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.
Infinity is always going to be bigger than you think, because once you think of how big it is, it's already bigger than that.
24 is the highest number.
Cantor's diagonal argument
Basics of Discrete Math
Got a kitty bite there, I reckon.
I get that there are different types of infinity. But wouldn't they all be equal? I can't wrap my head around the idea of something being... Twice as endless as the next. If its endless, it should just be endless, right?
I would just like to give OP a hardy "fuck you" because I just wasted 2 hours watching numberphile videos.
But seriously, I feel smarter now.
also 8 minute video. if you tilt 8 you ge.........nevermind. im just gonna let myself out.
Wait, so why can't we just flip real numbers over the decimal point to list them, ie .1 = 1, .2 = 2, .3 = 3, .01 = 10, .314 = 413, .0314... = ...4130? Wouldn't the countability of the ... in the decimal be the same as the countability of the ... in the integer?