mean things like the biggest number possible

or the entirety of everything,

or the universe, or God, or forever.

In math, all a number needs to be infinite

is to be bigger than any finite number.

No infinite number is going to behave

like one of the badly named so called real numbers.

Uh! Who decided to call them that?

An infinite number can be just barely bigger

than any finite number

or it can be a whole lot bigger than that.

They don't only come in different sizes,

they come in completely different flavors.

In this video all I want to do is give you an overview

of the many flavors of infinity

that have been discovered so far.

I want to give you a feel for different infinities,

like you have a feel for the halfway fiveness of five,

the even twoness of two, the singularness of one.

Countable infinity is the infinity of ...

it's the infinity of forever, of and so on,

of adding up one plus one plus one plus one ...

to get infinity,

or adding a half plus a fourth plus an eight ...

to get one.

This one is still the result of adding infinitely

but one isn't a huge number.

The way I see it,

countable infinity isn't such a big deal either.

It's just that infinite plus ones seem more impressive

than it really is.

We use one to describe big real world ideas all the time,

one person, one hour, one photon

and accountably infinite amount of real world things

seems incomprehensible or impossible.

Math doesn't know or care what you apply numbers to.

You want to use finite numbers to represents units of time

and particles and stuff,

that's not infinity's problem.

Countable infinity is not a number,

it's a mathematical description

that applies to many different infinite numbers

and functions and things.

Aleph null on the other hand is a number,

a meta number of sorts.

It's the number of counting numbers.

It's the first infinite cardinal number

in an infinite series of infinite cardinal numbers.

It's the only countably infinite one.

It's the precise number of hours in forever,

the number of digits of pi.

If countable infinity

is a series of individual piercing lights

along an infinite shoreline,

aleph null is a reflection in the water

of the stabbing lights.

They wave and flow and reorder themselves

to do things like make aleph null plus equal aleph null,

and aleph null squared equals aleph null.

Aleph null is a number and you can do numbery things to it

but it's not going to react to those numbery thing

the same way a badly named so called real number would.

Then there's the ordinals, ordered infinity.

Another kind of number entirely

where the lights can't flow and reorder themselves,

they're in a swamp and the lights congeal

into puddles of infinite light,

the countably infinite ordinal omega

is an ordinal number with exactly as many lights

as aleph null.

All those infinite lights congeal into the same pool

and if you add a light to the beginning of the line

of course it can congeal right on to the pile

and it's still omega light.

When you add a light in the distance

after infinite other lights, omega plus one,

the light is trapped behind the horizon.

It's stuck in order

beyond the last of these infinite lights.

It can't just glom on to the light pile

after the last of these infinite lights

because there is no last light.

This is infinite so it just hangs out there.

Omega plus one is larger than omega

and larger than one plus omega.

Obviously, infinite congealing swamp lights

are non-cumulative.

Those infinite countably infinite ordinals

and each different infinite ordinal

is a different pattern of congealed light.

Ordinals behave a little more like real numbers,

omega plus one plus two equals omega plus three.

But two plus omega plus three equals omega plus three.

The non-cumulativity lets you play with different shapes

of countable infinity

without accidentally making one equal two or something.

For omega plus three plus omega,

the three gloms on to the second omega

and then you get omega times two

which is different from two times omega

where they just meld in to each other.

You can do things like omega to the omega,

to the omega, to the omega ...

Okay, I'm getting destructed.

Anyway, ordinals are cool.

There are bigger cardinal numbers,

infinities that are fundamentally provably bigger

than the infinity you get by counting

which are cleverly called uncountable infinities.

The infinity that a ... can't even begin to approach.

First, the uncountable infinity of the real numbers,

smooth but individual,

a dense sea of things,

but any two no matter how close

are still measurably different,

they don't get stuck to each other.

They can be ordered into a line

yet they cannot be lined up one by one.

The cardinality of the reals,

which may or may not be aleph one

independent of standard axioms,

can be congealed into whole new bunches or ordinal numbers.

Then there's bigger transfinite cardinals,

bigger boxes containing bigger infinities.

In fact, there's an infinite amount of cardinals,

infinite sizes of infinity, aleph one, aleph two,

aleph omega, an infinite ordinals

with each of those cardinalities,

omega one's, omega two's.

I hear omega three's are good for your brain,

but if there's infinite kinds of infinity

it should make you wonder

just what kind of infinity amount of infinities are there?

Well, more than countable, more than uncountable,

that number is big they're infinite.

But the number of kinds of infinities

is too big to be a number.

If you took all the cardinal numbers

and put them in a box,

you can't because they don't fit in a box.

Each greater aleph allows infinite omegas

and each greater omega provides infinite greater alphas.

It's like how you can try to have a theoretical box

that contains all boxes,

but then it can't because the box can never contain itself.

So you make a bigger box to contain it

but then that box doesn't contain itself

just like the number of finite numbers

is bigger than any finite number.

The number of infinite numbers is bigger

than any infinite number and is also not a number,

or at least no one has figured out a way

to make it work without breaking mathematics.

Infinity isn't just about ordinals and cardinals either.

There's the infinities of calculus,

useful work courses treated delicately

like special cases.

Flaring up and dying down like virtual particles

with the sole purpose of leading some finite numbers

to their limits.

Your everyday infinities inherent in so much of life

but they get so little credit.

And there's hyper real numbers

that extend the reals to include infinite decimals,

close and soft, no drift of tiny numbers

on your other numbers that they're almost indistinguishable.

Hyper reals can describe any number system

that adds in infinite decimals to the reals

which you can do to varying degrees.

The fun part is that hyper reals,

unlike ordinals and cardinals,

follow the ordinary rules or arithmetic

which means you can do things like division.

If you divide one by a number that's infinite

decimally close to zero

you get a number thrown wide into the infinite.

Likewise, you can divide finite numbers

by infinitely large numbers

to get infinitely small but still non-zero numbers.

The super real numbers do similar sorts of things

but more so than there's the achingly beautiful

surreal numbers.

Open, vast, stretching in all dimensions

then flowering again in newly created dimensions,

filling every possible space

and then unfolding impossible space.

The surreals encompass the reals, the hyper reals,

the super reals and also all the infinite infinities

of the ordinals

which means the surreal numbers contain every cardinality.

There is no set of all surreal numbers

because there's too many to fit in this set.

They're basically the most numbery numbers possible.

They're provably the largest ordered field

depending on your axioms and they still act like numbers.

You can do arithmetic to them,

add, subtract, multiply, divide, cumulative, associative,

multiplicative and additive identities.

You can do things like infinity minus one

and infinity divided by two,

everything works except dividing by zero.

You can divide one by infinitely small numbers

you get infinitely large ones

but you still can't divide by zero

without ruining everything.

Which is why zero is a much weirder number

than any of these infinities.

There's other sorts of infinities

using other different ways of thinking about numbers.

If you think of the individual numberiness

of the natural counting numbers

as coming from the unique number of plus ones

they have in them,

then defining infinity

as being an infinite amount of plus ones makes sense.

Each natural counting number

also has a unique prime factorization.

Many people think of the prime factors of a number

has being what makes up its individual numberiness.

There's a way in which 16 is much closer to 32

than it is to 17, and they're the supernatural numbers.

The supernatural numbers are a system that decided,

"Yup, the natural numbers get their numberiness

"from their unique combination of prime factors."

What if you allow infinite prime factors

in a plus one sort of number definition?

Two times two times two times two ...

is the same as seven times seven times seven

times seven ...

In a prime factor definition

these two infinities are fundamentally different,

they feel different.

Supernatural numbers can be multiplied and divided

and you can find the greatest common factor

of infinite supernatural number A

and infinite supernatural number B.

What you can't do is add them.

They don't really work unless you let go of the idea

that 16 and 17 are plus one buddies.

In fact, you can't really tell whether one infinite

supernatural number is bigger than another.

Sure, those supernatural numbers

include the natural numbers.

But in the supernatural version

the natural numbers don't belong in this plus one order.

You can, however, order them all in a p-adic way

which does put 16 closer to 32 than 17.

It's funny because the p-adic numbers

are like the most infinite looking numbers ever.

They have their digits going infinitely to the left,

but this is just a notation thing.

P-adic numbers are, "Okay, rational numbers makes sense.

"Now, let's complete the number system

"to be this weird alternative

"to the badly named so called real numbers."