There are as many even numbers as there are numbers.

「偶數的個數和正整數的個數一樣多。」

Really? I thought.

「真的嗎？」我心想。

Well, yeah. There are infinitely many of both.

噢對！兩個都是無限多個。

So I suppose there are the same number of them.

所以我覺得他們一樣多。

But on the other hand, the even numbers are only part of the whole numbers.

但另一方面，偶數只是正整數的一部份。

All the odd numbers are left over.

而奇數就是剩下的部份。

So, there's got to be more whole numbers than even numbers, right?

所以正整數應該要比偶數還多，對吧？

To see what my teacher was getting at,

要了解老師那段話的道理，

Let's first think about what it means for two sets to be the same size.

我們要知道兩個集合一樣大是什麼意思。

What do I mean when I say I have the same number of fingers on my right hand

當我說我左手的手指 和右手的手指一樣多時，

as I do on my left hand?

是什麼意思？

Of course, I've five fingers on each. But it's actually simpler than that.

當然，兩隻手都是五根手指， 但是可以更簡單一些。

I don't have to count, I only need to see that I can match them up one to one.

我不用去算，我只要知道 我能夠將它們「一對一」對應起來。

In fact, we think that some ancient people,

事實上，我們認為古代

who spoke languages that didn't have words for numbers greater than three,

那些語言裡數字只到三的人們

used this sort of matching.

就是用這個技倆。

For instance, if you let your sheep out of a pen to graze,

如果你把你的羊從羊圈裡放出去吃草，

you can keep track of how many went out by setting aside a stone for each one

你可以隨時知道有幾隻羊跑出去，你只要在羊出去時將一顆石子放旁邊，

and then putting those stones back one by one when the sheep return,

然後在羊回來的時候 再把石子放回來就好。

so that you know if any are missing without really counting.

這樣你就不會亂掉，儘管你沒有真的去算羊的數目。

As another example of matching being more fundamental than counting,

另一個「一對一」的例子比計數更單純一些。

if I'm speaking to a packed auditorium,

如果在一個擁擠的禮堂裡，

where every seat is taken and no one is standing,

每個位子都有人坐而且沒人站著，

I know that there are the same number of chairs as people in the audience,

這樣我就知道人數跟椅子數一樣多，

even though I don't know how many there are of either.

雖然說我並不知道這兩者的個數。

So what we really mean when we say that two sets are the same size

所以，我們說兩個集合一樣大時真正的意思

is that the elements in those sets can be matched up one by one in some way.

就是兩集合裡的元素 有辦法「一對一」對應在一起。

So my 4th grade teacher showed us the whole numbers laid out in a row and below each we have its double.

所以國小老師將正整數寫成一列，並將數字的兩倍寫在下面。

As you can see, the bottom row contains all the even numbers,

你可以看到，底部那列包含了所有的偶數，

and we have a one-to-one match.

這樣就有了「一對一」的對應。

That is, there are as many even numbers as there are numbers.

也就是說，偶數和正整數一樣多。

But what still bothers us is our distress over the fact that the even numbers seem to be only part of the whole numbers.

但依舊苦惱我們的是偶數只是正整數的一部份這件事實。

But does this convince you that I don't have the same number of fingers

不過這樣能說服你

on my right hand as I do on my left?

我左右手手指數目不一樣嗎？

Of course not!

當然不行！

It doesn't matter if you try to match the elements in some way and it doesn't work.

就算有的方法配對失敗，那也沒關係，

That doesn't convince us of anything.

因為這並沒說服我們什麼。

If you can find one way in which the elements of two sets do match up,

如果你可以找到一種方法讓兩邊元素配對起來，

then we say those two sets have the same number of elements.

那我們就說這兩個集合個數一樣。

Can you make a list of all the fractions?

你有辦法將分數像正整數那樣列出來嗎？

This might be hard. There are a lot of fractions.

可能有點難，分數有很多！

and it's not obvious what to put first,

而且不太明顯哪個要放前面，

or how to be sure all of them are on the list.

或是怎樣把它們串起來。

Nevertheless, there is a very clever way that we can make a list of all the fractions.

不過，有一個辦法我們可以把所有分數依序串起來。

This was first done by Georg Cantor in the late 1800s.

這是十九世紀末數學家康托爾的貢獻。

First, we put all the fractions into a grid.

首先，我們把分數上下左右對好。

They're all there.

全部的分數都在這。

For instance, you can find, say, 117 over 243

比如說，你可以找到 117/243，

in the 117th row and 243rd column.

它在第 117 列第 243 行。

Now, we make a list out of this by starting at the upper left, and sweeping back and forth diagonally,

現在我們要把它們串起來，從左上開始，然後斜對角地串下來、串上去，

skipping over any fraction, like 2/2,

其中像 2/2 這類之前已經算過的分數就把它跳掉。

that represents the same number as one we've already picked.

因此我們就把分數串成一串了。

And so we get a list of all the fractions,

這意思是分數，

which means we've created a one-to-one match between the whole numbers and the fractions,

和正整數有「一對一」的對應，

despite the fact that we thought maybe there ought to be more fractions.

雖然我們直覺是分數比較多個。

OK. Here's where it gets really interesting.

好，這就是有趣的地方了。

You may know that not all real numbers – that is, not all the numbers on a number line – are fractions.

你也許知道用分數沒辦法表示所有的實數 ──也就是那些數線上的數。

The square root of two and pi, for instance.

像是根號 2、還有圓周率這些。

Any number like this is called "irrational".

這類的數字叫作「無理數」。

Not because it's crazy or anything,

不只是因為它們很難懂，

but because the fractions are ratios of whole numbers,

而是因為分數包含了所有整數的「比率」，

and so are called 'rationals,' meaning the rest are non-rational, that is, irrational.

所以被叫「可比的」，而剩的就被叫作「不可比的」，也就是「無理的」。

Irrationals are represented by infinite, non-repeating decimals.

無理數可以用無窮小數表示，而且各位數沒有規律。

So can we make a one-to-one match between the whole numbers and the set of all the decimals?

那麼，我們可以將正整數和小數「一對一」對應嗎？

Both the rationals and the irrationals?

所有無理、有理的小數？

That is, can we make a list of all the decimal numbers?

也就是，我們可以將所有小數串起來嗎？

Cantor showed that you can't.

康托爾證明了這行不通。

Not merely that we don't know how, but that it can't be done.

不只想不到辦法，而是真的沒辦法。

Look, suppose you claim you have made a list of all the decimals.

你看，如果你聲稱你把小數串好了。

I'm going to show you that you didn't succeed,

我要來告訴你這是不可能的，

by producing a decimal that's not on your list.

因為我要找一個你那串那面沒有的小數。

I'll construct my decimal one place at a time.

我要在小數點後一個一個位數決定。

For the first decimal place of my number,

為了決定我的第 1 位數，

I'll look at the first decimal place of your first number.

我要用你那串的第 1 個數字的第 1 位數。

If it's a 1, I'll make mine a 2.

如果它是 1，我的就是 2；

Otherwise, I'll make mine a 1.

否則我的就是 1。

For the second place of my number,

那我的第 2 位數，

I'll look at the second place of your second number.

我會用到你的第 2 個數字的第 2 位數。

Again, if yours is a 1, I'll make mine a 2,

一樣，如果你的是 1，我的就是 2；

and otherwise i'll make mine a 1.

否則我的就是 1。

See how this is going?

看出怎麼算下去了嗎？

The decimal I produce can't be on your list.

我找到的這個小數，不可能在你那串裡。

Why? Could it be, say, your 143rd number?

為什麼？比如說，它和你的第 143 個數會一樣嗎？

No, because the 143rd place of my decimal

不可能，因為第 143 位數裡

is different from the 143rd place of your 143rd number.

你的和我的不一樣。

I made it that way.

這是我特別挑的。

Your list is incomplete, it doesn't contain my decimal number.

你沒串成功，沒有串到所有小數。

And no matter what list you give me, I can do the same thing,

而不論你怎麼串，我都可以做同樣的事，

and produce a decimal that's not on that list.

然後找到一個你那串裡沒出現的小數。

So we're faced with this astounding conclusion:

所以我們得到了令人訝異的結論：

the decimal numbers cannot be put on a list.

所有小數沒辦法串成一串。

They represent a bigger infinity than the infinity of whole numbers.

它的「無限大」比正整數的「無限大」還大。

So even though we're familiar with only a few irrationals,

所以，儘管你只熟悉幾個無理數，

like square root of two and pi,

像是根號 2 和圓周率，

The infinity of irrationals is actually greater than the infinity of fractions.

無理數的「無限大」實際上也比 分數的「無限大」還要大。

Someone once said that the rationals – the fractions – are like the stars in the night sky.

有人曾這樣比喻： 有理數，或者說分數，就像天空中的星星；

The irrationals are like the blackness.

而無理數就像是無盡的黑暗。

Cantor also showed that for any infinite set,

康托爾同時也證明任何無窮大的集合，

forming a new set made of all the subsets of the original set

只要把它的所有子集都蒐集起來，

represents a bigger infinity than that original set.

新的集合的「無限大」就比原本的還大。

This means that once you have one infinity,

意思是說，只要你有一種「無限大」

you can always make a bigger one by making a set of all subsets of that first set.

那你就可以用它的所有子集來做出比它更「無限大」的集合。

And then an even bigger one

接著再用這集合做出更加「無限大」的集合。

by making a set of all subsets of that one, and so on.

不斷做下去。

And so, there are an infinite number of infinities of different sizes.

所以，「無限大」之間也是有分不同的大小。

If these ideas make you uncomfortable, you're not alone.

如果你覺得這令人想吐，並不奇怪。

Some of the greatest mathematicians of Cantor's day were very upset with this stuff.

一些康托爾那年代的偉大數學家也對這觀念非常反感。

They tried to make these different infinities irrelevant,

他們試著要把無限這觀念抽離，

to make mathematics work without them somehow.

讓數學可以沒有無限也能運作。

Cantor was even vilified personally,

康托爾甚至受到人身攻擊，

and it got so bad for him that he suffered severe depression.

嚴重到讓他飽受沮喪之苦。

He spent the last half of his life in and out of mental institutions.

並且在精神療院渡過後半餘生。

But eventually, his ideas won out.

不過他的想法最終得到肯定。

Today they are considered fundamental and magnificent.

今天，這觀念被認為是基礎並重要的。

All research mathematicians accept these ideas,

所有數學研究者都接受這觀念，

every college math major learns them,

每個數學系都也都在教，

and I've explained them to you in a few minutes.

而我剛剛已經花了幾分鐘來解釋。

Someday, perhaps, they'll be common knowledge.

也許有一天，這會變成大家的常識。

There's more.

還有一點。

We just pointed out that the set of decimal numbers –

我們剛剛指出小數，

that is, the real numbers – is a bigger infinity than the set of whole numbers.

也就是實數，比正整數的「無限大」還多。

Cantor wondered if there are infinities of different sizes between these two infinities.

康托爾在想兩個「無限大」之間是否還有不同層級的「無限大」。

He didn't believe there were, but couldn't prove it.

我們不這麼認為，但也沒辦法證明。

Cantor's conjecture became known as the continuum hypothesis.

康托爾的猜想變成有名的「連續統假說」。

In 1900, the great mathematician David Hilbert

在 1900 年，大數學家希爾伯特把連續統假說

listed the continuum hypothesis as the most important unsolved problem in mathematics.

列為數學裡最重要的未解問題。

The 20th century saw a resolution of this problem,

這問題在 20 世紀露出一些端倪，

but in a completely unexpected, paradigm-shattering way.

但是結果和超乎預期、並跌破大家眼鏡。

In the 1920s, Kurt Godel showed that you can never prove that the continuum hypothesis is false.

在 1920 年代，哥德爾證明了你不可能證明連續統假說是錯的。

Then in the 1960s, Paul J. Cohen showed that you can never prove that the continuum hypothesis is true.

接著在 1960 年代，寇恩證明了你不可能證明連續統假說是對的。

Taken together, these results mean that there are unanswerable questions in mathematics,

合在一起，這些結果告訴你數學裡也有一些不能回答的問題，

a very stunning conclusion.

這是一個很令人震驚的結論。

Mathematics is rightly considered the pinnacle of human reasoning,

數學被公認是人類邏輯的結晶，

but we now know that even mathematics had its limitations.

但現在我們知道就算是數學也有它的極限。

Still, mathematics has some truly amazing things for us to think about.

還有就是，數學裡有一些值得我們思考、而且很令人著迷的道理。