Voiceover: Any sequence you can come up with,
whatever pattern looks fun.
All your favorite celebrities birthdays lead into end
followed by random numbers, whatever.
All of that plus every sequence you can't come up with,
each of those are the decimal places
of a badly named so called real number
and any of those sequences
with one random digit changed
is another real number.
That's the thing most people don't realize
about the set of all real numbers.
It includes every possible combination of digits
extending infinitely among aleph null decimal places.
There's no last digit.
The number of digits is greater than any real number,
any counting number
which makes it an infinite number of digits.
Just barely an infinite number of digits
because it's only barely greater
than any finite number
but even though it's only the smallest possible
infinity of digits.
This infinity is still no joke.
It's still big enough, that for example
point nine repeating is exactly precisely one
and not epsilon less.
You don't get that kind of point nine repeating
equals one action
unless your infinity really is infinite.
You may have heard that some infinities
are bigger than other infinities.
This is metaphorically resonant and all
but whether infinity really exists
or if anything can last forever
or whether a life contains infinite moments.
Those aren't the kind of questions
you can answer with math
but if life does contain infinite moments,
one for each real number time,
that you can do math to.
This time, we're not just going to do metaphors.
We're going to prove it.
Understanding different infinities
starts with some really basic questions
like is five bigger than four.
You learned that it is
but how do you know?
Because this many is more than this many,
they're both just one hand equal to each other
except to fold it into slightly different shapes.
Unless you're already abstracting out
the idea of numbers and how you learn
they're suppose to work
just as you learned a long life
is supposed to be somehow more than a short life
rather than just a life equal to any other
but folded into a different shape.
Yeah, metaphorically resonate that.
Is five and six bigger than 12?
Five and six is two things after all
and twelve is just one thing and what about infinity?
If I want to make up a number bigger than infinity,
how would I know whether it really is bigger
and not just the same infinity
folded into a different shape?
The way five plus five
is just another shape for 10.
One way to make a big number
is to take a number of numbers, meta numbers.
This is where a box containing five and six
has two things and is actually bigger than a box
with only the number 12.
You could take the number of numbers from one to five
and put them in a box
and you'd have a box set of five
or you could take the number of numbers
that are five which is one
or you could take the number of counting numbers
or the number of real numbers.
It's kind of funny that the number of counting numbers
is not itself a counting number
but an infinite number often referred to
as aleph null.
This size of infinity is usually called countable infinity
because it's like counting infinitely
but I like James Grime's way of calling it
listable infinity because the usual counting numbers
basically make an infinite list
and many other numbers of numbers are also listable.
You can put all positive whole numbers
on an infinite list like this.
You can put all whole numbers including negative ones
You can list all whole numbers
along with all half way points between them.
You can even list all the rational numbers
by cleverly going through all possible combinations
of one whole number divided by another whole number.
All countably infinite numbers of things,
all aleph null.
Countable infinity is like saying
if I make an infinite list of these things,
I can list all the things.
The weird thing is that it seems like this definition
should be obvious that no matter how many things there are,
of course you can list all of them.
If your list is literally infinite
but nope so back to the reals.
Say you want to list all the real numbers.
If you did, it could start something like this
but the specifics don't matter
because we're about to prove
that there's too many real numbers to fit
even on an infinite list
no matter how clever you are at list finding.
What matters is the idea that you can create
any real number you want,
out of an infinite sequence of digits
and we're going to use this power to create a number
that couldn't possibly be on the list
no matter what the list is
even though the list is infinite.
All we need to do that is construct a real number
that isn't the first number on the list
and isn't the second number on the list
and isn't any number on the list
no matter what the list is.
Here's where I'm sure some of you are like "Yes!"
Cantor's diagonal proof.
Indeed my friends, that's what's going down.
In the first number on the list,
the first digit is one.
If I make a new number with a first digit is three
then even the rest of the digits are the same,
there's no way my new number
is equal to the first number on the list
though the rest of the digits
probably aren't all the same anyway.
The second number on the list does start with a three.
We don't know if this new number is the same or not yet
but I can make sure my new constructed number
is not the second number on the list
by making the second digit five
or eight or whatever
and I can make my number not be the third number
on the list
by making the third digit five
instead of three again.
I mean the new number was already different
from the third number on the list
but I don't even have to check the other digits
as long as I know that one of them
definitely conflicts which comes in handy
when I get to the 20 billion and oneth
number on the list
and I don't have to check the first 20 billion digits
against the 20 billion digits
I've constructed so far to be sure
that my new number is not the same
as the 20 billion and oneth number.
There's one digit in my number
for every number on the list
which means I can make a way for my new number
to not match every single number on the list
no matter what the list is.
Which means there's more real numbers than fit
on an infinite list.
This works no matter what the list is.
Take the diagonal and add two to every digit
or add five or whatever.
You can't actually sit down and write an infinite list
or infinite number though.
Here's another way to think about what's really going on.
We're trying to create a function that maps
one set of numbers to another.
You can map all the counting numbers
to all the whole numbers with a simple
minus one function or to all the even numbers
with a times two function
and map all the even numbers back
to all the counting numbers with the inverse function
division by two.
You can map all the real numbers between zero and one
to all the real numbers between zero and 10
by doing a times 10 function
and find every number has a place to go,
they match one to one.
The question is, is there a function
that maps every real number
or even just the real numbers between zero and one
to a unique counting number and vice versa.
Cantor's diagonal proof shows that any function
that claims to math counting numbers
and reals to each other must fail somewhere.
In fact you're not just missing one more real number
than fits on an infinite list
or else you could just add it to the beginning.
There's not just another infinite list
of numbers you're missing
or else you could zipper the lists together.
You could take every digit on the diagonal
and add either two or four
to get infinite combinations of numbers
that aren't on the list
and you can make a function that maps
those missing numbers to the real numbers in binary.
Of course the binary numbers
are just another way of writing the reals
which means an infinite list of real numbers
will quite literally be missing all of them.
Next to the infinity of the reals,
the infinity of infinite list
is actually mathematically nothing which is nuts
because countable infinity is still super huge,
The infinity of the reals is beyond
that what can we indicated with a simple dot, dot, dot,
a bigger infinity, a greater cardinal number.
We've gone beyond aleph null.
This is aleph one maybe.
Yeah funny thing that turns out
there's no way to tell how much bigger this infinity is
than aleph null.
Just that it's bigger like it could be the next step up
or there might be other sorts of infinites in between
but which one of those is the case?
It's kind of independent of standard axioms.
But whatever it is,
those are just two relatively small infinities
out of an infinite number of aleph numbers.
For every aleph number, there's infinite ordinal numbers
which I guess kind of are like
infinities folded into different shapes
and don't forget hyper real supernaturals,
or reals, etcetera.
I guess you could squeeze some Beth numbers in there
if you're into those axioms.
I don't judge.
Some of my best friends use the axiom of choice.
Anyway, some infinities are bigger than other infinities
but a mathematician would probably say
something more like I don't know,
there exists an aleph alpha
and aleph beta where aleph alpha
is greater than aleph beta or something
which is perfectly true.
Whether those different sorts of infinities
apply to something like moments of time is unknown.
What we do know is that if life has infinite moments
or infinite love or infinite being
then a life twice as long
still has exactly the same amount.
Some infinities only look bigger than other infinities
and some infinities that seem very small
are worth just as much as infinities 10 times their size.