, like,
I think about it as like a dying man
scratching a clue in the dirt, telling us…
"General relativity doesn't work here."
I'm of the opinion
that there is a new physics, actually, in a black hole.
[tense music plays]
I think that one thing that could be happening there
is that there's some sort of portal…
into a new realm.
Maybe a new universe, for example.
There's something I'm working on now
that seems to show that that could happen if the math works out.
You could have certain portals
where you can actually go through the black hole.
We call these things wormholes.
[tense music intensifies]
[Anthony] A wormhole is something where,
at some point, I say, "Here's a sphere."
"And I have a sphere inside this one that's got a bigger area."
Okay?
And then inside that one is another sphere with a bigger area than that.
So the-- At some level, you know, we are confined by
our normal understanding of space and time, right?
Something inside another thing is smaller.
But that bigger thing is… in here.
If I went in there, I would be in that bigger thing.
And yet I'm just seeing the sort of outside of it.
[producer] Do you understand it?
Uh, it's totally clear mathematically,
and when I look at this, I have no idea how that could be.
[chuckles]
[tense music continues]
Okay.
[producer] So I want you to think about this
and tell me what it makes you think about with infinity.
Well, um,
I would say that
if I were some microscopic being living at the surface of this… ball,
I may conclude that this surface is infinite,
'cause it would take an infinite amount of time, for example,
in my imagination of infinity,
to go from one side of the ball to the other side.
But, you know, it's not infinite, right? For me, it's not infinite.
[producer] So you're not holding infinity in your hand?
[laughs]
Well, poetically, I am.
I'm holding infinity in my hand.
I see everything in this room reflected in this sphere.
If we didn't have walls in this room,
I'd be able to see the entire universe in this sphere.
Well, the entire universe that's behind me,
because there's a path from everything to this sphere and then to my eye.
The one thing we cannot do in space-time
is look down on the universe like this.
And this is one of the, in some terms, mistakes we make,
uh, when we try to imagine a finite universe.
We imagine being in a space higher up,
an extra dimension out, and looking down on it.
But, of course, that would be part of the universe.
If light could get to me, that's part of the universe.
So there is never such a thing.
You can't jump out of the universe and look down on it. Ever.
[producer] You're holding infinity. It's real.
It's not.
[Steven chuckles]
Do you know the old story from Plato, the prisoners in the cave?
Right. So it's an old bit of philosophy,
that the prisoners are trapped in the cave,
their backs are to the opening of the cave,
light is streaming into the cave,
but all the prisoners can see is shadows on the wall of the cave
of things happening out there in the world.
To me, this is the shadow.
This is not the real sphere. This is not the perfect sphere.
The perfect sphere would have infinitely many points on its surface and inside.
This thing is made of atoms.
There's a lot of 'em, but not infinitely many.
It's a shadow of infinity.
I love this shadow… [laughs]
…because it gives me a glimpse of infinity.
[somber music plays]
[Anthony] It… feels almost inconceivable.
We're finite creatures, we have access to finite things.
We can only do so much in a lifetime.
[static]
Could we somehow nonetheless
get a glimpse into something that is truly infinite?
And I think this is not impossible.
An example that I've spent quite a lot of time thinking about is,
" What would happen to a physical system
if you just wait an infinite amount of time?"
So let's imagine we take a box. It's an excellent box.
Nothing can come in. Nothing can go out.
So we put an apple in the box, close the box.
We might come back in a month
and the apple is looking kind of mealy and decayed.
Come back in a year, the apple is a real mess.
The apple is rotted. Bacteria have done their thing.
Come back in, you know, a hundred years…
the apple is probably a kind of dust.
The apple contains chemical energy,
the same kind of energy you'd get if you ate the apple or burned an apple.
That energy will eventually come out,
and so the apple inside the box will get very hot,
uh, probably thousands of degrees.
Those particles can start to nuclear-fuse together.
[quiet thumps]
This will take a really long time,
because nuclear reactions happen incredibly slowly at thousands of degrees,
but eventually it will happen.
Your apple has turned into millions of degree plasma of fundamental particles
and, you know, burning into higher and higher things.
Eventually, you'll end up with probably some iron nuclei
and lots of photons.
[static]
Billions and billions of years later…
neutrons will decay into protons and other fundamental particles,
and then it'll sit there for a very, very long time.
Let's think about what the particles, the protons and neutrons, and stuff.
What do they-- How do they experience this?
They are just cranking along, obeying the laws of physics.
The state of the box is changing from one to the other, to the other.
So, if there's 10 to the 24 particles in an apple,
there's something like 10 to the 10 to the 24 different states
that those particles can be in.
That's a gigantic number.
But it isn't infinite.
And what that means is that,
if you let the box sit there for an infinite amount of time,
it will use them all up.
It will go through every possible state that it can,
all 10 to the 10 to the 24 or whatever of them.
And eventually, it will start having to reuse
states that it's been in before
because there just aren't any more that it can evolve into.
And so, eventually, if you wait a long time,
something will happen.
And this is the power that infinity has over the finite.
At some point,
you could open the box
and there's your apple again.
How did that happen? How did this hot gas turn into an apple?
But eventually, it has to.
In fact, every possible thing that could exist in the box will exist.
[fast motion sound]
And they will each exist an infinite number of times.
[slow motion sound]
Why do we care?
Well, we might be in the box.
[tense music plays]
[Brian] In any finite region of space,
like the observable universe that we now inhabit,
there's a finite amount of energy,
which is carried by a finite number of particles.
And those finite number of particles
can only be arranged in finitely many distinct patterns.
[tense music plays]
Because there are only finitely many distinct ways
that the particles can be arranged,
if space does go on infinitely far,
the particle pattern has to ultimately repeat.
[tense music continues]
And that would mean there'd be copies of us out there.
There'd be copies of us.
-An infinite number of copies. -Out there.
-An infinite number of copies. -Of us.
Infinitely many of us at very far parts of the universe
doing exactly the same thing that this copy of us is doing.
Copies of me that continue, and copies of me that die at any moment.
Anything that can happen will happen an infinite number of times.
Suddenly, we're in a wild…
-So somewhere there's an Earth… -…inhuman…
-…where I'm having this conversation… -…place.
Where instead of being very hot right now, it would be air-conditioned.
An infinite number of copies where…
An elephant suddenly appears in front of me.
-[laughs] -There could be
our exact universe, but with a different history.
Hillary Clinton won the election. Germany won the war.
And on that Earth, the dinosaurs might still rule.
[Anthony] I think when a lot of people hear some of these ideas,
they're like, "God, they were just up late drinking and having a good time
and came up with this crazy idea
'cause they wanted to think it in science fiction."
[roars]
But this is reality-based. [laughs]
[Brian] If the universe is infinite in extent,
there would be an infinite number of Einsteins in our universe.
And some of them would be talking to you
and probably be giving much better answers that I'm giving.
[Einstein echoing] According to the general theory of relativity,
it is probable that the universe is not infinite
but closed in upon itself.
Something like the surface of a sphere.
One thing I have learned in a long life…
[Brian] But those other Einsteins,
they're very far away.
[Einstein talks indistinctly]
[Brian] We may never be in contact with them,
because, according to his theory of relativity,
nothing can go faster than 186,000 miles per second,
which is the speed of light.
[mellow music plays]
We have this view of reality that--
We look at it in front of us, right?
I mean, I see you,
I see that white screen,
I see the camera, as they are now, right?
Which means if I see them as all you are now,
I see immediately.
So the light that comes from all these objects to me
arrives instantaneously at infinite speed.
Once upon a time, we did think that perhaps things happened instantaneously
and that information between one point and another
could be conveyed in the blink of an eye, in a single instance.
Well, no, of course. It doesn't come at infinite speed.
It takes some time to come.
So I don't see the now. I see the past in reality.
Why? Because there's no infinite speed.
[jaunty jazz music plays]
[announcer] Ladies and gentlemen, there he is.
The whizzingest wave, the peppiest particle. Philo T. Photon!
Move over, Magellan.
Philo is primed to perform his most fabulous feat of derring-do yet.
Circumnavigating the globe eight times in a single second.
[crowd cheers]
On your mark, Philo.
[old Western music plays]
[crowd cheering]
He did it, ladies and gentlemen. The son of a gun did it!
Oh…
[crowd cheers and applauds]
Let's see that again in real-time.
Do it again! Encore!
And now I'm sure Philo is pooped.
What's this?
He's going to fly to the next galaxy and back?
No one has ever attempted a stunt like this before.
Ladies and gentlemen, I can't watch!
[jaunty jazz music plays]
[Delilah] The speed of light is the maximum speed
at which anything can travel in the universe.
[somber music plays]
[clock ticks]
[Carlo] The speed of light, it's…
it's incredibly fast.
And the speed of light is horrendously slow.
If we wanna travel in the galaxy, it's hard enough, uh,
but if you want to travel in the universe from galaxy to galaxy,
we just cannot, because we cannot-- We're too slow.
And light is too slow. Incredibly slow.
[somber music continues]
[Delilah] So is the universe infinite?
[Carlo] It's hard to think what is the universe.
We know that we see the universe as many billion light years,
uh, and we know the universe is larger than what we see.
We have indications of that.
But it could be maybe ten times larger. Maybe a hundred times larger.
There are parts of the universe most probably
in which, even if we send a signal now, it will never get there.
So the universe is very, very, very, very, very big.
And it makes our heads spin.
[somber music continues]
[somber music ends]
[piano music plays]
[Alan] I was looking up at the sky one night
when I was about ten years old.
And…
I felt like my life didn't matter.
And I guess it was converting large space to large time.
One star after another star after another star…
[water dripping]
…and wondering whether that would keep going…
[wind chimes tinkling]
…forever.
I had this sense that the universe, it existed a long time before I was born…
and it would exist a long time after I was dead.
And I was just a speck… that didn't matter.
I don't matter. My parents don't matter.
Nothing matters.
We're all just specks.
We're just living in this brief moment.
None of us were here a million years ago.
None of us will be here a million years from now.
And the universe doesn't care.
It just goes on and on and on.
So…
why are we wasting time,
you know, going to school, having dentist appointments?
All of that.
Why are we wasting our time? Because none of it matters.
And then…
[birds singing]
…I fell in love.
And that changed everything.
That mattered. [laughs]
Even though we might both be specks in the cosmos.
[mellow music plays]
[Carlo] One of the oldest questions that thinkers puzzled on
is what happens in the universe if I could fly forever?
What would happen?
In the past, it seemed that there were two alternatives.
Both too strange.
One is that the universe is infinite and I could go forever.
And the other is that it's finite and there is a wall.
But if there is a wall,
then I could go through the wall,
and so what's next?
And Einstein published this spectacular paper.
He said, "No, no, no. The universe can be finite,
but without walls, because if I go in one direction,
I keep going, and I come back from the other direction."
Like it happens on the Earth.
If I walk on the Earth towards east,
I walk, walk, walk, walk. Do I find a wall? No.
Is Earth infinite? No. What happened? I just come back from the other side.
[somber music plays]
And it's very possible today
that the universe in fact has this shape.
Namely, has no boundary, there's no wall at the end of the universe,
but it's finite.
Huge, but finite.
[Rebecca] There's actually way more ways to make the universe finite,
just geometrically, abstractly playing around with whatever you want,
than there are to make it infinite.
In fact, there's an infinite number of ways to make the universe finite,
and there's really only a couple of ways to make it infinite.
[upbeat music plays]
[Delilah] We do know that the universe could be infinite,
but we don't actually have a mechanism to measure an infinite amount of length.
And because light travels at a finite speed
and the age of the universe is finite,
there will only ever be a finite subset of the universe
that we'll ever be able to observe.
Whether beyond that it's finite or infinite,
we just will never be able to experimentally know.
[upbeat music ends]
When we talk about infinity,
we can certainly talk about infinity in the extent of the spatial domain.
We can also talk about infinity in the temporal domain.
The duration of the universe.
It's possible that the universe will continue on expanding
into an infinite time in the future.
That's what the equations predict, actually.
[Brian] In 1998,
we discovered that the universe is not only expanding,
but the rate of expansion is accelerating.
[tense music plays]
The galaxies are moving away from each other with increasing speed.
And what this means
is that, eventually, we will be cut off
from other galaxies.
And so, as the stars in our galaxy eventually burn out…
as they will,
'cause all of them have a limited amount of nuclear fuel…
there eventually will not be any new energy sources in our galaxy.
[ethereal music plays]
And since we will be cut off from all other galaxies eventually,
maybe about 100 billion years from now…
there will not be any energy sources at all
and life will completely…
end.
And so, roughly 100 billion years from now…
that will be the end of life.
And if we keep on going along the cosmological timeline,
galaxies, planets, black holes,
everything that we know about, it will all disintegrate.
So all that would be left at that point
are particles wafting through the darkness.
[wind howling]
[tense music plays]
Now, if the universe exists in an infinite amount of time,
as we now think that it will,
the era of life is just a sliver of time
when you look at the full unfolding of the universe and time.
Even though 100 billion years seems like a long time,
it's nothing compared to infinity.
[Janna] Even if human beings make it past this particular… [laughs]
…immediate crisis, or other sentient life emerges,
there will be a last sentient being.
There will be a last living creature.
Even if the universe is infinite,
there will be a last thought.
[Delilah] A lot of people have kind of this visceral anxiety about not existing.
But, personally, I don't have this feeling.
I'm like, you know, once I didn't exist, before I was born.
There will be a point in time when I don't exist because I'll die.
And the same is true for our species.
And all life.
[drum music plays]
Um,
It's, it's-- You know, even physicists have feelings. [laughs]
And so we can ask, "How does that make you feel?"
"Are you-- As a physicist, is it daunting?"
"Is it, you know, an existential crisis?"
I… I have never--
I've always had the opposite reaction,
even to some of the most apocalyptic predictions
for the future of the universe.
It gives me a great sense of meaning and connectedness
to appreciate that you're part of this grand picture.
Nothing is permanent in that sense.
And to… to my mind, that's freeing.
It frees us from this focus on the permanent
as the place where value ultimately resides,
to a focus on the brief moment that we have,
in which we can understand things, and create beauty,
and experience wonder, regardless of how fleeting that experience may be.
That the universe itself gets to have its window of life,
its window of consciousness and beauty and love,
and then, poof.
It's sort of like the universe as a whole is living the way we do.
We're only here for a short time compared to infinity,
and, to me, that's a holy thought.
I mean, that's… as close as I can get to being religious.
The gift that you have of consciousness for the short time that you're here.
[majestic music plays]
[majestic music ends]
Wait, what was I trying to say? Something popped into my head.
[eerie music plays]
[Eugenia] There are interesting concepts in abstract math
where you can take something that appears to have one characteristic,
but you can put it inside something else or put it around something else
and it will look different from that point of view.
And, in a way, I think that the universe
is infinite in some sense, compared with our own lives,
because we just won't be there.
We won't be there to see it.
Whether it's really infinite or not,
I don't know, and I think that's wonderful.
[Delilah] Not knowing doesn't make me sad.
As a scientist, for me, not knowing is exciting.
[Stephon] There are things that I believe that our minds can't know,
but they are real and they exist.
If we want to call that the infinite, if we want to call that spirit,
if we want to call that God, whatever you want to call that thing,
I believe that thing is for real,
but not knowable.
[crickets chirping]
Infinity's very large. It's too big, infinity.
To me, infinity is an emotion that we get in front of the immensity of nature.
We are little teeny things.
We do science. It's great. I love to do science.
Try to figure out the quantum properties of gravity,
trying to figure out what is the shape of the cosmos.
But the reality is
we are, you know, like a little cat
trying to understand quantum mechanics.
The cat isn't going to understand quantum mechanics
because he has the brain it has.
A poor human like me won't understand everything about the universe
because of the poor brain I have.
The number of neurons in our brain
is comparable to the number of stars in the galaxies.
It is immense.
Which means that the space of thinkable thoughts
is a fantastically big number.
It's a one followed by billions of digits.
It's an incredibly large number.
Larger than anything we have encountered.
But, look, our brain,
it's a kilogram of meat
which can be in one configuration or the other, and that's it.
Uh, you can list all of them in principle.
Once again, there's no Infinity here,
and we have to confront the fundamental finiteness,
not of nature, but of ourselves.
[girl] One, two, three, four, five…
[producer] Do you think that human creativity is infinite?
-[girl] …eight, nine, ten… -[laughs] Uh…
[girl] …eleven, twelve, thirteen…
I sort of recoil at that suggestion.
I don't know. Oh, good gracious. I…
I don't know. I don't know that we are infinite.
[girl] Twenty-one, twenty-two, twenty-three. Seven, eight, nine,
ten, eleven, twelve.
Ninety-one, ninety-two, ninety-three, ninety-four, ninety-five, ninety-six…
[girl continues to count indistinctly]
[Steven] For some reason, thinking about anything about human beings being infinite
doesn't seem right to me.
We feel very bounded, to me.
Bounded in our rationality, bounded in our creativity.
[girl continues to count indistinctly]
[Delilah] Yeah, I don't even really know how to visualize infinity.
[girl] One, two, three…
[Kenny] Can I even picture a thousand?
[girl] …nine, five, two…
[Kenny] Even that number is too big for me to properly imagine.
[girl continues to count indistinctly]
I find that…
it's hard for me to really imagine numbers bigger than ten.
[upbeat music plays]
♪ Is it real or fantasy ? ♪
♪ Is it just a game we play ♪
♪ To pass the time while slips away In front of me ♪
♪ Possibilities to find As we walk the line ♪
♪ Faster and slower And over and over ♪
♪ We tumble and we intertwine ♪
♪ Solving the riddle And little by little we find ♪
♪ It goes on and on ♪
♪ Forever and ever and ever and ever ♪
♪ Love it's all that matters in the end ♪
♪ At the end of forever ♪
♪ Space is staring back at me ♪
♪ It's not easy to explain Where all just links in the daisy chain ♪
♪ Will you recognize just what you see? ♪
♪ Search and you won't miss the signs ♪
♪ The ones the universe left behind ♪
♪ From the beginning We're spinning and spinning ♪
♪ And spiraling out of control ♪
♪ It's never-ending ♪
♪ We never know where we will go ♪
♪ It's goes on and on ♪
♪ Forever and ever and ever and ever ♪
♪ Faster and slower And over and over ♪
♪ We tumble and we intertwine ♪
♪ It's never-ending ♪
♪ We never know what we'll find ♪
♪ It goes on and on ♪
♪ Forever and ever and ever and ever ♪
♪ Love isall that matters in the end ♪
♪ At the end of forever ♪
[echoing] ♪ Forever ♪
[upbeat music ends]
[somber music plays]
[somber music ends]