無邊際有多少種？ (How many kinds of infinity are there?)

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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
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."
Let's face it, the badly named so called real numbers
aside from rationals are a kind of super-weird themselves.
There's tons of awesome number systems
that don't contain infinite numbers
like all the types of hyper complex numbers.
Though you can apply the surreal numbers
to the complex numbers,
you get the surcomplex numbers
which seem maximal if you want numbers
that are cumulative and associative and add
and divide and stuff.
There's something like the ordinal octonions
would be tempting.
Anyway, there's lots of infinite numbers
but those are all numbers.
What about infinite space, geometric infinities?
What about the line with a point at infinity
that turns it into a circle?
That's the thing,
positive infinity equals negative infinity, no problem.
There's only all of projective geometry
where you casually treat infinity
just like any other point.
You can also divide by zero if you really want.
Yey! Projective geometry.
In projective geometry parallel lines do meet
at infinity.
While infinite space could refer to how your big
your space is in a distance sort of way,
it could also refer to the number of dimensions.
Infinite dimensional space is totally a thing.
A Hilbert space is the name for Euclidean space
with an arbitrary number of dimensions
and it could be countably infinite dimensions
or any cardinal number of dimensions.
You could take any ordinal number
and turn it into an ordinal space
which brings us to topology.
Topology deals with stretching and bending things
like lines and spheres and mobius strips
to figure out how they connect to themselves
with no regard for distance.
You can imagine taking a short line segment
and stretching it into a longer one
and then stretching it infinitely to get an infinite line.
Or you could take an infinitely long line
and contract it down to a point.
This line segment looks fundamentally smaller
than an infinite line.
There's a trick, this segment has no end points.
it's like the interval between zero and one
but not including zero and one.
If you were a point on this line
and wanted to get to the very end,
well, what's the last real number before one?
It's smaller than .9 repeating which is equal to one,
so you can just keep going and going and going
higher and higher and you'll never get to the end
which is the exactly the same thing that happens
when travelling along an infinitely long lines
that topology sees this line segment
as exactly the same thing as an infinite line
but your usual infinite line is only the usual infinite.
In topology there's a longer kind of line
very appropriately, yet confusingly called the long line.
The long line is so long that you can't stretch
a finite line segment to be a long line
even if you stretch it infinitely long.
Likewise, the long line is not contractable.
Here's one part of the long line and here's another,
but even if you grab them
and stretch them towards each other forever
you can never get them from here to there.
Infinite stretching isn't enough.
These points are part of the same line
but they don't connect to each other.
That's how long the long line is,
it's really long
which is just as ridiculous and awesome
and entirely mathematically provable
as any of these other stuff.
Finally, there is the not quite mathematical concept,
big omega, absolute infinity.
This biggest infinity would necessarily
have to contract itself
like the box containing all boxes.
Even though it's not a mathematically consistent thing,
some mathematicians still believe in it in a,
"I know this isn't really a thing
"and I can't do math to it, but it's still a thing."
sort of way.
Those are just the many different mathematical
meanings of infinity that I know of
and I probably made some mistakes
describing the things outside my usual areas
because these things are in such different fields
of mathematics.
Cardinals come from set theory.
Surreals come out of game theory.
Supernaturals comes from field theory.
Ordinal spaces are used in topology.
Hilbert spaces are used in analysis and quantum physics.
Infinity lets you do the projections of projective geometry
and infinite decimals let you avoid zeroes
in computational geometry.
In fact, it's finite mathematics that's rare
[unintelligible] or course of arithmetic.
A version of arithmetic that doesn't allow infinities
can have its strength as a mathematical system
quantified by applying ordinal analysis
a part of proof theory.
PRA only has a proof theoretic ordinal omega to the omega.
That is some weak sauce.
Anyway, I'm going to stop before this video
becomes infinite.
Let me know if know of any other sorts of infinities
and I'll make a list because apparently
no one has done that before.