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Infinity isn’t a number. It's something much weirder.
Infinity isn’t a number. It's something much weirder.
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Subtitles (269)
0:00
Thank you to AnyDesk for supporting PBS.
0:03
This is a problem that can break
0:04
the entire idea of infinity.
0:06
It starts with a vase and infinitely many numbered balls.
0:11
(balls falling)
0:14
Let's start one minute before noon.
0:16
I put balls one through 10 into the vase.
0:18
Then I remove ball number one.
0:21
30 seconds before noon, balls 11 through 20 go in
0:24
and I take ball two out.
0:25
15 seconds later, half way to noon.
0:29
21 through 30 go in, ball number three out.
0:31
We keep going, each step half as long as the last forever.
0:35
At exactly noon, after infinitely many steps
0:38
of adding 10 balls and removing one,
0:41
how many balls are in the vase?
0:43
The answer is infinite… and also zero.
0:47
Both are correct at the same time.
0:49
This is a video about how infinity
0:51
doesn't mean what we think it does
0:54
and what happens when our brains try to make sense
0:56
of more than everything.
0:58
(music)
1:04
Hey, smart people, Joe here.
1:05
How long does it take for infinity to happen?
1:09
That question sounds kind of ridiculous,
1:11
like if you ask ChatGPT to do a Vsauce impression.
1:15
I can never do that.
1:17
But to prove it, just imagine walking across a room.
1:20
First you have to cross half the room,
1:22
then half of what's left, then half of that.
1:26
Infinite steps, each step half as close to the finish,
1:29
but never there yet.
1:31
Here I am across the room.
1:33
This very problem was devised
1:35
by the Greek philosopher Zeno 2,500 years ago.
1:38
He wanted to prove that motion is impossible,
1:42
which we know is silly because it is obviously possible.
1:47
Look.
1:48
Zeno was right that there are infinitely many steps,
1:51
but he was wrong that they add up to infinite time.
1:54
You see, if each step keeps getting shorter fast enough,
1:57
infinite things can fit in finite time.
2:01
Half a minute plus a quarter minute
2:03
plus an eighth of a minute, so on forever.
2:05
That adds up to exactly one minute.
2:08
Infinite steps, finite time.
2:11
And completing finitely many operations
2:13
inside of a finite window?
2:15
That's what mathematicians call a supertask.
2:17
You can't do a supertask with your real human hands,
2:21
but the mathematics works.
2:22
And that math forces us to ask
2:24
some pretty weird questions about
2:26
how infinity is different from, you know, non-infinity.
2:32
When you take supertasks seriously,
2:34
infinity starts revealing some deeply strange stuff.
2:37
Let's say that you run a hotel with infinite rooms
2:41
and every single room was occupied.
2:44
A new guest shows up at the front desk.
2:46
Instead of telling them no vacancy,
2:48
just ask the guest in room one to move to room two,
2:51
room two to room three, and so on forever.
2:54
Now, room one is free.
2:56
The hotel is still completely full,
2:58
but you've still added a guest.
3:00
The rooms represent an infinite set.
3:02
You can add to the whole without changing
3:05
the size of the hole, and both are still infinite.
3:08
This sounds ridiculous, but this isn't a glitch in the math.
3:13
So what happens if 10 guests arrive?
3:15
You don't need to break a sweat,
3:16
you just get on the intercom.
3:17
And ask every current guest to move to the room
3:20
10 numbers higher than theirs.
3:22
Room 1 to 11, room two to room 12,
3:25
and so on forever down the hall of infinite rooms.
3:29
But now rooms one through 10 are empty
3:32
and all 10 new guests get a bed
3:34
and the hotel is still completely full
3:36
just as it was before.
3:37
So what if an infinite bus arrives carrying infinite guests?
3:41
Trickier, but you have a move for this too.
3:44
Just ask every current guest to move to the room
3:47
that's double their current number.
3:49
Room one goes to room two.
3:51
Room two goes to room four.
3:53
Room three to room six, and so on.
3:56
Now every odd numbered room
3:58
in the entire infinite hotel is empty.
4:01
And since there are infinitely many odd numbers,
4:03
every passenger on the infinite bus gets a room.
4:06
We fit infinity inside of infinity.
4:11
Now, let's say inside every one of those infinite rooms
4:14
is a lamp.
4:15
Then one minute before noon, you flip the switch to on.
4:18
30 seconds before noon, lamp off.
4:21
15 seconds before noon, on again.
4:23
The flipping doubles in pace forever.
4:25
We flip the switch infinite times before noon arrives.
4:29
So at exactly noon, is the lamp on or off?
4:33
There is no right answer,
4:36
not because we haven't figured it out yet.
4:38
There literally isn't an answer.
4:40
When we get to noon, it doesn't end on, it doesn't end off.
4:46
It just ends.
4:47
This is one of the things that makes supertasks so weird.
4:50
They can end without a final ending state.
4:54
Problems like these introduce ideas that are hard to accept
4:57
because they don't match our experience in the real world,
5:01
but they are still just as true according to mathematics.
5:04
Infinity isn't one thing,
5:07
and some infinities are bigger than others.
5:09
The whole numbers go on forever.
5:11
One, two, three, four, et cetera,
5:14
and the fractions go on forever too.
5:16
But we can pair every fraction to a whole number one to one
5:20
with nothing left over in either set.
5:22
So these infinities are the same size,
5:25
but the decimals don't work that way.
5:28
There are too many of them to match these infinities.
5:31
In the 1870s, mathematician Georg Cantor proved this,
5:35
and his argument is an elegant one.
5:37
Suppose that you claim to have a complete list
5:39
of every decimal.
5:41
This one, this one, and so on.
5:43
We can attempt to match every decimal number one for one
5:47
with every whole number,
5:48
but I can always find a decimal that's not on your list.
5:53
Take the first digit of your first number and change it.
5:56
Take the second digit of your second number and change it.
5:58
The third digit of the third and so on.
6:01
the decimal I've built differs from every number
6:04
on your list in at least one digit position.
6:07
So your complete list of decimals
6:10
isn't so complete after all.
6:11
Cantor showed that whatever infinity you're looking at,
6:15
you can always construct a larger one.
6:17
The decimal infinity is bigger
6:19
than the whole numbered infinity.
6:21
But there's an infinity bigger than that one too,
6:23
and one bigger than that.
6:25
The ladder of infinities goes up forever.
6:28
Infinity also breaks arithmetic in a way
6:31
that honestly kind of bothers me.
6:33
Take this infinite series.
6:35
One minus one, plus one, minus one, plus one, and so on.
6:40
What does that equal?
6:41
Well group the terms one way and you get zero.
6:44
But start from a different grouping and you get one.
6:48
We can even rearrange the math in a different way.
6:50
Let's subtract this series from one.
6:53
Do some basic algebra to rearrange things,
6:56
and the answer ends up being one half.
7:01
It's an equally valid answer,
7:03
but for certain infinite sums,
7:05
the order that we add things up changes the result,
7:09
which brings us back to the vase.
7:11
This thought experiment was originally devised
7:13
by mathematicians John Littlewood and Sheldon Ross.
7:15
It's known as the Ross-Littlewood paradox,
7:18
and it's a paradox
7:19
because two completely true lines of reasoning
7:22
can give us completely opposite answers.
7:25
Recall that at every step we add 10 balls and remove one,
7:29
and each step is done in half as much time
7:32
as the one before.
7:33
A net gain each step of nine balls.
7:36
So after 10 steps, roughly 90 balls.
7:39
After 100 steps, 900.
7:41
run that out over infinitely many steps
7:44
and the number of balls in the vase is infinity.
7:47
Then again, maybe the vase is empty.
7:50
At noon, which balls are in the vase?
7:53
Let's pick any ball.
7:54
Ball 47.
7:56
That went in during step five
7:58
and it came out again during step 47.
8:01
Ball one left in step one, ball 1,000 left in step 1,000.
8:07
Name any ball that I can name the exact step
8:10
when it was removed.
8:11
So at noon after infinite steps,
8:14
there is no ball in the vase
8:16
that wasn't eventually taken out.
8:18
But suppose instead of removing the lowest numbered ball,
8:21
I change which ball I take out each step.
8:24
Same deal, 10 balls in, one ball out.
8:27
But now instead of removing ball one in step one,
8:31
I remove ball 10.
8:33
You're the last one that I just added.
8:34
Step two, balls 11 through 20 go in, ball 11 out.
8:38
Step 23, ball 21-30 are in.
8:41
Ball 12 out and so on.
8:43
Now, which balls are left at noon?
8:45
Ball 10 was removed in step one.
8:48
Ball 11 in step two, every subsequent ball
8:51
gets pulled out in some future step
8:53
except balls one through nine.
8:56
Nobody ever touches them.
8:58
No step exists where they get removed.
9:00
Exactly nine balls remain at noon.
9:03
We did the same process at the same rate,
9:05
10 in, one out every single step.
9:09
But by changing which balls I remove,
9:11
the answer went from infinity to zero to nine.
9:16
The vase doesn't have a true answer
9:19
because the answer is whatever the process says it is.
9:22
All of these arguments have sound logic.
9:25
The secret is that each scenario in our paradox
9:29
is actually tracking different things.
9:31
One watches the size of the set at each step
9:33
and then extrapolates forward.
9:35
The second traces the fate of individual balls.
9:39
And the third shifts how we group things
9:41
to get a totally different answer.
9:43
In truth, we can't have an answer for what happens
9:46
after infinite steps because there is no final step at all.
9:50
Asking what's in the vase at noon
9:52
is like asking what's north of the North Pole?
9:55
The question makes grammatical sense,
9:58
but it doesn't refer to anything real.
10:00
The Vase and Balls paradox doesn't really have a solution
10:05
because the fact that it has many solutions IS the solution.
10:08
This is what infinity actually is.
10:11
Not a very large number, not the number
10:13
at the end of the number line.
10:16
I've learned to think of it as a direction, a process,
10:20
something that you can approach forever
10:22
without ever arriving.
10:24
Treat infinity like a number and it will fool you.
10:27
Mathematics can describe what happens every individual step.
10:32
It's a beautiful machine that can take our brains
10:34
right to the edge and then show us
10:37
that the edge isn't there.
10:39
Stay curious.
10:41
And thank you to AnyDesk for supporting PBS.
10:45
AnyDesk was created to provide fast, secure remote access,
10:48
whether you're working from home, on the road
10:50
or managing systems across multiple locations.
10:53
With AnyDesk, you can remotely access computers
10:55
and servers directly, enabling secure file transfer
10:58
without relying on consumer cloud storage.
11:01
And there's no unnecessary data collection.
11:03
It's lightweight and privacy-respecting.
11:05
Plus AnyDesk is built with their proprietary codec desk RT,
11:09
and runs on servers that use airline technology.
11:11
This helps ensure minimal latency and crisp visuals
11:14
even over slow or unstable internet connections.
11:17
AnyDesk is available for personal use,
11:18
but if your research requires a lot of global coordination,
11:21
AnyDesk offers custom tailored business plans too.
11:24
Learn more about AnyDesk.
11:25
Go to AnyDesk.com/besmart
11:28
or check out the link in the description.
11:30
And as always, thank you to everyone
11:32
who supports the show on Patreon,
11:34
including these fine folks at our top tier
11:36
and everyone else who's part of our community.
11:40
You help make this show possible in a very real way.
11:44
Videos like this take a lot of resources and time
11:47
and energy to put together,
11:48
and we could not do it without you.
11:51
I am infinitely grateful.
11:56
See you in the next video.
11:57
More than everything.
11:59
Rest of the episode. Paint it out.
12:02
It will fool you.
12:04
Don't kick the stool in the middle of your line.
12:06
(stool creaking)
