ADM-201 dump PMP dumps pdf SSCP exam materials CBAP exam sample questions

无穷究竟有多大? – 译学馆
未登录,请登录后再发表信息
最新评论 (0)
播放视频

无穷究竟有多大?

How big is infinity? - Dennis Wildfogel

当我在四年级时,有一天我的老师对我们说:
When I was in fourth grade, my teacher said to us one day:
偶数个数与整数个数是一样多的
There are as many even numbers as there are numbers.
”真的吗?”我想着。
“Really?” I thought.
哦,是的,这两者都有无穷多
Well, yeah, there are infinitely many of both,
所以我料想它们是一样多的。
so I suppose there are the same number of them.
但另一方面,偶数只是整数的一部分
But on other hand even numbers are only part of the whole numbers,
所有奇数都被剩下了
all the odd numbers are left over,
所以整数应该比偶数多,对吧?
so there’s got to be more whole numbers than even numbers, right?
为了领会我老师的意思
To see what my teacher was getting at,
让我们先想想两个集合大小一样是指什么意思
let’s first think about what it means for two sets to be the same size.
我说我左手和右手的手指一样多
What do I mean when I say I have the same number of fingers
是什么意思呢?
on my right hand as I do on left hand?
当然。我两边都有五个手指,但这其实比前面的情况简单。
Of course, I have five fingers on each, but it’s actually simpler than that.
我不用数,我只需要看看它们是不是能一一配对。
I don’t have to count, I only need to see that I can match them up, one to one.
实际上,我们认为一些古代人
In fact, we think that some ancient people
语言里都没有超过数字3的词
who spoke languages that didn’t have words for numbers greater than three
就使用了这种技巧
used this sort of magic.
举个例子,如果你把羊放出栅栏去放牧
For instance, if you let your sheep out of a pen to graze,
你可以在一旁用扔下石头的办法记录有多少走出来的羊
you can keep track of how many went out by setting aside a stone for each one,
然后在羊回来的时候把石头收起来
and putting those stones back one by one when the sheep return,
这样不用去数羊就知道有没有丢
so you know if any are missing without really counting.
另一个比数数更基本的例子
As another example of matching being more fundamental than counting,
如果我说满座的观众席
if I’m speaking to a packed auditorium,
每个位子都有一个人坐着并且没有一个人站着
where every seat is taken and no one is standing,
我就知道椅子和观众人数是一样的
I know that there are the same number of chairs as people in the audience,
即使我不知道两者数量
even though I don’t know how many there are of either.
所以,我们真正的意思是当我们说两个集合一样大时
So, what we really mean when we say that two sets are the same size
是说在里面的元素
is that the elements in those sets
可以以某种方式一一对应。
can be matched up one by one in some way.
我的四年级老师展示给我们的是
So my fourth grade teacher showed us
所有的整数排成一排,下面是对应的加倍值
the whole numbers laid out in a row, and below each we have its double.

就像你看到的那样,底下的一排都是偶数
As you can see, the bottom row contains all the even numbers,
所以我们有了一一对应关系
and we have a one-to-one match.
那就是说,偶数和整数一样多
That is, there are as many even numbers as there are numbers.
但是依然困扰着我们的是
But what still bothers us is our distress
偶数似乎只是整数一部分的令人悲伤的事实
over the fact that even numbers seem to be only part of the whole numbers.
但是这能让你信服吗?
But does this convince you
我左手的手指数和右手的手指数
that I don’t have the same number of fingers
不是一样多的
on my right hand as I do on my left?
当然不能
Of course not.
你配对元素的方法行不通
It doesn’t matter if you try to match
并不重要
the elements in some way and it doesn’t work,
这不能说服我们什么
that doesn’t convince us of anything.
如果你能找到一种办法
If you can find one way
让两个集合中的元素确实相配对了
in which the elements of two sets do match up,
这样我们就说着两个集合有相同数量的元素
then we say those two sets have the same number of elements.
你能为分数列个单子吗?
Can you make a list of all the fractions?
这可能很困难。有那么多分数呢!
This might be hard, there are a lot of fractions!
而且哪个放在前面
And it’s not obvious what to put first,
怎么保证它们都在列表中也不明晰
or how to be sure all of them are on the list.
虽然如此,有一个非常聪明的办法
Nevertheless, there is a very clever way
来写一个包含所有分数的列表
that we can make a list of all the fractions.
十八世纪晚期乔治·康托儿是第一个做这件事的人
This was first done by Georg Cantor, in the late eighteen hundreds.
首先,我们把所有的分数放到格子里
First, we put all the fractions into a grid.
他们都在这儿
They’re all there.
举个例子,你可以找到,117/243
For instance, you can find, say, 117/243,
在第117行和第223列中
in the 117th row and 223rd column.
现在我们列了一个列表
Now we make a list out of this
通过从左上角开始,来来回回向着对角扫描
by starting at the upper left and sweeping back and forth diagonally,
跳过任何像2/2的分数
skipping over any fraction, like 2/2,
大小相同的分数我们已经选择了一个。
that represents the same number as one the we’ve already picked.
我们把所有分数列成一个表格
We get a list of all the fractions,
这意味着我们已经在整数和分数之间
which means we’ve created a one-to-one match
建立了一对一的匹配
between the whole numbers and the fractions,
尽管我们以为应该有更多的分数
despite the fact that we thought maybe there ought to be more fractions.
好吧,这正是它有趣的地方。
OK, here’s where it gets really interesting.
你可能知道不是所有都是实数
You may know that not all real numbers
也就是说,不是所有分数都在坐标轴上
— that is, not all the numbers on a number line — are fractions.
用2π的平方根来举例
The square root of two and pi, for instance.
像这种数叫做无理数。
Any number like this is called irrational.
不是因为它很疯狂或其他的,
Not because it’s crazy, or anything,
而是因为是整数的比率的分数,
but because the fractions are ratios of whole numbers,
被称为有理数。
and so are called rationals;
意思就是剩下的都是无理数
meaning the rest are non-rational, that is, irrational.
无理数代表着无穷,不循环小数
Irrationals are represented by infinite, non-repeating decimals.
所以我们能一对一匹配吗?
So, can we make a one-to-one match
在整数和所有小数之间
between the whole numbers and the set of all the decimals,
包括有理数和无理数?
both the rationals and the irrationals?
也就是说,我们能列一个包括所有小数的表?
That is, can we make a list of all the decimal numbers?
康托尔演示过,你做不到
Candor showed that you can’t.
不仅仅是因为我们不知道怎么做,更因为它不可能完成
Not merely that we don’t know how, but that it can’t be done.
看,假设你提出你列了一个包括所有小数的表格
Look, suppose you claim you have made a list of all the decimals.
我会通过产生一个不在你列表里的小数
I’m going to show you that you didn’t succeed,
来证明你不可能成功
by producing a decimal that is not on your list.
我一次一位的建立我的小数。
I’ll construct my decimal one place at a time.
我的小数的第一个小数位,
For the first decimal place of my number,
我会看你的列表里的第一个小数的第一个小数位
I’ll look at the first decimal place of your first number.
如果他是1,我会让我的是2
If it’s a one, I’ll make mine a two;
如果他是2,我会让我的是1。
otherwise I’ll make mine a one.
在我的小数的第二个小数位
For the second place of my number,
我要看看你的第二个数字的第二个小数位的数字。
I’ll look at the second place of your second number.
跟前面一样,如果你的是1,我会让我的是2
Again, if yours is a one, I’ll make mine a two,
否则我会让我的是1
and otherwise I’ll make mine a one.
想想看这样继续下去会怎样?
See how this is going?
我生产的小数不可能在你的列表。
The decimal I’ve produced can’t be on your list.
为什么?有没有可能,跟你的第143个小数一样?
Why? Could it be, say, your 143rd number?
不,因为我的小数的第143位数位上的数字
No, because the 143rd place of my decimal
不同于你的第143个小数的第143位。
is different from the 143rd place of your 143rd number.
我就是这样做的
I made it that way.
你的清单是不完整的
Your list is incomplete.
它不包含我的小数
It doesn’t contain my decimal number.
而且不管你给我什么列表,我都可以做同样的事情
And, no matter what list you give me, I can do the same thing,
并产生一个不在列表里的小数
and produce a decimal that’s not on that list.
所以我们面对着这个令人震惊的结论:
So we’re faced with this astounding conclusion:
小数不能被全部放在一个列表中
The decimal numbers cannot be put on a list.
他们代表一个比整数的无穷更大的无穷
They represent a bigger infinity that the infinity of whole numbers.
所以,即使我们熟悉只有像根号2和π
So, even though we’re familiar with only a few irrationals,
这样的几个无理数
like square root of two and pi,
无理数个数的无穷大
the infinity of irrationals
实际上是比分数个数的无穷大更大的无穷。
is actually greater than the infinity of fractions.
有人曾经说过有理数
Someone once said that the rationals
——分数——就像夜空的星星。
— the fractions — are like the stars in the night sky.
无理数就像黑暗。
The irrationals are like the blackness.
康托尔也表明,对于任何无限集,
Cantor also showed that, for any infinite set,
形成新的由原集的所有子集构成的集合
forming a new set made of all the subsets of the original set
代表一个比原来更大的无穷集。
represents a bigger infinity than that original set.
这意味着,一旦你有一个无穷,
This means that, once you have one infinity,
你总是可以通过构造这个集合的所有子集
you can always make a bigger one
来构造一个更大的无穷集
by making the set of all subsets of that first set.
然后又可以通过构造上一个无穷集的所有子集
And then an even bigger one
来构造一个更大的无穷集
by making the set of all the subsets of that one.
以此类推
And so on.
所以,有无穷多个无穷大的尺寸。
And so, there are an infinite number of infinities of different sizes.
如果这些想法让你不舒服,你并不孤单。
If these ideas make you uncomfortable, you are not alone.
一些最伟大的数学家康托尔
Some of the greatest mathematicians of Cantor’s day
对这个东西非常不满。
were very upset with this stuff.
他们试图让这个不同的无穷大无关,
They tried to make this different infinities irrelevant,
使数学没有这些莫名其妙。
to make mathematics work without them somehow.
康托尔甚至诋毁自己,
Cantor was even vilified personally,
对他来说这如此糟糕以至于他患上了严重的抑郁症
and it got so bad for him that he suffered severe depression,
这让他的后半生都是在精神病院度过的。
and spent the last half of his life in and out of mental institutions.
但最终,他的想法胜利了
But eventually, his ideas won out.
今天,这些理论被认为是基础和宏伟的
Today, they’re considered fundamental and magnificent.
所有研究数学家都接受了这些想法
All research mathematicians accept these ideas,
每个大学数学专业都学习它们
every college math major learns them,
我已经用几分钟把它解释给你
and I’ve explained them to you in a few minutes.
也许有一天,他们会变成常识
Some day, perhaps, they’ll be common knowledge.
还有更多的
There’s more.
我们只是指出,小数的集合
We just pointed out that the set of decimal numbers
——也就是说,实数——是一个比整数
— that is, the real numbers — is a bigger infinity
更大的集合。
than the set of whole numbers.
康托尔想知道两个无穷大之间
Cantor wondered whether there are infinities
是否存在尺寸不同或尺寸无穷
of different sizes between these two infinities.
他相信没有,但无法证明这一点
He didn’t believe there were, but couldn’t prove it.
康托尔的猜想被称为连续统假设
Cantor’s conjecture became known as the continuum hypothesis.
在1900年,一个伟大的数学家,大卫·希尔伯特
In 1900, the great mathematician David Hilbert
将连续介质假设列为
listed the continuum hypothesis
数学中尚未解决的最重要问题。
as the most important unsolved problem in mathematics.
20世纪看到了解决这一问题的方法
The 20th century saw a resolution of this problem,
通过一个完全意想不到的,具有突破性的方式
but in a completely unexpected, paradigm-shattering way.
在1920年代,库尔特·哥德尔展示了
In the 1920s, Kurt Gödel showed
你永远无法证明连续介质假设是错误的
that you can never prove that the continuum hypothesis is false.
然后,在1960年代,保罗·j·科恩展示
Then, in the 1960s, Paul J. Cohen showed
你永远无法证明连续介质假设是正确的。
that you can never prove that the continuum hypothesis is true.
综上所述,这些研究结果意味着
Taken together, these results mean
在数学方面总有无法回答的问题。
that there are unanswerable questions in mathematics.
一个非常惊人的结论。
A very stunning conclusion.
数学一向被认为是人类推理的顶峰,
Mathematics is rightly considered the pinnacle of human reasoning,
但我们现在知道甚至数学都有其局限性。不过,数学有一些值得思考的真正让人惊叹的事情。
but we now know that even mathematics has its limitations.Still, mathematics has some truly amazing things for us to think about.

发表评论

译制信息
视频概述
听录译者

收集自网络

翻译译者

收集自网络

审核员

自动通过审核

视频来源

https://www.youtube.com/watch?v=UPA3bwVVzGI

相关推荐