## 字幕列表 影片播放

• When I was in 4th grade, my teacher said to us one day:

我在四年級的時候，國小老師有一天跟我們說：

• 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 numbersthat is, not all the numbers on a number lineare 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 rationalsthe fractionsare 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 numbersis 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。

When I was in 4th grade, my teacher said to us one day:

B1 中級 中文 美國腔 TED-Ed 整數 無限 分數 位數 數學

# 【TED-Ed】無限有多大？ How Big Is Infinity? - Dennis Wildfogel

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Furong Lai 發佈於 2014 年 01 月 11 日