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- Hello, welcome to "Up and Atom."
I'm Jade and I have a thought experiment for you.
Imagine you have an infinitely large vase
and an infinite number of balls
all labeled one, two, three, and so on.
So obviously this is not an infinitely large vase
and this is not an infinite number of balls.
I'm on a budget, but this is a thought experiment.
And we can do things here
that we can't usually do in the real world.
The time is exactly one minute before noon.
You add balls one to 10 and remove ball ten.
At exactly half a minute until noon,
you add balls 11 to 20 and remove ball 20.
At one quarter of a minute until noon.
you add balls 21 to 30 and remove both 30.
You keep going in this fashion, adding more balls
and removing one at each time division until you reach noon.
Now the question is, how many balls are in the vase at noon?
If you'd like to think about this yourself first,
pause the video now and write your answer in the comments.
So if I were to try and answer this question,
I'd probably go something like this.
For every 10 balls you put in
you're only taking one ball out.
All you're really doing is putting in nine balls
at every step, right?
So at noon, we can agree that there will definitely
be some number of balls in the vase, but how many?
Well, how many times do you put balls in?
If we figure out the number of times you put the balls in
or time steps, we'll call them,
we can just multiply that by nine
and that should tell us how many balls there are.
Same simple enough.
The pattern is that each time step takes place
in half the amount of time as the last.
At 60 seconds, then 30 seconds,
then at 15 seconds and so on.
If we trade time as infinitely divisible,
then there's always enough time for the next step.
Whatever amount of seconds we end up with
we can always just divide it in half again.
Therefore there are an infinite number of time steps.
Infinity times nine is infinity.
So there should be infinitely many balls left
in the vase at noon.
(balls falling)
- My vase is empty.
- What?
- Yeah, it's empty.
Maybe I did it wrong.
- Yeah have probably.
- I thought you said to remove ball one
at the first time step and ball two at the second time step
and ball three at the third time step and so on.
Well, that shouldn't change anything
because we're still putting in nine balls
for everyone we're taking out.
- Yeah, but if it's the first time step we take out ball one
and the second time step we take at ball two
and the third time step we take at ball three,
and the billionth times step we take out ball 1 billion.
Then if there are an infinite number of timestamps
we take out an infinite number of balls.
- What about ball 3,436,000, is that still in there?
- Well, no, that was taken out on the 3,436,000 step.
Seriously, for every ball there was a timestamp
where I took it out.
The vase is empty.
This is called the Ross-Littlewood paradox
We have two arguments that each same reasonable
but that come to two completely different conclusion.
How?
Let's dive in.
Have we messed up somewhere in our counting?
Come to think of it.
What really is counting?
Let say, you've totally forgotten how to count.
It's just one of those days.
You're also surrounded by a bunch of cows,
again, one of those days.
Now each cow wants a milkshake
and you need to know whether to make more.
So how would you go about doing this?
Well, you can simply give the milkshakes out,
one milkshake per cow.
If there are cows without a milkshake
then there were more cows than milkshakes.
If there are extra milkshakes
there were more milkshakes than cows.
If we run out of cows and milkshakes at the same time,
then every cow has exactly one milkshake
and every milkshake belongs to exactly one cow.
And the set of cows was the same size
as the set of milkshakes.
This is called a one-to-one correspondence
and can be thought of as a way to count
without actually counting.
If we can place the elements of a set
in a one-to-one correspondence
with the elements of another set,
the two sets of the same size.
That is they have the same number of elements in them.
This is a pretty straightforward idea but the thing is
there were a finite amount of cows and milkshakes.
Will it help us with our problem where we have
an infinite amount of objects?
Well, let's see.
Let's test it out on two infinite sets of numbers.
Let's take the set of all the counting numbers.
One, two, three, four, and five, and so on,
and the set of counting numbers
without the first 100 elements.
So it starts at 101, then 102, 103, 104, and so on.
We'll call the set of all counting numbers, X
and the set of all counting numbers
minus the first 100 elements, Y.
So if I'd heard this for the first time,
I'd think, well, Y is definitely smaller than X
after all the two sets have all the same things in them
except that Y is missing the 100 smallest elements of X.
So why should be smaller, right?
Well, let's try a one-to-one correspondence idea to see.
Take the element one from X and pair it with 101 from Y.
Match two from X with 102 from Y.
Three from X with 103 from Y.
It looks like every element in X
gets paired with an element from Y.
Now because we know that the counting numbers go on forever,
we never going to run out of elements from either set.
Because of the pairing we using
we won't skip any either.
We can even say exactly which element of X
is paired with a given element of Y and vice versa.
For example, element 109,732 from X
is paired with element 109,832 from Y.
Because every element from each set is paired with exactly
one element from the other set,
this is a one-to-one correspondence
and the two sets have the same size.
That's a little bit surprising
but does it help us solve our paradox?
In the Ross-Littlewood paradox with the balls and the vase,
we aren't just putting in 100 extra balls
we're putting in nine balls for every ball with taking out.
So it seems like there should be nine times as
many more balls in the vase than we took out.
What could be an example that deals
with something more like that?
How about this time, we compare the infinite set
of counting numbers X, with the infinite set
of even counting numbers, we'll call it E.
We can tell just by looking
that E has half as many elements as X, right?
Well, let's see what happens
if we try a one-to-one correspondence.
Let's try matching two and X with two and E,
then four and X with four and E,
six and X with six and E, and so on.
This pairing doesn't seem to work.
We're missing all the odd numbers in X
which isn't a one-to-one correspondence.
So does that mean the sets aren't the same size?
X really is twice as big as E.
Well, before we jump to any conclusions,
let's try another pairing.
What if instead we match one from X, with two from E,
then match two from X with four from E,
then three from X with six from E, and so on?
With this pattern, we never run out of elements
from either set.
And every element from Y is paired with exactly one element
from E and vice-versa, this is a one-to-one correspondence.
This means that the two sets are in fact the same size.
So with the wrong pairing, it looked like X has twice
as many elements as E, but by finding the right pairing
we managed to show that they are in fact, the same size.
The same logic works for a set that looks like it has
nine times as many elements as another.
We can always find a one-to-one correspondence.
So what does this mean for our paradox?
How many balls there in the vase at noon?
Who was right me or Blade?
Well, the answer is,
(tapping table)
we were both right.
By choosing to take out balls one, two, three, and so on,
Blade managed to create a one-to-one correspondence
between the number of balls taken out
and the number of time steps.
That means all the balls were taken out.
So at noon we're left with zero balls.
I however, failed to create a one-to-one correspondence.
By taking out balls 10, 20, 30 and so on the remaining balls
never had a chance to be mapped to a specific time step.
And so we're never taken out.
So at noon, we left with an infinite number of balls.
In fact, depending on which balls you decide to remove
any chosen number of balls can be left in the vase at noon.
Can you see why?
Let me know your thoughts in the comments below.
This actually isn't a real paradox.
There is no logical inconsistency.
The way infinity behaves just goes
against our intuition about numbers.
Both explanations are perfectly reasonable.
Basic rules of arithmetic, like subtraction
and multiplication, are different for infinity.
I find this really interesting
because these concepts same fundamental to our reality
and the way we experienced the world.
Earlier in the video,
I said that nine times infinity is infinity
but that reasoning doesn't work because we're trying
to apply the usual laws of arithmetic
which are defined for numbers, not for infinity.
Isn't it weird that such fundamental ideas
don't apply to everything the same way?
Why not?
What makes infinity different from the usual objects
we deal with in everyday life?
The idea of infinity seems so natural to us
from the time we land to count.
The next question is, well
what would happen if I counted forever?
How can there be a biggest number
when I can always add one more?
But even everyday symbols of feelings of endlessness,
for example the sign of infinity,
which is called a lemniscate,
perfectly resembles the idea it represents
as one can trace it forever without end.
The idea of something going on forever and ever
fascinates and mesmerizes us.
Infinity seems like a straightforward continuation
of the basic idea of counting.
So then why doesn't it behave the same way?
Is it just some purely abstract entity with different rules
than mathematics based in the real world?
Well, infinity may not be so separate from the real world.
The very idea of forever is based on one of the most
fundamental concepts in physics, time.
Many cosmologists believe that our universe will one day
stop expanding and eventually fall in on itself.
But some believe that it will simply keep expanding forever.
But even if the universe did end, would time end?
Did time exist before the big bang?
It's not so straightforward to dismiss
the idea of infinity in a description of time.
Mathematics and therefore physics would be extremely limited
without the concept of infinity.
Even if this is just a mind tool we've created to
help us understand our world,
it's interesting that it works so well
given that infinity seems to obey different rules.
But back to the question of why the infinite
and finite behave differently in the first place
where are the boundaries
between the physical world and the mathematical world
and the way we experience the world?
I don't bring up these questions to be mystical or deep.
I just wanna point out that the nature of reality
is more complicated and fascinating than we might think.
Infinity is right on the threshold of one of the most
interesting questions in philosophy
is math invented or discovered?
Would the concept of infinity exists?
if us humans, weren't there to conceive of it?
Did we invent it to solve problems?
Or was it always there and we just discovered it?
This question fascinated me so much
that I made a feature video about it on Nebula.
Nebula is a streaming platform made by a bunch
of us educational, YouTube creators, where we can explore
making different and experimental content.
Oh, and it's now being nominated for a streaming award
which is pretty cool.
CuriosityStream a streaming service with award-winning
documentaries, is so supportive of our new venture
that they're offering Nebula completely free
when you sign up with them.
In fact, there's a really cool documentary
by {Hannah Fry} about this exact question
of where the math is an invention or a discovery.
It's full of trivia visuals and loads of mathematical ideas
but presented in a very easy to understand way.
I watched the entire three hour series in one go.
So if you do get the two for one deal,
make sure to watch Hannah Fry's "Magic Numbers."
And also my video on Nebula 'cause I think it's pretty good.
Sign up with the link in the description
or go to curiositystream.com/upandatom.
That's it for me and I'll see you in the next video.
Bye.
(upbeat music)