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  • 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:

我在四年級的時候,國小老師有一天跟我們說:

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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
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