The Infinite Hotel Paradox - Jeff Dekofsky

无限旅馆悖论 | Jeff Dekofsky | Ted-ed

翻译人员: JIN锦금 Qian钱전 校对人员: kiki zhang

In the 1920's,

在1920年，

the German mathematician David Hilbert

德国数学家David Hilbert

devised a famous thought experiment

设计了一个著名的思维实验

to show us just how hard it is

向我们展示了

to wrap our minds around the concept of infinity.

深入思考无限的理念到底有多难。

Imagine a hotel with an infinite number of rooms

想象一个酒店有无限数量的房间

and a very hardworking night manager.

和一个认真工作的夜班经理。

One night, the Infinite Hotel is completely full,

一天晚上，无限酒店满房了，

totally booked up with an infinite number of guests.

被无限数量的客人全部预定了。

A man walks into the hotel and asks for a room.

一个男士走进了酒店

并且要求一个房间。

夜班经理并没有拒绝他，

Rather than turn him down,

the night manager decides to make room for him.

而是决定给他一个房间。

How?

怎么可能？

Easy, he asks the guest in room number 1

很简单，他让1号客房的客人

to move to room 2,

搬到了2号客房，

the guest in room 2 to move to room 3,

2号客房的客人搬到3号客房，

and so on.

以此类推。

Every guest moves from room number "n"

每个客人从“n”号房间

to room number "n+1".

搬入“n+1”号房间。

Since there are an infinite number of rooms,

因为那里有一个无限个房间，

there is a new room for each existing guest.

总有一个新房间给每一个已有的客人

This leaves room 1 open for the new customer.

这样1号房间就留给了新的客人。

The process can be repeated

这个过程可以被重复

for any finite number of new guests.

给任何有限数量的新客人们。

If, say, a tour bus unloads 40 new people looking for rooms,

假设一个观光大巴

40人下车要找房间，

then every existing guest just moves

那么每个已在的客人只要

from room number "n"

从“n"号房间

to room number "n+40",

搬到“n+40"号房间，

thus, opening up the first 40 rooms.

因此，就能打开新的40个房间。

但是现在有一个无限大的巴士

But now an infinitely large bus

with a countably infinite number of passengers

拉了可数的无限多的乘客

pulls up to rent rooms.

来租房间。

countably infinite is the key.

可数的无限是关键。

现在，极大的巴士的无限的乘客

Now, the infinite bus of infinite passengers

perplexes the night manager at first,

一开始为难了夜店经理，

but he realizes there's a way

但是他意识到有一个方法

to place each new person.

来安置每一个新人。

He asks the guest in room 1 to move to room 2.

他让1号房间的客人

搬到了2号房间。

He then asks the guest in room 2

他然后让2号房间的客人

to move to room 4,

搬到了4号房间，

the guest in room 3 to move to room 6,

3号房间的客人

搬到6号房间，

and so on.

以此类推。

Each current guest moves from room number "n"

每一个当前的客人从”n"号房间

to room number "2n" --

搬到了“2n“号房间，

filling up only the infinite even-numbered rooms.

填补了只有无限的偶数号房间。

By doing this, he has now emptied

通过这个，他现在清空了

all of the infinitely many odd-numbered rooms,

所有的无限的奇数号房间，

which are then taken by the people filing off the infinite bus.

无限大的巴士的乘客们

将占用这些奇数房间。

每个人的开心和酒店的生意

Everyone's happy and the hotel's business is booming more than ever.

达到了从未有过的兴荣。

Well, actually, it is booming exactly the same amount as ever,

好吧，事实上，它只是和以前一样

一直在兴荣，

banking an infinite number of dollars a night.

在一夜之间把无数的美元存入银行。

关于这家惊人的酒店的消息传开了。

Word spreads about this incredible hotel.

People pour in from far and wide.

人们从世界各地蜂拥而来。

One night, the unthinkable happens.

一天晚上，意外发生了。

The night manager looks outside

夜班经理看了外面

and sees an infinite line of infinitely large buses,

并且看到了由无限大巴们

组成的一个无限的排列，

each with a countably infinite number of passengers.

每个大巴都有一个可数的无限多的客人。

What can he do?

他能干些什么？

If he cannot find rooms for them, the hotel will lose out

如果他不能给他们找到房间，

这个酒店可能会

on an infinite amount of money,

失去一大笔无数的钱，

and he will surely lose his job.

并且他肯定会失去他的工作。

Luckily, he remembers that around the year 300 B.C.E.,

幸运的是，他记得

在公元300年前，

Euclid proved that there is an infinite quantity

Euclid证明了质数的

of prime numbers.

一个无穷量。

So, to accomplish this seemingly impossible task

所以，为了完成这个看上去不可能的任务

of finding infinite beds for infinite buses

找到无数的床

给无数的大巴上的

of infinite weary travelers,

无数的疲倦的旅客们，

the night manager assigns every current guest

夜店经理安排给每个当前的客人

第一个质数，2，

to the first prime number, 2,

raised to the power of their current room number.

幂指数为他们当前的房间号。

So, the current occupant of room number 7

因此，当前所居住房间号为7

goes to room number 2^7,

那么就要住到房间号为2的7次方的房间里，

which is room 128.

也就是128号房。

夜班经理然后带领

The night manager then takes the people on the first of the infinite buses

在第一个超级大巴们上的人们

and assigns them to the room number

并且安排了房间号给他们

of the next prime, 3,

下一个质数，3，

raised to the power of their seat number on the bus.

幂指数为他们在大巴的座位号。

So, the person in seat number 7 on the first bus

因此，座位号为7的第一辆大巴上的人

goes to room number 3^7

到房间号为3的七次方

or room number 2,187.

即2187号房间去。

This continues for all of the first bus.

这个过程持续给第一辆大巴上的所有人。

The passengers on the second bus

第二辆大巴上的乘客们

are assigned powers of the next prime, 5.

被安排到了下一个质数，5的幂.

The following bus, powers of 7.

接下的大巴，7的幂。

Each bus follows:

每辆大巴如下：

powers of 11, powers of 13,

11的幂，

13的幂，

powers of 17, etc.

17的幂，等等。

Since each of these numbers

因为这些数字每一个

only has 1 and the natural number powers

都只有1和它们的幂本身

of their prime number base as factors,

作为因数，

there are no overlapping room numbers.

因此就没有重叠数字号的房间。

All the buses' passengers fan out into rooms

所有大巴乘客们呈扇形散开到各自房间去

using unique room-assignment schemes

利用独特的房间安置计划

based on unique prime numbers.

基于独特的质数们。

In this way, the night manager can accommodate

这样一来，夜班经理能够安排

every passenger on every bus.

每辆大巴的每位乘客入住。

Although, there will be many rooms that go unfilled,

尽管，还有许多房间是空的，

像是6号房

like room 6,

因为6不是任何质数的幂。

since 6 is not a power of any prime number.

Luckily, his bosses weren't very good in math,

幸运的是，他的老板数学不是很棒，

so his job is safe.

所以他的工作是安全的。

The night manager's strategies are only possible

夜班经理策略的实现是可能的

because while the Infinite Hotel is certainly a logistical nightmare,

仅仅因为infinite酒店

一定是难办之事，

it only deals with the lowest level of infinity,

它只能处理最低水平的无穷数，

mainly, the countable infinity of the natural numbers,

主要是可数的

无限自然数

1, 2, 3, 4, and so on.

1，2，3，4.等等。

Georg Cantor called this level of infinity aleph-zero.

George Cantor称这个水平为无穷大阿列夫零.

We use natural numbers for the room numbers

我们使用自然数为房间号

同时也是大巴的座位号。

as well as the seat numbers on the buses.

If we were dealing with higher orders of infinity,

如果我们处理更高级顺序的无穷数，

such as that of the real numbers,

比如实数，

these structured strategies would no longer be possible

这些结构策略

便是不可能的

as we have no way to systematically include every number.

因为我们没有办法

系统地包含每一个数字。

实数Infinite酒店

The Real Number Infinite Hotel

has negative number rooms in the basement,

有负数号的房间在地下室，

fractional rooms,

分数号的房间，

so the guy in room 1/2 always suspects

因此房间号为二分之一的人总怀疑

he has less room than the guy in room 1.

他的房间小于1号房的人。

Square root rooms, like room radical 2,

平方根的房间，像房间号为根号2的房间

and room pi,

和房间号为圆周率的房间，

where the guests expect free dessert.

这些乘客期待免费的点心。

What self-respecting night manager would ever want to work there

而什么样的自重的夜班经理

会想要在那工作

even for an infinite salary?

甚至是为了无穷的薪水？

But over at Hilbert's Infinite Hotel,

但是再看看Hilbert的Infinite酒店，

where there's never any vacancy

永远都不会有空缺

and always room for more,

并且总是有房间给更多的人，

永远的勤劳还有可能太热情好客的夜班经理

the scenarios faced by the ever-diligent

and maybe too hospitable night manager

面临着这样的场景

serve to remind us of just how hard it is

是在提醒我们

我们这样相对有穷的思维

for our relatively finite minds

想掌握一个像无穷数一样大的概念。

to grasp a concept as large as infinity.

是有多么困难

Maybe you can help tackle these problems

可能你在好好睡了一晚后

能解决这些问题

after a good night's sleep.

But honestly, we might need you

但是老实说，我们可能需要你

to change rooms at 2 a.m.

在凌晨2点换房间。