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“无穷”比“无穷”大

How to Count Infinity

今天我们要数到无穷大
Today we’re going to count infinity.
数数看起来像基本演绎法 比如 当我们说 我们有5只羊的时候
Now counting may seem elementary, like, when we say that we have five sheep, what we mean
意思是我们有编号从1到5的五只羊
is that we have one sheep for every number from one to five.
十只羊就是编号从1到10 或者从2到11的十只羊
And ten sheep means one for every number from one to ten… or two to eleven.
所以说两个集合有同样的元素 简单来说 如果画一条线
So we say that two sets have the same number of things in them, simply if you can draw
连接两个集合中的元素 反之亦然 但每个元素只能连一条线
a line relating every item in one set to something in the other, and vice versa, exactly once.
它们是一对!
They’re partners!
这与1+2=3一样 但3≠4
It’s the same when we say that two plus one equals three, or three doesn’t equal four:
我们仅仅是用线将一个集合中的元素与另一个集合中的元素连起来
we’re just describing the lines you draw to relate one set of things to another.
不管怎样 数羊太无聊了 除非你想数到无穷大
But either way, counting sheep is boring, that is, unless you want to count INFINITELY
无数的羊
many sheep.
假设你在0到2之间有无数只羊
Like, if you had a sheep for every number between 0 and 2, would that be more sheep
它们会比0到1之间的羊多吗?
than if you had one for every number between 0 and 1?
不可能!
Nope!
因为0到1之间的每一只羊都能对应0到2之间的两只
Because you can relate every number between 0 and 1 to its double, giving you every number
(如果你想“拆开” 可以把0到2之间的每个数字
between 0 and 2 (and if you want to “undo,” you can just divide every number between 0
删去一半 这样就和0到1之间的数字一样多了)
and 2 in half to get back all the numbers between 0 and 1).
但是 0到1之间的数要比
But there are more real numbers between 0 and 1 than there are in the infinite set of
无穷大的整数集合中的数还要多
integers 1, 2, 3, 4, and so on.
究竟怎样才能证明呢?
How on earth do we know that?
只要画一些线
Just draw some lines.
从1开始 画一条线来连接1和0到1中的某个数
For “1”, draw a line to a number between 0 and 1.
然后是2 画一条线来连接2和0到1中的某个数
And for “2”, draw a line to another number between zero and one.
然后是3 连接3和0到1中的某个数
For “3”, draw a line to a number between… zero and one.
以此类推
And so on.
但是 不管我们用线连接了多少0到1中的数
BUT, no matter what numbers between 0 and 1 that we’ve drawn lines to, we can always
还是能不断的写出新的数字 和这里的第17个数字不同
write down a number between 0 and 1 that disagrees with the first digit here, and the second
和第2个不同 和第3个不同…新写出来的数字就和之前
digit here, and the third digit here, and so on… so this new number will be different
所有数字都不一样了
from ALL of the other numbers we’ve drawn lines to.
每个整数都被连接起来了
But we’ve already drawn a line from every integer, so there’s no one left to be this
没有这个和这个数字配对的了!
number’s partner!
然而 由于构建数字的方式很机智 我们能找到一个例外
What’s more, because of the clever way we built it, we can find an extra, lonely number
不管我们选了哪个数字 都会出现这样一个单独的数字 也就意味着
like this no matter what other numbers we picked, which means we can NEVER draw lines
在每个整数只能连线一次的情况下 我们不能将整数与所有0到1之间的数字连线
from the integers to all of the numbers between 0 and 1 with only one line per integer…
也就是说 0到1之间确实存在着
And this means that there really are more real numbers between 0 and 1 than there are
比整数集合还要多的数字
in the infinite set of counting numbers 1, 2, 3, 4, and so on forever.
所以 黑兹尔·格蕾丝 某些无穷的数字集合确实要比另外一些无穷的数字集合大的
So, Hazel Grace, some infinities truly are bigger than other infinities.

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视频概述

本视频证明了某些无穷的集合确实要比另外一些无穷的集合大

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收集自网络

翻译译者

Iris

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W

视频来源

https://www.youtube.com/watch?v=A-QoutHCu4o

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