JAMES GRIME: We’re going to break a rule. We’re break one of the rules of Numberphile. We’re talking about something that isn’t a number. We’re going to talk about infinity. So infinity. Now like I said, infinity is not a number. It’s a idea. It’s a concept. It’s the idea of being endless, of going on forever. I think everyone’s familiar with the idea of infinity, even kids. You start counting 1, 2, 3, 4, 5– you might be five years old, but already you’re thinking, what’s the biggest number I can think of. And you go, oooh, it’s 20. You get a bit older, and you go, maybe it’s a million. It never ends, does it? ‘Cause you can keep adding 1. So that’s the idea of infinity. The numbers go on forever.

But I’m going to tell you one of the more surprising facts about infinity. There are different kinds of infinity. Some infinities are bigger than others. Let’s have a look. The first type of infinity is called countable. And I don’t like the name countable. And Brady gave me a little bit of a hmm, just then. Because if you’re talking about infinity, you can’t count infinity, can you? Because it goes on forever. I think it’s a terrible name. I prefer to call it listable. Can we list these numbers? All right.

Let’s do these simple numbers, 1, 2, 3–

BRADY HARAN: You’re not gonna do all of them, are you James?

JAMES GRIME: 4. How long have we got?

BRADY HARAN: (LAUGHING) 10 minutes.

JAMES GRIME: Right. 5, 6– so you can list the whole numbers. So this is called countable. Listable, I prefer. What about the integers? All the integers. That’s all the negative numbers as well. So there’s 0. Let’s have that. But there’s 1 and minus 1, there’s 2 and minus 2, there’s 3, and minus 3. Now, that is an infinity as well. And in some sense, it’s twice as big, because there seems to be twice as many numbers. But it is infinity as well. They’re both infinity, and they’re both the same type of infinity. They both can be listed.

Perhaps more surprisingly, the fractions can be listed as well. But you have to be a bit clever about this. Let’s try and list the fractions. I’m going to write out a rectangle. 1 divided by 1. That’s a fraction. [INAUDIBLE]. Let’s have 1 divided by 2, 1/3, 1/4, 1/7– OK, that goes on. Let’s do the next row and have two at the top. 2/1, 2/2, 2/3, 2/4. Let’s do the next one. 3/1, 3/2. 4/6, 4/7. That goes on and we can keep going. So here, I’ve made some sort of an infinite rectangle array of fractions. Now if I want to make it a list like this, though, If I went row by row, you’re going to have a problem. If you go row by row, I’ll go– there’s 1, 1/2, 1/3, 1/5, 1/6, 1/7– 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. Now, I can guarantee that every fraction will appear on one of those diagonal lines. And you list them diagonal by diagonal. So that’s the first diagonal. Then you list the second diagonal– there it is. Then you list the third diagonal, then you take the fourth diagonal, and the fifth. So eventually, you are going to do this every fraction. Every faction appears on a diagonal, and you’re going to list them. Now, if you take all the numbers, right? That’s the whole number line.

Let’s try that. Look, I’m going to draw it. It’s a continuous line of numbers. These are all your decimals. You’ve got 0 there in the middle, and you’ll go 1 and 2 and 3. But it has a 1/3. It will contain pi, and e, and all the irrational numbers as well. Can you list them? How do you list them? 0 to start with, and then 1? But hang on. We’ve missed a half. So we put in the half. Hang on, we’ve missed the quarter. We put in the quarter. But we’ve missed 0.237– so how do you list the real numbers? It turns out you can’t. In fact, rather remarkably, I can show you that we can’t list them, even though were talking about something so complicated as infinity.

BRADY HARAN: Do it, man!

JAMES GRIME: We need paper.

BRADY HARAN: We need an infinite amount of paper here, I think.

JAMES GRIME: (LAUGHING) It’s a big topic. Imagine we could list all the decimals, right? We can’t, actually. But pretend we can. What sort of– what would it look like? We’ll start with all the 0-point decimals. Let’s pick some decimals. 0.121– dot dot dot dot dot. Let’s pick the next one. Let’s say the next one is 0.221–. Next one, let’s do 0.31111129–. And let’s take another one, here. 0.00176–. Now I’m going to make a number. This is the number I’m going to make. I’m going to take the diagonals here.

I’m going to take this number and this number and this number and this number and this number. And I am going to write that down. So what’s that number I’ve made? It’s 0.12101– something, something, something. Now this is my rule. I’m going to make a whole new number from that one. This is the number I’m going to make. If it has a 1, I’m going to change it to a 2. And if it has a 2 or anything else, I will change it to a 1. So let’s try that. So I’m going to turn this into– 0-point. So if it has a 1, I’m going to turn it into a 2. If it’s anything else, I’m going to turn it into a 1. So that will be a 1. I’m going to change 1 here into a 2. I’m going to change that one into a 1. I’m going to change that one into a 2– that was my rule.

And I’ll make something new. That does not appear on the list. That number is completely different from anything else on the list, because it’s not the first number, because it’s different in the first place. It’s not the second number, because it’s different in second place. It’s not the third number, because it’s different in the third place. It’s not the fourth number because it’s different in the fourth place. It’s not the fifth number, because it’s different in the fifth place. You’ve made a number that’s not on that list. And so you can’t list all the decimals, in which case it is uncountable. It is unlistable. And that means it’s a whole new type of infinity. A bigger type of infinity.

BRADY HARAN: Surely we could, James, because all we’ve got to do is keep doing your game and making them and adding them to the list. And if we keep doing that, won’t we get there eventually?

JAMES GRIME: But you could then create another number that won’t be on that list. And so the guy who came up with is a German mathematician called Cantor. Cantor lived ’round about the turn of the 20th century. He was ridiculed for this. For this idea that there were different types of infinity, he was called a charlatan. And he was called– it was nonsense, it was called. And poor old Cantor was treated really badly by his contemporaries, and he spent a lot of his later life in and out of mental institutions, where he died, in the end. Near the end of his life, it was recognized. It was true. It was recognized. And he had all the recognition that he deserved.

BRADY HARAN: And now he’s on Numberphile.

JAMES GRIME: And now he’s on Numberphile, the greatest accolade of all. Georg Cantor.

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