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Practice English Speaking&Listening with: Proof some infinities are bigger than other infinities

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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

by alternating.

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,

it's infinite.

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.

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