When I was in fourth grade, my teacher said to us one day:
我在四年级的时候，小学老师有一天跟我们说：
'There are as many even numbers as there are numbers.' 'Really?', I thought.
“偶数的个数和正整数的个数一样多。”“真的吗？”我心想。
Well, yeah, there are infinitely many of both, so I suppose there are the same number of them.
噢对！两个都是无限多个，所以一样多。
But even numbers are only part of the whole numbers, all the odd numbers are left over,
但另一方面，偶数只是正整数的一部分，而奇数就是剩下的部分，
so there's got to be more whole numbers than even numbers, right?
所以正整数应该要比偶数还多，对吧？
To see what my teacher was getting at, let's first think about what it means for two sets to be the same size.
要了解老师那段话的道理，我们必须知道两个集合一样大是什么意思。
What do I mean when I say I have the same number of fingers on my right hand as I do on left hand?
当我说我左手的手指和右手的手指一样多时，这意味着什么？
Of course, I have five fingers on each, but it's actually simpler than that.
当然，两只手都是五根手指，但是可以更简单一些。
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，我只要知道我能够将它们“一对一”对应起来。
In fact, we think that some ancient people who spoke languages that didn't have words for numbers greater than three used this sort of magic.
事实上，我们认为古代那些语言里数字只到三的人们就是用这个伎俩。
For instance, if you let your sheep out of a pen to graze,
如果你把你的羊从羊圈里放出去吃草，
you can keep track of how many went out by setting aside a stone for each one,
你可以随时知道有几只羊跑出去，你只要在羊出去时将一颗石子放旁边，
and putting those stones back one by one when the sheep return, so you know if any are missing without really counting.
然后在羊回来的时候再把石子放回来就好。这样你就不会乱掉，尽管你没有真的去算羊的数目。
As another example of matching being more fundamental than counting,
另一个“一对一”的例子比计数更单纯一些。
if I'm speaking to a packed auditorium, where every seat is taken and no one is standing,
如果在一个拥挤的礼堂里，每个位子都有人坐而且没人站着，
I know that there are the same number of chairs as people in the audience,
这样我就知道人数跟椅子数一样多，
even though I don't know how many there are of either.
虽然说我并不知道这两者的个数。
So, what we really mean when we say that two sets are the same size
所以，我们说两个集合一样大时，
is that the elements in those sets can be matched up one by one in some way.
它真正的意思就是两集合里的元素有办法“一对一”对应在一起。
My fourth grade teacher showed us the whole numbers laid out in a row, and below each we have its double.
所以小学老师将正整数写成一列，并将数字的两倍写在下面。
As you can see, the bottom row contains all the even numbers, and we have a one-to-one match.
你可以看到，底部那列包含了所有的偶数，这样就有了“一对一”的对应。
That is, there are as many even numbers as there are numbers.
也就是说，偶数和正整数一样多。
But what still bothers us is our distress over the fact that even numbers seem to be only part of the whole numbers.
但依旧困扰着我们的是偶数只是正整数的一部份这件事实。
But does this convince you that I don't have the same number of fingers on my right hand as I do on my left? Of course not.
不过这样能说服你，我左右手手指数目不同吗？当然没有！
It doesn't matter if you try to match the elements in some way and it doesn't work, that doesn't convince us of anything.
就算有的方法配对失败，那也没关系，因为这并没说服我们什么。
If you can find one way in which the elements of two sets do match up,
如果你可以找到一种方法让两边元素配对起来，
then we say those two sets have the same number of elements.
那我们就说这两个集合个数一样。
Can you make a list of all the fractions?
你有办法将分数像正整数那样列出来吗？
This might be hard, there are a lot of fractions!
这可能有点难，分数有很多！
And it's not obvious what to put first, or how to be sure all of them are on the list.
而且不太明显哪个要放前面，或是怎样把它们串起来。
Nevertheless, there is a very clever way that we can make a list of all the fractions.
不过，有一个办法我们可以把所有分数依序串起来。
This was first done by Georg Cantor, in the late eighteen hundreds.
这是十九世纪末数学家格奥尔格·康托的贡献。
First, we put all the fractions into a grid. They're all there.
首先，我们把分数上下左右对好。
For instance, you can find, say, 117/243, in the 117th row and 223rd column.
全部的分数都在这。比如说，你可以找到117/243，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,
现在我们要把它们串起来，从左上开始，然后斜对角地串下来、串上去。
skipping over any fraction, like 2/2, that represents the same number as one the we've already picked.
其中像2/2这类之前已经算过的分数，就把它跳过。
We get a list of all the fractions, which means we've created a one-to-one match between the whole numbers and the fractions,
因此我们就把分数串成一串了，这意思是分数和正整数有“一对一”的对应，
despite the fact that we thought maybe there ought to be more fractions.
虽然我们直觉是分数比较多个。
OK, here's where it gets really interesting.
好，这就是有趣的地方了。
You may know that not all real numbers -- that is, not all the numbers on a number line -- are fractions.
你也许知道用分数没办法表示所有的实数--也就是那些数线上的数。

The square root of two and pi, for instance.
像是根号2，还有圆周率π这些。
Any number like this is called irrational.
这类的数字叫作“无理数”。
Not because it's crazy, or anything, but because the fractions are ratios of whole numbers, and so are called rationals;
不只是因为它们很难懂，而是因为分数包含了所有整数的“比率”，所以被叫“可比的”，
meaning the rest are non-rational, that is, irrational.
而剩的就被叫作“不可比的”，也就是“无理的”。
Irrationals are represented by infinite, non-repeating decimals.
无理数可以用无穷小数表示，而且各位数没有规律。
So, can we make a one-to-one match between the whole numbers and the set of all the decimals, both the rationals and the irrationals?
那么，我们可以将正整数和所有无理、有理的小数“一对一”对应吗？
That is, can we make a list of all the decimal numbers?
也就是，我们可以将所有小数串起来吗？
Candor showed that you can't.
康托证明了这行不通。
Not merely that we don't know how, but that it can't be done.
不只想不到办法，而是真的没办法。
Look, suppose you claim you have made a list of all the decimals.
来看看，如果你声称你把小数串好了。
I'm going to show you that you didn't succeed, by producing a decimal that is not on your list.
我要来告诉你这是不可能的，因为我要找一个你那串那面没有的小数。
I'll construct my decimal one place at a time.
我要在小数点后一个一个位数决定。
For the first decimal place of my number, I'll look at the first decimal place of your first number.
我要用你那串的第1个数字的第1位数，来决定我的第1位数。
If it's a one, I'll make mine a two; otherwise I'll make mine a one.
如果它是1，我的就是2；否则我的就是1。
For the second place of my number, I'll look at the second place of your second number.
再用你的第2个数字的第2位数来决定我的第2位数。
Again, if yours is a one, I'll make mine a two, and otherwise I'll make mine a one.
一样，如果你的是1，我的就是2；否则我的就是1。
See how this is going? The decimal I've produced can't be on your list. Why?
看出怎么算下去了吗？我找到的这个小数，不可能在你那串里。为什么？
Could it be, say, your 143rd number?
比如它和你的第143个数会一样吗？
No, because the 143rd place of my decimal is different from the 143rd place of your 143rd number.
不可能，因为第143位数里，你的和我的不一样。
I made it that way. Your list is incomplete. It doesn't contain my decimal number.
这是我特别挑的。你没串成功。没有串到所有小数。
And, no matter what list you give me, I can do the same thing, and produce a decimal that's not on that list.
而不论你怎么串，我都可以做同样的事，然后找到一个你那串里没出现的小数。
So we're faced with this astounding conclusion: The decimal numbers cannot be put on a list.
所以我们得到了令人讶异的结论：所有小数没办法串成一串。
They represent a bigger infinity that the infinity of whole numbers.
它的“无限大”比正整数的“无限大”还大。
So, even though we're familiar with only a few irrationals, like square root of two and pi,
所以，尽管你只熟悉几个无理数，像是根号2和圆周率π，
the infinity of irrationals is actually greater than the infinity of fractions.
无理数的“无限大”实际上也比分数的“无限大”还要大。
Someone once said that the rationals -- the fractions -- are like the stars in the night sky.
有人曾这样比喻：有理数，或者说分数，就像天空的星星；
The irrationals are like the blackness.
而无理数就像是无尽的黑暗。
Cantor also showed that, for any infinite set,
康托同时也证明任何无穷大的集合，
forming a new set made of all the subsets of the original set represents a bigger infinity than that original set.
只要把它的所有子集都搜集起来，新的集合的“无限大”就比原本的还大。
This means that, once you have one infinity, you can always make a bigger one by making the set of all subsets of that first set.
意思是说，只要你有一种“无限大”，那你就可以用它的所有子集来做出比它更“无限大”的集合。
And then an even bigger one by making the set of all the subsets of that one. And so on.
接着再用这集合做出更加“无限大”的集合。不断做下去。
And so, there are an infinite number of infinities of different sizes.
所以，“无限大”之间也是有分不同的大小。
If these ideas make you uncomfortable, you are not alone.
如果你觉得这令人不适，这并不奇怪。
Some of the greatest mathematicians of Cantor's day were very upset with this stuff.
康托那个年代的一些伟大数学家也对这观念非常反感。
They tried to make this different infinities irrelevant, to make mathematics work without them somehow.
他们试着要把无限这观念抽离，让数学可以没有无限也能运作。
Cantor was even vilified personally, and it got so bad for him that he suffered severe depression,
康托甚至受到人身攻击，严重到让他饱受忧郁之苦，
and spent the last half of his life in and out of mental institutions.
并且在精神疗院渡过后半余生。
But eventually, his ideas won out. Today, they're considered fundamental and magnificent.
不过他的想法最终还是得到了肯定。今天，这观念被认为是基础并重要的。
All research mathematicians accept these ideas, every college math major learns them,
所有数学研究者都接受这观念，每个数学系都也都在教，
and I've explained them to you in a few minutes.
而我刚刚已经花了几分钟来解释。
Some day, perhaps, they'll be common knowledge.
也许有一天，这会变成大家的常识。
There's more. We just pointed out that the set of decimal numbers-- that is, the real numbers
还有一点。我们刚刚指出，小数，也就是实数，
is a bigger infinity than the set of whole numbers.
比正整数的“无限大”还多。
Candor wondered whether there are infinities of different sizes between these two infinities.
康托在想两个“无限大”之间是否还有不同层级的“无限大”。
He didn't believe there were, but couldn't prove it.
我们不这么认为，但也没办法证明。
Candor's conjecture became known as the continuum hypothesis.
康托的猜想变成了有名的“连续统假说”。
In 1900, the great mathematician David Hilbert listed the continuum hypothesis as the most important unsolved problem in mathematics.
在1900年，大数学家大卫·希尔伯特把连续统假说列为数学里最重要的未解问题。
The 20th century saw a resolution of this problem,
这问题在20世纪露出一些端倪，
but in a completely unexpected, paradigm-shattering way.
但是结果和超乎预期，并跌破大家眼镜。
In the 1920s, Kurt Godel showed that you can never prove that the continuum hypothesis is false.
在20世纪20年代，库尔特·哥德尔证明了你不可能证明连续统假说是错的。
Then, in the 1960s, Paul J. Cohen showed that you can never prove that the continuum hypothesis is true.
接着在20世纪60年代，保罗·J·寇恩证明了你不可能证明连续统假说是对的。
Taken together, these results mean that there are unanswerable questions in mathematics. A very stunning conclusion.
合在一起，这些结果告诉你数学里也有一些不能回答的问题。这是一个很令人震惊的结论。
Mathematics is rightly considered the pinnacle of human reasoning,
数学被公认是人类逻辑的结晶，
but we now know that even mathematics has its limitations.
但现在我们知道，就算是数学也有它的极限。
Still, mathematics has some truly amazing things for us to think about.
还有就是，数学里有一些值得我们思考、而且很令人着迷的道理。