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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.