How real are the real numbers, really?

Say Achilles wants to run a race.
First he’d have to run 1/2 way, then half the rest of the way then, then half of that
I just wanted to point out, to get to the end, Achilles had to run through every fraction
of the length at some point.
So here’s the question for this video.
Does Achilles just run from fraction to fraction?
If your first instinct is like mine, you might say, of course not.
He doesn’t jump from one spot to another; he smoothly goes along all the points.
But see, if he goes along all the fractions, he doesn’t have to jump.
He starts at 0, but what’s the closest fraction to 0?
Well, pick any really small fraction, there’s always another that’s closer, just pick
the one that’s half of it.
Then pick one closer than that by looking at the half way point again, and again and
again, to infinity.
This means, that there is never any distance between the fraction Achilles is on, and the
next one, so there’s never any jumps.
Ok, so it’s looks like we’re done, Achilles can move smoothly along the fractions.
But see how we’ve introduced infinity here?
The total distance is just a finite number, but for Achilles to move smoothly along it,
he has to along every fraction, and there are a lot of fractions.
It’s not just that there’s an infinite number of fractions along this line, but that,
even between any two fractions, no matter how tiny the distance is between them, there
is an infinite number of more fractions.
That’s an insane infinity.
But don’t worry, there is way way more.
When we think of writing a fraction out in decimal, it’s either some finite string
of digits, or sometimes and infinite one.
But not all infinite strings of numbers are fractions.
Of course there’s all the classic examples.
We’ll call these numbers irrational, the ones that can’t be written as fractions
but can be written as a huge string of numbers.
We said the fractions had no gaps right?
But consider this.
Pick some irrational number, any.
Let’s say mine starts off like this.
Then say that Achilles runs with a constant speed, but at first he runs .7 of the way,
which is a fraction, so he can do that.
Then he runs 0.09 more, again, a fraction, so it’s fine.
Then 0.003, etc etc.
Each bit of the race is a fraction, so he’s allowed to run that much.
He will stop when he reaches the end of this process- you might be saying but he never
will, there’s an infinite number of steps.
But don’t forget, he’s running at a constant speed.
That means the amount of time he takes to run these smaller amounts is tiny.
So he’ll get there in a finite amount of time, in fact in slightly less time than it
would take him to run .8 of the way.
So gets there alright.
But the question is, where is there?
It’s this irrational number.
If he could only land on fractions, we would be fine through this process since he’s
always landing on a fraction, but gets to the end and is on a point we didn’t think
existed- so does just dissapear?
Since that doesn’t seem reasonable, we’re forced to accept that all these irrational
numbers are there too, between the fractions numbers, even though we showed that there
are no gaps between the fractions.
The fractions and the irrationals are collectively called the real numbers, because they’re
the one’s we need for real world.
The fact that we need real numbers might seem obvious now, but it was a long long process
in maths to invent the right numbers to support reality.
This problem really puzzled mathematicians: Suppose that we want to know how fast achilles
is running.
If we just want his average, we’d just get the total distance, and divide it by his time.
But say Achilles tends to change his speed quite a lot during the race.
You want to know exactly what his speed is at a particular point.
Then what do you do?
A better estimate is start timing for a short amount of time after he passed that point,
and see how far he’s gone.
Then find the average speed from that.
But of course that’s not completely accurate either, his speed may have changed a bit even
in that smaller time.
You’d want to make it more accurate by picking a smaller time still, and doing the same thing.
But of course, no matter how small you pick the time, this is never 100% accurate... but
we kind of need to know he’s speed at that point in order to do physics.
Well, then Libniz and Newton came along with a great idea.
They said, why don’t you look at what happens when the time gap is really small.
Like basically zero.
But not zero.
But smaller than any number.
Let’s call it an infinitesimal.
Then do the same calculation with your infinitesimal this time.
Done.
Problem solved.
If that sounds like magic to you, then don’t worry, it is, but they cheerfully accepted
dividing by 0 because, well it worked.
No one knew why it worked though.
But it eventually started to really bother mathematicians.
They didn’t like these weird infinitesimals, and how they weren’t zero sometimes and
they were later- I mean, that’s not very rigourous and mathematical sounding.
So they went and started upheaving big chunks of maths to make it more rigourous, in the
hope they would eventually be able to explain calculus without the magic.
Their hunt lead them all the way back to the numbers themselves.
They realised that what they needed was the real numbers.
See this property that the real numbers had, that if a sequence is getting closer and closer
together, that it is going to converge to something, that’s exactly what you need.
Let’s see why.
Back to our speed calculation.
If you do that process of averaging, each time halfing the time that you use, and repeat
to infinity, you get a long sequence of average times.
Let’s say something like this.
You can see that the numbers get closer.
But since we’re dealing with real numbers, we know that this is all going to some number.
Whatever that number is, define that to be the speed of Achilles at the point.
So real numbers saved the day.
But not without a price.
We saw that, even in a finite amount of space, there is an infinite number of fractions,
and then an infinite number of irrationals slotted between them.
That is much more infinity then you might have imagined fitting in there.
But the plot thickens...
See, mathematicians where happy to discard all those nasty infinitesimals from maths.
But in the 60s, a mathematician discovered that you can reformulate all of calculus completely
rigourously... using infinitesimals.
Using that method, that dividing by zero but not really dividing by zero thing is completely
valid.
But instead of getting rid of real numbers to do this, you actually add in another infinite
set of numbers, called the hyperreals, that slot into the gaps between the reals.
But that’s a story for another time.
The conclusion of this video is that we have an infinite number of numbers but
But, just because your universe is finite in size, doesn’t mean it’s totally finite.
Physics still needs a lot of infinity to support it.
I’ll explain why getting Achilles from A to B requires you to assume there are way
more numbers in this distance than you might have imag