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