How big is infinity? - Dennis Wildfogel

无限有多大？- 丹尼斯·瓦弗杰

翻译人员: Geoff Chen 校对人员: Zhiting Chen

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."

“偶数的个数 和正整数的个数一样多。”

"Really?", I thought.

“真的吗？”我心想。 噢对！两个都是无限多个，所以一样多。

Well, yeah, there are infinitely many of both,

so I suppose there are the same number of them.

But even numbers are only part of the whole numbers,

但另一方面，偶数只是正整数的一部份， 而奇数就是剩下的部份，

all the odd numbers are left over,

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,

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。

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

当我说我左手的手指 和右手的手指一样多时，这意谓着什么？

What do I mean when I say I have the same number of fingers

on my right hand as I do on left hand?

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。

我不用去算，我只要知道 我能够将它们“一对一”对应起来。

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

In fact, we think that some ancient people

事实上，我们认为古代那些 语言里数字只到三的人们

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

就是用这个技俩。 如果你把你的羊从羊圈里放出去吃草，

used this sort of magic.

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

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,

然后在羊回来的时候 再把石子放回来就好。

so you know if any are missing without really counting.

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,

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，

where every seat is taken and no one is standing,

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， 它真正的意思就是

is that the elements in those sets

两集合里的元素 有办法“一对一”对应在一起。

can be matched up one by one in some way.

My fourth grade teacher showed us

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。

the whole numbers laid out in a row, and below each we have its double.

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。

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

and we have a one-to-one match.

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

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。

over the fact that even numbers seem to be only part of the whole numbers.

不过这样能说服你 我左右手手指数目不同吗？

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.

当然没有！就算有的方法 配对失败，那也没关系，

It doesn't matter if you try to match

the elements in some way and it doesn't work,

因为这并没说服我们什么。

that doesn't convince us of anything.

如果你可以找到一种方法 让两边元素配对起来，

If you can find one way

in which the elements of two sets do match up,

那我们就说这两个集合个数一样。

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

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？ 这可能有点难，分数有很多！

This might be hard, there are a lot of fractions!

而且不太明显哪个要放前面， 或是怎样把它们串起来。

And it's not obvious what to put first,

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

不过，有一个办法 我们可以把所有分数依序串起来。

Nevertheless, there is a very clever way

that we can make a list of all the fractions.

这是十九世纪末 数学家康托尔的贡献。

This was first done by Georg Cantor, in the late eighteen hundreds.

首先，我们把分数上下左右对好。 全部的分数都在这。比如说，你可以找到 117/243

First, we put all the fractions into a grid.

They're all there.

For instance, you can find, say, 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

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。

by starting at the upper left and sweeping back and forth diagonally,

skipping over any fraction, like 2/2,

其中像 2/2 这类之前已经算过的分数 就把它跳掉。

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

We get a list of all the fractions,

因此我们就把分数串成一串了， 这意思是分数

which means we've created a one-to-one match

between the whole numbers and the fractions,

和正整数有“一对一”的对应， 虽然我们直觉是分数比较多个。

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

好，这就是有趣的地方了。

OK, here's where it gets really interesting.

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。

You may know that not all real numbers

-- that is, not all the numbers on a number line -- are fractions.

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”。 不只是因为它们很难懂， 而是因为分数包含了

Not because it's crazy, or anything,

but because the fractions are ratios of whole numbers,

所有整数的“比率”，所以被叫“可比的”， 而剩的就被叫作“不可比的”， 也就是“无理的”。

and so are called rationals;

meaning the rest are non-rational, that is, irrational.

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

那么，我们可以将正整数和 所有无理、有理的小数

between the whole numbers and the set of all the decimals,

both the rationals and the irrationals?

“一对一”对应吗？ 也就是，我们可以将所有小数串起来吗？

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

康托尔证明了这行不通。 不只想不到办法，而是真的没办法。

Cantor showed that you can't.

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

来看看，如果你声称你把小数串好了。 我要来告诉你这是不可能的，

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

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

by producing a decimal that is not on your list.

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,

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。

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

If it's a one, I'll make mine a two;

如果它是 1，我的就是 2；否则我的就是 1。

otherwise I'll make mine a one.

For the second place of my number,

再用你的第 2 个数字的第 2 位数 来决定我的第 2 位数。

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

一样，如果你的是 1，我的就是 2； 否则我的就是 1。

Again, if yours is a one, I'll make mine a two,

and otherwise I'll make mine a one.

See how this is going?

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。

The decimal I've produced can't be on your list.

为什么？比如它和你的 第 143 个数会一样吗？ 不可能，因为第 143 位数里，

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

No, because the 143rd place of my decimal

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

你的和我的不一样。 这是我特别挑的。

I made it that way.

Your list is incomplete.

你没串成功。 没有串到所有小数。

It doesn't contain my decimal number.

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

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。

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

So we're faced with this astounding conclusion:

所以我们得到了 令人讶异的结论：

The decimal numbers cannot be put on a list.

所有小数没办法串成一串。 它的“无限大”比正整数的“无限大”还大。

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

所以，尽管你只熟悉几个无理数， 像是根号 2 和圆周率 π，

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

like square root of two and pi,

the infinity of irrationals

无理数的“无限大”实际上也比 分数的“无限大”还要大。

is actually greater than the infinity of fractions.

Someone once said that the rationals

有人曾这样比喻： 有理数，或者说分数，就像天空的星星；

-- the fractions -- are like the stars in the night sky.

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,

康托尔同时也证明任何无穷大的集合， 只要把它的所有子集都搜集起来，

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

新的集合的“无限大”就比原本的还大。 意思是说，只要你有一种“无限大”

represents a bigger infinity than that original set.

This means that, once you have one infinity,

you can always make a bigger one

那你就可以用它的所有子集 来做出比它更“无限大”的集合。

by making the set of all subsets of that first set.

And then an even bigger one

by making the set of all the subsets of that one.

接着再用这集合做出更加“无限大”的集合。 不断做下去。

And so on.

所以，“无限大”之间也是有分不同的大小。

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

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。 一些康托尔那年代的伟大数学家

Some of the greatest mathematicians of Cantor's day

were very upset with this stuff.

也对这观念非常反感。 他们试着要把无限这观念抽离，

They tried to make these different infinities irrelevant,

to make mathematics work without them somehow.

让数学可以 没有无限也能运作。

康托尔甚至受到人身攻击， 严重到让他饱受忧郁之苦，

Cantor was even vilified personally,

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

并且在精神疗院渡过后半余生。

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

不过他的想法最终得到肯定。 今天，这观念被认为是基础并重要的。

But eventually, his ideas won out.

Today, they're considered fundamental and magnificent.

All research mathematicians accept these ideas,

所有数学研究者都接受这观念， 每个数学系都也都在教，

every college math major learns them,

而我刚刚已经花了几分钟来解释。

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

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.

还有一点。我们刚刚指出 小数，也就是实数，

We just pointed out that the set of decimal numbers

-- that is, the real numbers -- is a bigger infinity

than the set of whole numbers.

比正整数的“无限大”还多。 康托尔在想两个“无限大”之间

Cantor wondered whether there are infinities

是否还有不同层级的“无限大”。 我们不这么认为，但也没办法证明。

of different sizes between these two infinities.

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

康托尔的猜想变成 有名的“连续统假说”。

Cantor's conjecture became known as the continuum hypothesis.

在 1900 年，大数学家希尔伯特 把连续统假说列为

In 1900, the great mathematician David Hilbert

listed the continuum hypothesis

as the most important unsolved problem in mathematics.

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， 但是结果和超乎预期，并跌破大家眼镜。

but in a completely unexpected, paradigm-shattering way.

In the 1920s, Kurt Gödel showed

在 1920 年代，哥德尔证明了 你不可能证明连续统假说是错的。

that you can never prove that the continuum hypothesis is false.

Then, in the 1960s, Paul J. Cohen showed

接着在 1960 年代，寇恩证明了 你不可能证明连续统假说是对的。

that you can never prove that the continuum hypothesis is true.

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。

Taken together, these results mean

that there are unanswerable questions in mathematics.

A very stunning conclusion.

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。

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。

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