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  • 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.

JAMES GRIME: We're going to break a rule.

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Infinity is bigger than you think - Numberphile

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    VoiceTube posted on 2012/12/30
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