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# How many kinds of infinity are there?

Video transcript

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.