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- This is a problem that can break the entire idea of infinity.

It starts with a vase and infinitely many numbered balls.

(balls ping) Let's start one minute before noon.

I put balls one through 10 into the vase.

Then I remove ball number one.

30 seconds before noon, balls 11 through 20 go in and I take ball two out.

15 seconds later, half the way to noon.

21 through 30 go in, ball number three out.

We keep going, each step half as long as the last forever.

At exactly noon, after infinitely many steps of adding 10 balls and removing one, how many balls are in the vase?

The answer is infinite and also zero.

Both are correct at the same time.

This is a video about how infinity doesn't mean what we think it does and what happens when our brains try to make sense of more than everything.

(gentle music) Hey, smart people, Joe here.

How long does it take for infinity to happen?

That question sounds kind of ridiculous, like if you ask ChatGPT to do a Vsauce impression.

Can never do that.

But to prove it, just imagine walking across a room.

First you have to cross half the room, then half of what's left, then half of that.

Infinite steps, each step half as close to the finish, but never there yet.

Here I am across the room.

This very problem was devised by the Greek philosopher Zeno 2,500 years ago.

He wanted to prove that motion is impossible, which we know is silly because it is obviously possible.

Look.

Zeno was right that there are infinitely many steps, but he was wrong that they add up to infinite time.

You see, if each step keeps getting shorter fast enough, infinite things can fit in finite time.

Half a minute plus a quarter minute plus an eighth of a minute, so on forever.

That adds up to exactly one minute.

Infinite steps, finite time.

And completing finitely many operations inside of a finite window?

That's what mathematicians call a super task.

You can't do a super task with your real human hands, but the mathematics works.

And that math forces us to ask some pretty weird questions about how infinity is different from, you know, non-infinity.

When you take super tasks seriously, infinity starts revealing some deeply strange stuff.

Let's say that you run a hotel with infinite rooms and every single room was occupied.

A new guest shows up at the front desk.

Instead of telling them no vacancy, just ask the guest in room one to move to room two, room two to room three, and so on forever.

Now, room one is free.

The hotel is still completely full, but you've still added a guest.

The rooms represent an infinite set.

You can add to the whole without changing the size of the hole, and both are still infinite.

This sounds ridiculous, but this isn't a glitch in the math.

So what happens if 10 guests arrive?

You don't need to break a sweat, you just get on the intercom.

And ask every current guest to move to the room 10 numbers higher than theirs.

Room 1 to 11, room two to room 12, and so on forever down the hall of infinite rooms.

But now rooms one through 10 are empty and all 10 new guests get a bed and the hotel is still completely full just as it was before.

So what if an infinite bus arrives carrying infinite guests?

Trickier, but you have a move for this too.

Just ask every current guest to move to the room that's double their current number.

Room one goes to room two.

Room two goes to room four.

Room three to room six, and so on.

Now every odd numbered room in the entire infinite hotel is empty.

And since there are infinitely many odd numbers, every passenger on the infinite bus gets a room.

We fit infinity inside of infinity.

Now, let's say inside every one of those infinite rooms is a lamp.

Then one minute before noon, you flip the switch to on.

30 seconds before noon, lamp off.

15 seconds before noon, on again.

The flipping doubles in pace forever.

We flip the switch infinite times before noon arrives.

So at exactly noon, is the lamp on or off?

There is no right answer, not because we haven't figured it out yet.

There literally isn't an answer.

When we get to noon, it doesn't end on, it doesn't end off.

It just ends.

This is one of the things that makes super tasks so weird.

They can end without a final ending state.

Problems like these introduce ideas that are hard to accept because they don't match our experience in the real world, but they are still just as true according to mathematics.

Infinity isn't one thing, and some infinities are bigger than others.

The whole numbers go on forever.

One, two, three, four, et cetera, and the fractions go on forever too.

But we can pair every fraction to a whole number one to one with nothing left over in either set.

So these infinities are the same size, but the decimals don't work that way.

There are too many of them to match these infinities.

In the 1870s, mathematician Georg Cantor proved this, and his argument is an elegant one.

Suppose that you claim to have a complete list of every decimal.

This one, this one, and so on.

We can attempt to match every decimal number one for one with every whole number, but I can always find a decimal that's not on your list.

Take the first digit of your first number and change it.

Take the second digit of your second number and change it.

The third digit of the third and so on.

the decimal I've built differs from every number on your list in at least one digit position.

So your complete list of decimals isn't so complete after all.

Cantor showed that whatever infinity you're looking at, you can always construct a larger one.

The decimal infinity is bigger than the whole numbered infinity.

But there's an infinity bigger than that one too, and one bigger than that.

The ladder of infinities goes up forever.

Infinity also breaks arithmetic in a way that honestly kind of bothers me.

Take this infinite series.

One minus one, plus one, minus one, plus one, and so on.

What does that equal?

Well group the terms one way and you get zero.

But start from a different grouping and you get one.

We can even rearrange the math in a different way.

Let's subtract this series from one.

Do some basic algebra to rearrange things, and the answer ends up being one half.

It's an equally valid answer, but for certain infinite sums, the order that we add things up changes the result, which brings us back to the vase.

This thought experiment was originally devised by mathematicians John Littlewood and Sheldon Ross.

It's known as the Ross-Littlewood paradox, and it's a paradox because two completely true lines of reasoning can give us completely opposite answers.

Recall that at every step we add 10 balls and remove one, and each step is done in half as much time as the one before.

A net gain each step of nine balls.

So after 10 steps, roughly 90 balls.

After 100 steps, 900. run that out over infinitely many steps and the number of balls in the vase is infinity.

Then again, maybe the vase is empty.

At noon, which balls are in the vase?

Let's pick any ball.

Ball 47.

That went in during step five and it came out again during step 47.

Ball one left in step one, ball 1,000 left in step 1,000.

Name any ball that I can name the exact step when it was removed.

So at noon after infinite steps, there is no ball in the vase that wasn't eventually taken out.

But suppose instead of removing the lowest numbered ball, I change which ball I take out each step.

Same deal, 10 balls in, one ball out.

But now instead of removing ball one in step one, I remove ball 10.

You're the last one that I just added.

Step two, balls 11 through 20 go in, ball 11 out.

Step 23, ball 21-30 are in.

Ball 12 out and so on.

Now, which balls are left at noon?

Ball 10 was removed in step one.

Ball 11 in step two, every subsequent ball gets pulled out in some future step except balls one through nine.

Nobody ever touches them.

No step exists where they get removed.

Exactly nine balls remain at noon.

We did the same process at the same rate, 10 in, one out every single step.

But by changing which balls I remove, the answer went from infinity to zero to nine.

The vase doesn't have a true answer because the answer is whatever the process says it is.

All of these arguments have sound logic.

The secret is that each scenario in our paradox is actually tracking different things.

One watches the size of the set at each step and then extrapolates forward.

The second traces the fate of individual balls.

And the third shifts how we group things to get a totally different answer.

In truth, we can't have an answer for what happens after infinite steps because there is no final step at all.

Asking what's in the vase at noon is like asking what's north of the North Pole?

The question makes grammatical sense, but it doesn't refer to anything real.

The Vase and Balls paradox doesn't really have a solution because the fact that it has many solutions is the solution.

This is what infinity actually is.

Not a very large number, not the number at the end of the number line.

I've learned to think of it as a direction, a process, something that you can approach forever without ever arriving.

Treat infinity like a number and it will fool you.

Mathematics can describe what happens every individual step.

It's a beautiful machine that can take our brains right to the edge and then show us that the edge isn't there.

Stay curious.

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See you in the next video.

More than everything.

Rest of the episode.

Paint it out.

It will fool you.

Don't kick the stool in the middle of your line.

(stool creaking)