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Proof some infinities are bigger than other infinities

Created by Vi Hart.
Video transcript
Voiceover: Any sequence you can come up with, whatever pattern looks fun. All your favorite celebrities birthdays lead into end followed by random numbers, whatever. All of that plus every sequence you can't come up with, each of those are the decimal places of a badly named so called real number and any of those sequences with one random digit changed is another real number. That's the thing most people don't realize about the set of all real numbers. It includes every possible combination of digits extending infinitely among aleph null decimal places. There's no last digit. The number of digits is greater than any real number, any counting number which makes it an infinite number of digits. Just barely an infinite number of digits because it's only barely greater than any finite number but even though it's only the smallest possible infinity of digits. This infinity is still no joke. It's still big enough, that for example point nine repeating is exactly precisely one and not epsilon less. You don't get that kind of point nine repeating equals one action unless your infinity really is infinite. You may have heard that some infinities are bigger than other infinities. This is metaphorically resonant and all but whether infinity really exists or if anything can last forever or whether a life contains infinite moments. Those aren't the kind of questions you can answer with math but if life does contain infinite moments, one for each real number time, that you can do math to. This time, we're not just going to do metaphors. We're going to prove it. Understanding different infinities starts with some really basic questions like is five bigger than four. You learned that it is but how do you know? Because this many is more than this many, they're both just one hand equal to each other except to fold it into slightly different shapes. Unless you're already abstracting out the idea of numbers and how you learn they're suppose to work just as you learned a long life is supposed to be somehow more than a short life rather than just a life equal to any other but folded into a different shape. Yeah, metaphorically resonate that. Is five and six bigger than 12? Five and six is two things after all and twelve is just one thing and what about infinity? If I want to make up a number bigger than infinity, how would I know whether it really is bigger and not just the same infinity folded into a different shape? The way five plus five is just another shape for 10. One way to make a big number is to take a number of numbers, meta numbers. This is where a box containing five and six has two things and is actually bigger than a box with only the number 12. You could take the number of numbers from one to five and put them in a box and you'd have a box set of five or you could take the number of numbers that are five which is one or you could take the number of counting numbers or the number of real numbers. It's kind of funny that the number of counting numbers is not itself a counting number but an infinite number often referred to as aleph null. This size of infinity is usually called countable infinity because it's like counting infinitely but I like James Grime's way of calling it listable infinity because the usual counting numbers basically make an infinite list and many other numbers of numbers are also listable. You can put all positive whole numbers on an infinite list like this. You can put all whole numbers including negative ones by alternating. You can list all whole numbers along with all half way points between them. You can even list all the rational numbers by cleverly going through all possible combinations of one whole number divided by another whole number. All countably infinite numbers of things, all aleph null. Countable infinity is like saying if I make an infinite list of these things, I can list all the things. The weird thing is that it seems like this definition should be obvious that no matter how many things there are, of course you can list all of them. If your list is literally infinite but nope so back to the reals. Say you want to list all the real numbers. If you did, it could start something like this but the specifics don't matter because we're about to prove that there's too many real numbers to fit even on an infinite list no matter how clever you are at list finding. What matters is the idea that you can create any real number you want, out of an infinite sequence of digits and we're going to use this power to create a number that couldn't possibly be on the list no matter what the list is even though the list is infinite. All we need to do that is construct a real number that isn't the first number on the list and isn't the second number on the list and isn't any number on the list no matter what the list is. Here's where I'm sure some of you are like "Yes!" Cantor's diagonal proof. Indeed my friends, that's what's going down. In the first number on the list, the first digit is one. If I make a new number with a first digit is three then even the rest of the digits are the same, there's no way my new number is equal to the first number on the list though the rest of the digits probably aren't all the same anyway. The second number on the list does start with a three. We don't know if this new number is the same or not yet but I can make sure my new constructed number is not the second number on the list by making the second digit five or eight or whatever and I can make my number not be the third number on the list by making the third digit five instead of three again. I mean the new number was already different from the third number on the list but I don't even have to check the other digits as long as I know that one of them definitely conflicts which comes in handy when I get to the 20 billion and oneth number on the list and I don't have to check the first 20 billion digits against the 20 billion digits I've constructed so far to be sure that my new number is not the same as the 20 billion and oneth number. There's one digit in my number for every number on the list which means I can make a way for my new number to not match every single number on the list no matter what the list is. Which means there's more real numbers than fit on an infinite list. This works no matter what the list is. Take the diagonal and add two to every digit or add five or whatever. You can't actually sit down and write an infinite list or infinite number though. Here's another way to think about what's really going on. We're trying to create a function that maps one set of numbers to another. You can map all the counting numbers to all the whole numbers with a simple minus one function or to all the even numbers with a times two function and map all the even numbers back to all the counting numbers with the inverse function division by two. You can map all the real numbers between zero and one to all the real numbers between zero and 10 by doing a times 10 function and find every number has a place to go, they match one to one. The question is, is there a function that maps every real number or even just the real numbers between zero and one to a unique counting number and vice versa. Cantor's diagonal proof shows that any function that claims to math counting numbers and reals to each other must fail somewhere. In fact you're not just missing one more real number than fits on an infinite list or else you could just add it to the beginning. There's not just another infinite list of numbers you're missing or else you could zipper the lists together. You could take every digit on the diagonal and add either two or four to get infinite combinations of numbers that aren't on the list and you can make a function that maps those missing numbers to the real numbers in binary. Of course the binary numbers are just another way of writing the reals which means an infinite list of real numbers will quite literally be missing all of them. Next to the infinity of the reals, the infinity of infinite list is actually mathematically nothing which is nuts because countable infinity is still super huge, it's infinite. The infinity of the reals is beyond that what can we indicated with a simple dot, dot, dot, a bigger infinity, a greater cardinal number. We've gone beyond aleph null. This is aleph one maybe. Yeah funny thing that turns out there's no way to tell how much bigger this infinity is than aleph null. Just that it's bigger like it could be the next step up or there might be other sorts of infinites in between but which one of those is the case? It's kind of independent of standard axioms. Awkward. But whatever it is, those are just two relatively small infinities out of an infinite number of aleph numbers. For every aleph number, there's infinite ordinal numbers which I guess kind of are like infinities folded into different shapes and don't forget hyper real supernaturals, or reals, etcetera. I guess you could squeeze some Beth numbers in there if you're into those axioms. I don't judge. Some of my best friends use the axiom of choice. Anyway, some infinities are bigger than other infinities but a mathematician would probably say something more like I don't know, there exists an aleph alpha and aleph beta where aleph alpha is greater than aleph beta or something which is perfectly true. Whether those different sorts of infinities apply to something like moments of time is unknown. What we do know is that if life has infinite moments or infinite love or infinite being then a life twice as long still has exactly the same amount. Some infinities only look bigger than other infinities and some infinities that seem very small are worth just as much as infinities 10 times their size.