Ordinal Numbers - Numberphile

Key Takeaways

Okay, so let's talk about the concept of a length, but not just the normal, you know, how long is this pen or how long is this piece of paper? Let's talk about the length of a queue. We're in Britain, people love queuing for stuff. Let's say that we have something really nice here. Maybe it's a party. Maybe it's a Taylor Swift concert. I don't know. >> It's a really nice new box of Lego. And then there's a queue for people who want to play with it, right? >> You don't play with Lego. You build it. You construct it. >> It's constructed. >> Yeah. Yeah. Okay. All right. There's a queue of people that want the Lego. >> Yeah. Okay. There's the first person in the queue. And there's going to be another one here and another and another. So on and so forth. So how many people are ahead of this guy right here? >> None. >> None. Zero. How many people are ahead of this one? Just one. Two. And three. And you can see roughly where it's going, right? If we had n then this person has to wait for n people to play with the Lego set before they get to could be a long time especially since you're not coming out anytime soon with finite length we get something interesting if I ask you how many people are in the queue the answer is n maybe n was five five people are in the queue if I ask you how long is the queue you'd say five right in that sense cardinals how many people and ordinals how long is the queue but the order are the same. Now imagine if you will that this is you know in the lobby of Hilbert's hotel famously infinite keep going n+1 n plus2 and so on and so on and so on eventually this q becomes infinite let me even use infinity as a symbol here just for this sense how many people are in this queue >> uh an infinite number of people >> an infinite number of people that's that's not I'm not trying to be clever here just infinite number of people how long is the queue >> it's infinitely long >> it's infinitely there. But let's look closer at this cube. Despite being infinitely long, if I pick any one person here, right? Maybe it was 100 to the power of n factorial with respect to this n. There's a lot of people ahead. You know, the universe will decay before this person gets to play with the Lego. But we're in a mathematical universe, so that's fine. But there's only finely many. A lot of them, but finally many. Imagine if you will this poor guy here at zero. He gets a phone call and so he is stepping out from the queue. Okay. Now everybody moves by one. So one becomes zero and this goes to one. This goes to two. This goes to n minus one. This goes to 100 to n factorial minus one and so on. The q as a q looks exactly the same. We have just matched the two qes. They look exactly the same. It's still infinitely long. There's still infinitely many people. But now this guy here at zero is stepping back in the queue. We're in civilized society. So it's not just pushing back in. No. No. It's going back here. >> Back at the queue. >> Back at the queue. How many people are in here? Still infinitely many. It's the same set of people even. Right. Same. Exactly. >> It's not the same set of people. Cuz if that person was Assaf. >> Yeah, sure. >> Who was in the set at one point and he's not in the set at another point. There is one extra person in the queue. >> So when I stepped out, the set of people had changed. But now I'm stepping back in. I simply moved myself and put myself back. That's all I've done. >> Yeah. >> Right. If I look at all the natural numbers and I take zero out, I have all the natural numbers without zero and I put zero back in, I get all the natural numbers again. >> Yeah. So when I step back to the back of the queue, it's the same set of people as we had previously. >> But when you were out of the queue, while you were on that phone call, >> it changed. >> It did change. >> It did change though. Yes. >> Okay. >> How long is the queue now? >> Now you're back in it. >> Yes. >> Same same length as before. >> Ah, it's not the same length as before. It has the same number of people. The cardinality of the queue is the same, but the ordinal number that this Q represents is now different. >> Why? Why? Because look at me. I'm this poor guy at the back. I don't wait for finely many people. I have to wait for infinitely many people to play with the Lego before I get to, you know, construct and and admire it. Previously, there was nobody who had to wait for an infinitely long time to get to the Legos. Everybody had to wait for some amount of time, some amount of finite time, but nobody had to wait for an infinite long time. Oh, okay. Then in this queue, >> yeah, >> who's in front of you? >> Everybody else now that you're at the back. >> So, there's no one person who's ahead of me, right? So, the question is, >> but what Okay, before you stepped back into the queue. >> Mhm. >> When you were on the phone. >> Yeah. >> Who was last in the queue? >> There was no last person. >> But now there is a last person. >> Now I'm the last person in the queue. >> Okay. So ordinal numbers measure the length of a queue like this. Now to put this in kind of mildly programming terms, normally when you think about cues, you're thinking about a linked list. Every person knows who's immediately before and immediately after. That's all they care about. And this will give you the order just fine. Once you reach infinitely long cues, you actually need a different kind of information. What you need to know is just everybody who's ahead of you or everybody who's behind you. One of those will be enough. But you need to know all of them. Just the one is not going to be enough. So what happened here? We have shifted people from the queue and we got a different length of a queue. We've got infinity + one. >> Okay. >> Right. >> Infinity plus one. >> Yeah. Now, you know, >> you you've used an analogy here of people waiting for Lego. That was good fun. >> Yeah. >> How does this work in real numbers and mathematics and things like that? What's an example of this that's not just an analogy? >> Right. So, as I said, oral numbers represent this concept of a queueing, right? They have this really nice property that we can use them to define things one step at a time by induction by recursion. We can do this even though there's an infinite point and a jump can still use recursion and induction on these kind of things. They will come up perhaps if you talk about the hydra problem or all kind of computational games that you do. You end up having to study this concept of ordinal numbers. It will tell you to how what length of time do you need to go? How long is the process going to take? If you measure it in steps, one step at a time before you will finish. >> So ordinal numbers are ordinal numbers like a kind of number like prime numbers or three like is three an ordinal number? >> Yes. So if you think about ordinal numbers as quite literally measuring the length of cubes and a professor I used to work with actually introduced oral numbers by first talking about Q's. Three is an ordinal number. It measures the length of a queue that has three people in it. >> Okay? >> Right? Pi on the other hand is not an ordinal number because there's no fractions. You know that you get 1 2 3 and you know a about a seventh of a person. That's not how it works. >> Okay. So numbers generalize the the natural numbers in a different way. Now they have arithmetic. You can take two Q's and you can put them together. That's addition. And you can take two Q's and you can multiply them in a certain way and you can even do exponentiation. And even more than that, >> what can't you do with ordinal numbers? >> Uh you can't really divide them and you can't really subtract them, right? Because if I look at this, this is what we had here. We had this infinitely long Q and we subtracted a point, but we got the same thing, right? So, you can't quite do that. It It doesn't fully work and the vision gets even worse. >> So, when you came out of the queue from the front and you went to the back into this terrible position, your ordinal number felt like it was zero there. >> Yeah. >> What's your ordinal number now? You're at the back. >> Omega. Because you're asking how many people are ahead. And so this infinity in set theory is denoted by omega >> as as opposed to the other infinity symbol. >> Yes. >> So it's a this symbol has a specific meaning just like pi has a specific meaning when thinking about real numbers versus using it to to denote something else otherwise. Right? So in set theory omega represent this length of the first infinity. And so now this is omega. This point is is really omega because it's at the end of the cube, but the length is omega + one. And now maybe somebody else stepped out and joined here. We have omega + 2. So on and so forth. And maybe all the people at even coordinates had to step out and they all stepped back. And now you get omega plus omega. You get two copies of this infinite queue, one after the other. This omega plus omega. >> If everybody in the original infinite queue got a phone call at the same time, right, and stepped out and then stepped back in. >> Yeah. >> Would that be the original queue again or would it be >> So if they stepped back in at exactly the same order, yes. Even if they step in with some kind of, you know, variation. Yes. But let's suppose that all the people that previously had an even number stepped in and only then the people who had powers of three stepped in and only then people with mult powers of five step and so on. Then you get copies of omega stacked on top of each other and this is how you get omega times omega right you get omega many kind of points and in each one has its own kind of copy of omega these are really cool numbers and they don't behave like we expect from just the usual counting numbers or the real numbers but they work to do these kind of things and they're really fun to play with. One of the things we can really enjoy doing with them is kind of going back to the Hydra game, right? And doing things like that. Or if you know about good sequences, you can use ordinal numbers to kind of prove that good good sequences terminate because you kind of say, okay, let me kind of like extend it and plug in omega and and everything works fine. >> There's one half, third, fourth, fifth, sixth, seventh, and I'll keep going forever. And I'm never going to reach the second row. I can't list them. Not that way. You can't list them that way. You'll never reach the second row. This is how you list them. Slightly more clever than that. You take the diagonal lines.