Subtitles section Play video Print subtitles 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