the German mathematician David Hilbert

德國數學家大衛‧希爾伯特

devised a famous thought experiment

設計了一個聞名的思考實驗

to show us just how hard it is

向世人展示要了解「無限」的概念

to wrap our minds around the concept of infinity.

究竟有多麼困難。

Imagine a hotel with an infinite number of rooms

試想一間擁有無限多間客房的旅館

and a very hardworking night manager.

和一名非常努力工作的夜班經理。

One night, the Infinite Hotel is completely full,

有一晚，這間旅館的房間均已客滿

totally booked up with an infinite number of guests.

入住了無限名房客。

A man walks into the hotel

一個男人走進了旅館，

and asks for a room.

想要入住

Rather than turn him down,

與其拒絕他，

the night manager decides to make room for him.

這名夜班經理決定為他騰出一間空房。

How?

怎麼做呢？

Easy, he asks the guest in room number 1

很簡單，經理先將原先入住1號房的客人

to move to room 2,

安置至2號房

the guest in room 2 to move to room 3,

將原先入住2號房的客人安置至3號房，

and so on.

依此類推。

Every guest moves from room number "n"

n號房間原有的客人

to room number "n+1".

皆安置到n+1號房間。

Since there are an infinite number of rooms,

由於旅館內有無限多個房間，

there is a new room for each existing guest.

每一位既有的房客都能夠搬到新的房間，

This leaves room 1 open for the new customer.

於是新的客人便得以入住1號房。

The process can be repeated

這個過程可不斷重複，

for any finite number of new guests.

只要新房客的數目是有限可數的。

If, say, a tour bus unloads

假如一輛遊覽車駛進旅館

40 new people looking for rooms,

車上下來了40人都想入住

then every existing guest just moves

那麼每一位既有的房客

from room number "n"

只要從原先入住的n號房

to room number "n+40",

搬到n+40號房

thus, opening up the first 40 rooms.

就能將前40間房都空出來給新客人。

But now an infinitely large bus

但假如有一輛無限大的遊覽車

with a countedly infinite number of passengers

載滿了無限多名的乘客

pulls up to rent rooms.

停進旅館要入住

Countedly infinite is the key.

「可數無限」便成了關鍵。

Now, the infinite bus of infinite passengers

這輛載滿無限多名乘客的無限大遊覽車

perplexes the night manager at first,

起初讓夜班經理相當苦惱

but he realizes there's a way

幸好他後來還是找到了

to place each new person.

能夠安頓每個新房客的方法。

He asks the guest in room 1

他將原先入住1號房的客人

to move to room 2.

安置於2號房；

He then asks the guest in room 2

再將2號房的客人

to move to room 4,

安置於4號房；

the guest in room 3

將3號房的客人

to move to room 6,

安置於6號房；

and so one.

依此類推。

Each current guest moves from room number "n"

每一位既有的房客都從原先入住的n號房

to room number "2n",

搬到2n號房

filling up only the infinite even-numbered rooms.

便只有無限間數的雙號房會有人住

By doing this, he has now emptied

如此一來，旅館中

all of the infinitely many odd-numbered rooms,

無限多間的單號房都是空的了

which are then taken by the people

而無限大的遊覽車上所乘載的無限多名乘客

filing off the infinite bus.

便可入住

Everyone's happy and the hotel's business

這不僅是皆大歡喜的結果

is booming more than ever.

旅館的生意也越發興隆了。

Well, actually, it is booming

不過實際上，旅館的生意

exactly the same amount as ever,

其實和先前相比沒有任何變化

banking an infinite number of dollars a night.

每一晚都賺進無限可數的利潤。

Word spreads about this incredible hotel.

這間不可思議的旅館名氣越來越響亮

People pour in from far and wide.

人們從各地蜂擁而至

One night, the unthinkable happens.

有一晚，意想不到的事發生了

The night manager looks outside

夜班經理向外頭一看

and sees an infinite line

發現有無限多輛大巴士

of infinitely large buses,

正在旅館前大排長龍

each with a countedly infinite number of passengers.

每輛巴士上都載滿了無限可數的乘客。

What can he do?

他該怎麼辦呢？

If he cannot find rooms for them,

如果他無法為這些人安排房間

the hotel will lose out

旅館便將損失

on an infinite amount of money,

無限大的一筆收入

and he will surely lose his job.

他也一定會丟了飯碗。

Luckily, he remembers

還好，他還記得

that around the year 300 B.C.E.,

大約在西元前300年時

Euclid proved that there is an infinite quantity

歐幾里得證明了一件事：

of prime numbers.

質數是無限的。

So, to accomplish this seemingly impossible task

於是為了達成這項看似不可能的任務

of finding infinite beds

找到無限多的床位

for infinite buses

給乘坐在無限多輛巴士上的

of infinite weary travelers,

的無限多名疲憊旅客

the night manager assigns every current guest

夜班經理將每一位既有房客

to the first prime number, 2,

都安置在第一個質數，2，

raised to the power of their current room number.

再依該房客原本入住的房間號碼次方的房間號碼。

So, the current occupant of room number 7

於是，原本入住7號房的客人

goes to room number 2^7,

就會改住（2的7次方）號房

which is room 128.

也就是第128號房。

The night manager then takes the people

接著，夜班經理將無限多輛巴士中

on the first of the infinite buses

第一輛巴士上的所有乘客

and assigns them to the room number

都安排在下一個質數3

of the next prime, 3,

再依每一名乘客在巴士上的座號次方

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

所對應的房號。

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

如此一來，第一輛巴士上坐在7號座位的客人

goes to room number 3^7

便會入住（3的7次方）號房

or room number 2,187.

也就是2187號房。

This continues for all of the first bus.

第一輛巴士上的客人皆經此安排。

The passengers on the second bus

第二輛巴士上的客人

are assigned powers of the next prime, 5.

所各自入住的房號則是下一個質數5的座號次方

The following bus, powers of 7.

下一輛巴士，7的次方

Each bus follows:

依序排列：

powers of 11,

11的次方、

powers of 13,

13的次方、

powers of 17, etc.

17的次方等等。

Since each of these numbers

由於這些數字

only has 1 and the natural number powers

只有1和數字本身的自然數次方

of their prime number base as factors,

為其公因數

there are no overlapping room numbers.

就不會有重複的房號產生。

All the buses' passengers fan out into rooms

所有的客人便遵循這從質數發展出來的

using unique room assignment schemes

獨特房間安排方式

based on unique prime numbers.

各自散開進入他們的房間。

In this way, the night manager can accomodate

如此一來，夜班經理便能將

every passenger on every bus.

每一輛巴士上的每一位客人都安置妥當。

Although, there will be many rooms that go unfilled,

雖然會因此有許多空房產生

like room 6

例如6號房

since 6 is not a power of any prime number.

因為6不是任何一個質數的次方

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

還好，夜班經理的老闆們並沒有很會算數學

so his job is safe.

他也因此保住了他的工作。

The night manager's strategies are only possible

這位夜班經理所想出的策略之所以可行

because while the Infinite Hotel

是因為當這間無限旅館

is certainly a logistical nightmare,

雖然從後勤上來講像是一場噩夢

it only deals with the lowest level of infinity,

卻也只處理無限的最低層級

mainly, the countable infinity

也就是可數的無限，

of the natural numbers,

像是自然數

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

1、2、3、4， 等等。

Georg Cantor called this level of infinity aleph-zero.

另一名數學家康托爾稱這種程度的無限為「阿列夫零」。

We use natural numbers for the room numbers

旅館的房號和巴士上的座號

as well as the seat numbers on the buses.

皆使用自然數。

If we were dealing with higher orders of infinity,

假如我們今天所面對的是更高等的無限

such as that of the real numbers,

例如實數的程度，

these structured strategies

此類架構之下的策略

would no longer be possible

便不再可行

as we have no way

因為我們將無法

to systematically include every number.

有系統的包含每一個數字。

The Real Number Infinite Hotel has

實數無限旅館將會有

negative number rooms in the basement,

負數房號在地下樓層、

fractional rooms,

和幾分之幾的房號

so the guy in room 1/2 always suspects

於是住在1/2號房的人便總會懷疑

he has less room than the guy in room 1.

自己的房間大小不如1號房

Square root rooms, like room radical 2

平方根號的房號，像是（2的開根號）號房

and room pi,

以及數學常數Pi號房，

where the guests expect free dessert.

入住此房的客人也許會期待被招待點心

What self-respecting night manager

有哪一個有自尊心的夜班經理

would ever want to work there

會想在這裡工作

even for an infinite salary?

即便薪水是無限高？

But over at Hilbert's Infinite Hotel,

但在希爾伯特的無限旅館

where there's never any vacancy

住房率總是百分之百

and always room for more,

卻總還是有空房可供入住

the scenarios faced by the ever diligent

這名勤奮並可能過於好客的夜班經理

and maybe too hospitable night manager

所面對的各種狀況

serve to remind us

可以提醒我們

of just how hard it is

理解「無限」這樣的概念

for our relatively finite minds

對人類相對有限的頭腦來說

to grasp a concept as large as infinity.

是多大的困難。

Maybe you can help tackle these problems

或許在好好睡一頓覺後，

after a good night's sleep.

你會有辦法解開這些難題。

But honestly, we might need you

但老實說，你可能會在半夜兩點時

to change rooms at 2 a.m.

被通知要換房間。