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• Voiceover: Some people's personal definitions of infinity

• 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