We're break one of the rules of Numberphile.

我們打破了Numberphile的規則之一。

We're talking about something that isn't a number.

我們'說的是一些不是'數字的東西。

We're going to talk about infinity.

我們'要談的是無窮大。

So infinity.

所以無窮大。

Now like I said, infinity is not a number.

就像我說的，無窮大不是一個數字。

It's a idea.

這'是一種想法。

It's a concept.

這是一個概念。

It's the idea of being endless, of going on forever.

它'是無窮無盡的想法，永無止境地進行下去。

I think everyone's familiar with the idea of

我想大家'的想法很熟悉。

infinity, even kids.

無限，甚至孩子。

You start counting 1, 2, 3, 4, 5--

你開始數1，2，3，4，5 --

you might be five years old, but already you're thinking,

你可能是五歲，但你已經'想。

what's the biggest number I can think of.

什麼'是我能想到的最大的數字。

And you go, oooh, it's 20.

你去，哦，是&#39；20。

You get a bit older, and you go, maybe it's a million.

你再大一點，你去，也許是'萬。

It never ends, does it? 'Cause you can keep adding 1.

它永遠不會結束，不是嗎'因為你可以繼續增加1。

So that's the idea of infinity.

所以，這就是無限的想法。

The numbers go on forever.

數字永遠在繼續。

But I'm going to tell you one of the more surprising facts

但我'要告訴你一個比較驚人的事實

about infinity.

關於無窮大。

There are different kinds of infinity.

無間有不同的種類。

Some infinities are bigger than others.

有的無限大，有的無限大。

Let's have a look.

讓我們'看看吧。

The first type of infinity is called countable.

第一種類型的無窮大叫做可數。

And I don't like the name countable.

而且我不喜歡可數這個名字。

And Brady gave me a little bit of a hmm, just then.

而布雷迪也給了我一個小小的暗示，就在這時。

Because if you're talking about infinity, you can't

因為如果你說的是無限大，你不能

count infinity, can you?

數無窮大，你會嗎？

Because it goes on forever.

因為它永遠在繼續。

I think it's a terrible name.

我認為這是一個可怕的名字。

I prefer to call it listable.

我更喜歡叫它可上市。

Can we list these numbers?

我們可以列出這些數字嗎？

All right.

好吧，我知道了

Let's do these simple numbers, 1, 2, 3--

讓我們來做這些簡單的數字，1，2，3------。

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

BRADY HARAN：你'不打算做所有的人，是詹姆斯？

JAMES GRIME: 4.

JAMES GRIME: 4.

How long have we got?

我們還有多少時間？

BRADY HARAN: (LAUGHING) 10 minutes.

十分鐘。

JAMES GRIME: Right.

對。

5, 6--

5, 6--

so you can list the whole numbers.

所以你可以列出整數。

So this is called countable.

所以這叫可數。

Listable, I prefer.

可上市，我更喜歡。

What about the integers?

那整數呢？

All the integers.

所有的整數。

That's all the negative numbers as well.

這'也都是負數。

So there's 0.

所以有'的0。

Let's have that.

讓'的有。

But there's 1 and minus 1, there's 2 and minus 2, there's

但是，有1和負1，有2和負2，有

3, and minus 3.

3，和負3。

Now, that is an infinity as well.

現在，這也是一個無窮大的問題。

And in some sense, it's twice as big, because there seems to

而在某種意義上，它的兩倍大，因為似乎有。

be twice as many numbers.

是數字的兩倍。

But it is infinity as well.

但它也是無窮大的。

They're both infinity, and they're both the

他們都是無限的，他們都是那個

same type of infinity.

同類型的無窮大。

They both can be listed.

他們都可以上市。

Perhaps more surprisingly, the fractions can

也許更令人驚訝的是，分數可以是

be listed as well.

也被列入。

But you have to be a bit clever about this.

但是，你必須要有一點聰明的做法。

Let's try and list the fractions.

讓我們'試著列出分數。

I'm going to write out a rectangle.

我'要寫出一個長方形。

1 divided by 1.

1除以1。

That's a fraction.

那是一小部分。

[INAUDIBLE].

[INAUDIBLE].

Let's have 1 divided by 2, 1/3, 1/4, 1/7--

讓我們有1除以2，1/3，1/4，1/7--。

OK, that goes on.

好，那就繼續。

Let's do the next row and have two at the top.

讓'我們做下一排，上面有兩個。

2/1, 2/2, 2/3, 2/4.

2/1, 2/2, 2/3, 2/4.

Let's do the next one.

讓我們'做下一個。

3/1, 3/2.

3/1, 3/2.

4/6, 4/7.

4/6, 4/7.

That goes on and we can keep going.

這樣下去，我們可以繼續走下去。

So here, I've made some sort of an infinite rectangle array

所以在這裡，我做了一個無限的矩形數組。

of fractions.

的分數。

Now if I want to make it a list like this, though, If I

現在，如果我想把它做成這樣的清單，雖然，如果我。

went row by row, you're going to have a problem.

了一排一排，你'會有問題。

If you go row by row, I'll go--

如果你一排一排地走，我就... ...

there's 1, 1/2, 1/3, 1/5, 1/6, 1/7-- and

有1，1/2，1/3，1/5，1/6，1/7-----------。

I'll keep going forever.

我'會一直走下去的。

And I'm never going to reach the second row.

而我'永遠也到不了第二排。

I can't list them.

我不能列出它們。

Not that way.

不是這樣的。

You can't list them that way.

你不能用這種方式列出它們。

You'll never reach the second row.

你'永遠也到不了第二排。

This is how you list them.

這就是你列出它們的方式。

Slightly more clever than that.

比這稍微聰明一點。

You take the diagonal lines.

你把對角線。

Now, I can guarantee that every fraction will appear on

現在，我可以保證每一個分數都會出現在。

one of those diagonal lines.

其中一條對角線。

And you list them diagonal by diagonal.

而你卻把它們一斜一斜地列出來。

So that's the first diagonal.

所以這就是第一條對角線。

Then you list the second diagonal-- there it is.

然後你列出第二條對角線... 就是這樣。

Then you list the third diagonal, then you take the

然後你列出第三條對角線，然後你以

fourth diagonal, and the fifth.

第四條對角線，和第五條。

So eventually, you are going to do this every fraction.

所以最終，你要做到每一個分數。

Every faction appears on a diagonal, and you're

每個派別都出現在對角線上，你'。

going to list them.

要把它們列出來。

Now, if you take all the numbers, right?

現在，如果你把所有的數字，對嗎？

That's the whole number line.

這'是整個數字線。

Let's try that.

讓我們試試吧。

Look, I'm going to draw it.

你看，我打算畫它。

It's a continuous line of numbers.

這是一條連續的數字線。

These are all your decimals.

這些都是你的小數。

You've got 0 there in the middle, and you'll

你有0在中間，你會

go 1 and 2 and 3.

去1和2和3。

But it has a 1/3.

但它有1/3。

It will contain pi, and e, and all the

它將包含pi，和e，以及所有的。

irrational numbers as well.

無理數也。

Can you list them?

你能列出他們嗎？

How do you list them?

你怎麼把它們列出來？

0 to start with, and then 1?

0開始，然後1?

But hang on.

但是等一下

We've missed a half.

我們'已經錯過了一半。

So we put in the half.

所以我們把在半。

Hang on, we've missed the quarter.

等等，我們已經錯過了這一季。

We put in the quarter.

我們把四分之一。

But we've missed 0.237--

但我們已經錯過了0.237----。

so how do you list the real numbers?

那你怎麼列出實數呢？

It turns out you can't.

事實證明，你不能'。

In fact, rather remarkably, I can show you that we can't

事實上，相當驚人的是，我可以告訴你，我們可以'不

list them, even though were talking about something so

列舉他們，即使是在談論這樣的事情。

complicated as infinity.

複雜如無窮大。

BRADY HARAN: Do it, man!

動手吧，夥計！

JAMES GRIME: We need paper.

我們需要紙。

BRADY HARAN: We need an infinite amount of

我們需要無限量的... ...

paper here, I think.

我想，這裡的紙。

JAMES GRIME: (LAUGHING) It's a big topic.

JAMES GRIME: (LAUGHING) It's a big topic.

Imagine we could list all the decimals, right?

想象一下，我們可以列出所有的小數，對嗎？

We can't, actually.

其實，我們不能'。

But pretend we can.

但是假裝我們可以。

What sort of--

什麼樣的...

what would it look like?

會是什麼樣子？

We'll start with all the 0-point decimals.

我們'將從所有的0點小數開始。

Let's pick some decimals.

讓我們'選擇一些小數。

0.121--

0.121--

dot dot dot dot dot.

點點滴滴。

Let's pick the next one.

讓我們'挑下一個。

Let's say the next one is 0.221--.

比方說下一個是0.221--。

Next one, let's do 0.31111129--.

下一個，讓我們做0.31111129--。

And let's take another one, here.

我們再來一個，這裡。

0.00176--.

0.00176--.

Now I'm going to make a number.

現在我'要做一個數字。

This is the number I'm going to make.

這是我要做的號碼。

I'm going to take the diagonals here.

我'要在這裡採取對角線。

I'm going to take this number and this number and this

我要把這個號碼和這個號碼和這個

number and this number and this number.

這個數字和這個數字，這個數字。

And I am going to write that down.

而我要把這些寫下來。

So what's that number I've made?

那麼，我做的那個號是什麼呢'？

It's 0.12101--

這是0.12101--。

something, something, something.

東西，東西，東西。

Now this is my rule.

這是我的原則。

I'm going to make a whole new number from that one.

我'要從那個號上做一個全新的號。

This is the number I'm going to make.

這是我要做的號碼。

If it has a 1, I'm going to change it to a 2.

如果它有1，我'要把它改成2。

And if it has a 2 or anything else, I will change it to a 1.

如果它有2或其他什麼，我會把它改成1。

So let's try that.

所以讓我們試試吧。

So I'm going to turn this into--

所以我打算把這個變成... ...

0-point.

0分

So if it has a 1, I'm going to turn it into a 2.

所以如果它有1，我'要把它變成2。

If it's anything else, I'm going to turn it into a 1.

如果它'別的，我'要把它變成1。

So that will be a 1.

所以這將是一個1。

I'm going to change 1 here into a 2.

我'要把這裡的1改成2。

I'm going to change that one into a 1.

我'要把這個改成1。

I'm going to change that one into a 2-- that was my rule.

我要把這個改成2--這是我的規則。

And I'll make something new.

而我'會做出一些新的東西。

That does not appear on the list.

這並沒有出現在名單上。

That number is completely different from anything else

這個數字和其他東西完全不同

on the list, because it's not the first number, because it's

因為它不是第一個數字，因為它是

different in the first place.

首先是不同的。

It's not the second number, because it's different in

這不是第二個數字，因為它的不同，在

second place.

第二名。

It's not the third number, because it's different in the

這不是第三個數字，因為它是不同的在。

third place.

第三名。

It's not the fourth number because it's different in the

這不是第四個數字，因為它是不同的在。

fourth place.

第四名。

It's not the fifth number, because it's different in the

這不是第五個數字，因為它是不同的在。

fifth place.

第五名。

You've made a number that's not on that list.

你'做了一個不在那個名單上的數字。

And so you can't list all the decimals, in which case it is

所以你不能列出所有的小數，在這種情況下，它是

uncountable.

不可計數的。

It is unlistable.

它是無法上市的。

And that means it's a whole new type of infinity.

而這意味著它'是一種全新的無窮大的類型。

A bigger type of infinity.

更大類型的無窮大。

BRADY HARAN: Surely we could, James, because all we've got

我們當然可以，詹姆斯，因為我們所有的'已經得到了

to do is keep doing your game and making them and adding

要做的是繼續做你的遊戲，並使他們和添加。

them to the list.

列表中。

And if we keep doing that, won't we get there eventually?

如果我們繼續這樣做，我們最終不會到達那裡嗎？

JAMES GRIME: But you could then create another number

但你可以再創造一個數字

that won't be on that list.

那不會在那個名單上。

And so the guy who came up with is a German mathematician

所以想出這個辦法的人是一個德國數學家。

called Cantor.

稱為康托爾。

Cantor lived 'round about the turn of the 20th century.

康托爾生活在'20世紀初左右。

He was ridiculed for this.

他是以受到了嘲笑。

For this idea that there were different types of infinity,

對於這種想法，有不同類型的無限性。

he was called a charlatan.

他被稱為金光黨。

And he was called-- it was nonsense, it was called.

他被稱為... ... 這是胡說八道，它被稱為。

And poor old Cantor was treated really badly by his

而可憐的老康托爾被他的家人虐待

contemporaries, and he spent a lot of his later life in and

同時代的人，而他晚年的很多時間都是在和。

out of mental institutions, where he died, in the end.

出了精神病院，在那裡他死了，最後。

Near the end of his life, it was recognized.

在他生命接近尾聲時，得到了認可。

It was true.

這是真的。

It was recognized.

它是公認的。

And he had all the recognition that he deserved.

而他也得到了所有他應得的認可。

BRADY HARAN: And now he's on Numberphile.

現在他在Numberphile上。

JAMES GRIME: And now he's on Numberphile, the greatest

現在他在《Numberphile》上，最偉大的。

accolade of all.

譽的所有。

Georg Cantor.

Georg Cantor.