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无限旅馆悖论 – 译学馆
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无限旅馆悖论

The Infinite Hotel Paradox - Jeff Dekofsky

在20世纪20年代 德国数学家大卫•希尔伯特
In the 1920’s, the German mathematician David Hilbert
设计了一个著名的思维实验
devised a famous thought experiment
来向我们表明
to show us just how hard it is
用人类的思维去把握“无限”这个概念究竟有多难
to wrap our minds around the concept of infinity.
假设有一个拥有无限多个房间的旅馆
Imagine a hotel with an infinitenumber of rooms
和一个非常勤劳的夜班经理
and a very hardworking night manager.
一天晚上 无限旅馆所有房间均已客满
One night, the Infinite Hotelis completely full,
被无限多个客人预订一空
totally booked upwith an infinite number of guests.
一位男士走进旅馆并且要求一间客房
A man walks into the hoteland asks for a room.
夜班经理并没有拒绝他
Rather than turn him down,
而是决定给他安排一间客房
the night manager decides to make room for him.
怎么办?
How?
很简单 他要求1号房间的客人
Easy, he asks the guest in room number 1
搬到2号房间
to move to room 2,
2号房间的客人搬到3号房间
the guest in room 2 to move to room 3,
以此类推
and so on.
每位客人都从“n”号房间
Every guest moves from room number”n”
搬到了“n+1”号房间
to room number”n+1″.
因为旅馆有无限个房间
Since there are an infinitenumber of rooms,
所以总有多余的房间给多出来的客人
there is a new roomfor each existing guest.
这样的话 1号房间就留给了新的客人
This leaves room 1 openfor the new customer.
这个方法可以重复用于安排任何有限个新客人
The process can be repeated for any finite number of new guests.
那么 假设
If, say,
一个观光大巴有40个乘客下车要找房间
a tour bus unloads 40 new people looking for rooms,
那么每个已住下的客人只要
then every existing guest just moves
从“n”号房间
from room number”n”
搬到“n+40 ”号房间
to room number”n+40″,
这样 就能有新的40个房间供入住
thus, opening up the first 40 rooms.
但是现在有一个无限长的巴士
But now an infinitely large bus
拉了可数的无限多位客人
with a countably infinitenumber of passengers
来订房间
pulls up to rent rooms.
可数无限是个关键
countably infinite is the key.
现在 这个无限长的巴士里面搭乘的无限多个客人
Now, the infinite busof infinite passengers
一开始为难了夜班经理
perplexes the night manager at first,
但是他找到一个方法来安置每一位新客人
but he realizes there’s a way to place each new person.
他让1号房间的客人搬到了2号房间
He asks the guest in room 1 to move to room 2.
他然后让2号房间的客人
He then asks the guest in room 2
搬到了4号房间
to move to room 4,
3号房间的客人搬到6号房间
the guest in room 3 to move to room 6,
以此类推
and so on.
让每一个原先入住的客人从”n”号房间
Each current guest movesfrom room number”n”
搬到了”2n”号房间
to room number”2n” —
于是只有无限多的偶数号房间里住了人
filling up only the infiniteeven-numbered rooms.
从而空闲下来的
By doing this, he has now emptied all
无限多个奇数号房间由新来的客人入住
of the infinitely many odd-numbered rooms,
这样 这辆无限长的巴士的乘客们将占用这些奇数房间
which are then taken by the people filing off the infinite bus.
皆大欢喜 旅馆的生意达到了空前的兴隆
Everyone’s happy and the hotel’s businessis booming more than ever.
那么 实际上呢 和以前一样
Well, actually, it is boomingexactly the same amount as ever,
旅馆一个晚上赚取了无数美钞
banking an infinite numberof dollars a night.
这个神奇的旅馆名声开始传播开来
Word spreads about this incredible hotel.
人们从世界各地蜂拥而来
People pour in from far and wide.
一天晚上 超乎想象的事情发生了
One night, the unthinkable happens.
夜班经理看了外面
The night manager looks outside
并且看到了无数辆无限长的大巴
and sees an infinite lineof infinitely large buses,
每辆大巴都载着可数的无限多位的客人
each with a countably infinitenumber of passengers.
他该怎么办呢?
What can he do?
如果他不能给这些顾客找到房间
If he can not find rooms for them,
这个旅馆就要失去
the hotel will lose out
无限的收入 并且他肯定会失去工作
on an infinite amount of money, and he will surely lose his job.
幸运的是 他记得在公元300年前
Luckily, he remembersthat around the year 300 B.C.E.,
欧几里得证明了
Euclid proved that thereis an infinite quantity
质数有无穷多个
of prime numbers.
所以 为了完成这个看上去不可能实现的任务
So, to accomplish thisseemingly impossible task
找到无限张床 给无限辆的大巴上的
of finding infinite bedsfor infinite buses
无限位疲倦的顾客 夜班经理这样安排原有的客人
of infinite weary travelers, the night manager assignsevery current guest
把第一个质数
to the first prime number,
2的当前房间号的指数幂作为新的房间号
2, raised to the power of their current room number.
比如 原先住在7号房间的客人
So, the current occupant of room number 7
就要搬到2的7次幂号房间里
goes to room number 2^7,
也就是128号房间
which is room 128.
然后 夜班经理这样安排
The night manager then takes the people
在第一个超级大巴上的客人
on the first of the infinite buses
以下一个质数3为底数
and assigns them to the room number
以他们在大巴的座位号为指数
of the next prime, 3,
来分配房间
raised to the power of their seatnumber on the bus.
因此 座位号为7的第一辆大巴上的人
So, the person in seatnumber 7 on the first bus
到房间号为3的7次方
goes to room number 3^7
即2187号房间去
or room number 2,187.
这个过程持续给第一辆大巴上的所有人
This continues for all of the first bus.
第二辆大巴上的乘客们被安排到了
The passengers on the second bus are assigned powers
下一个质数5为底数的房间
of the next prime, 5.
接下的大巴以 7 为底的幂
The following bus, powers of 7.
每辆大巴如下:11的幂 13的幂
Each bus follows: powers of 11, powers of 13,
17的幂 等等
powers of 17, etc.
因为所有的这些
Since each of these numbers only
以质数为底的幂
has 1 and the natural number powers
的因数都是这个质数本身
of their prime number base as factors,
所以房间号就不会重复
there are no overlapping room numbers.
也就利用了这种质数的独特性
All the buses’ passengersfan out into rooms
安置分配方案将所有大巴里的乘客
using unique room-assignment schemes
散开分配到到各自房间
based on unique prime numbers.
这样一来 夜班经理能够安排
In this way, the nightmanager can accommodate
每辆大巴的每位乘客入住
every passenger on every bus.
尽管 还有许多房间是空的
Although, there will bemany rooms that go unfilled,
像是6号房间 因为6不是任何质数的幂
like room 6, since 6 is not a power of any prime number.
不过幸运的是 他的老板并不擅长数学
Luckily, his bossesweren’t very good in math,
所以夜班经理的工作还是保住了
so his job is safe.
夜班经理的策略之所以可行
The night manager’s strategiesare only possible
是因为尽管无限旅馆的后勤难如噩梦
because while the Infinite Hotelis certainly a logistical nightmare,
但它只涉及了最低等级的无限
it only deals with the lowestlevel of infinity,
主要是自然数的可数无限
mainly, the countable infinityof the natural numbers,
比如1 2 3 4 等等
1, 2, 3, 4, and so on.
格奥尔格•康托尔称这个等级的无限为阿列夫零
Georg Cantor called this levelof infinity aleph-zero.
我们使用自然数为房间号
We use natural numbersfor the room numbers
同时也是使用自然数来对应大巴的座位号
as well as the seat numbers on the buses.
如果我们处理更高级的无穷数
If we were dealingwith higher orders of infinity,
比如实数的无限情况
such as that of the real numbers,
前面这些结构策略就不行了
these structured strategies would no longer be possible
因为我们没有办法系统地包含每一个数字
as we have no wayto systematically include every number.
实数的无限旅馆
The Real Number Infinite Hotel has
在地下室有负数号的房间
negative number rooms in the basement,
还有分数号的房间 因此1/2号房间的客人总是怀疑
fractional rooms, so the guy in room 1/2 always suspects
他的房间比1号房间的面积小
he has less room than the guy in room 1.
还有平方根的房间 比如根号2的房间
Square root rooms, like room radical 2,
还有π号房间 这里的房客想要吃免费甜点
and room pi, where the guests expect free dessert.
哪位有自尊的夜班经理愿意在那里工作呢?
What self-respecting night managerwould ever want to work there
就算薪水无限也不干吧
even for an infinite salary?
但在希尔伯特的无限旅馆那边
But over at Hilbert’s Infinite Hotel, where
永远没有空房 又总是有地方剩余
there’s never any vacancy and always room for
勤勉不休而又热情好客
more, the scenarios faced by the ever-diligent
甚至热情过头的夜班经理所面对的情况告诉我们
and maybe too hospitable night manager serve to remind us
我们这样相对的有限的思维
of just how hard it is
想掌握无限那么大的概念
for our relatively finite minds
是多么的难
to grasp a concept as large as infinity.
也许你在好好睡了一晚后 能解决这些问题
Maybe you can help tackle these problems after a good night’s sleep.
不过老实说 我们可能需要你在凌晨两点钟换房间
But honestly, we might need you to change rooms at 2 a.m.

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视频概述

无限旅馆是由德国数学家大卫·希尔伯特(David Hilbert)创造的一个思想实验,说的是一间拥有无限数量客房的旅馆,客满后继续住客的问题。这个实验告诉我们,用人类的有限思维去把握“无限”这个概念究竟有多难。

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视频来源

https://www.youtube.com/watch?v=Uj3_KqkI9Zo

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