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In the 1920's, the German mathematician David Hilbert devised a famous thought experiment
在1920年，德国数学家戴维·希尔伯特设计了一个著名的思维实验，
to show us just how hard it is to wrap our minds around the concept of infinity.
向我们展示了深入思考无限的理念到底有多难。
Imagine a hotel with an infinite number of rooms and a very hardworking night manager.
想象一个酒店有无限数量的房间和一个认真工作的夜班经理。
One night, the Infinite Hotel is completely full, totally booked up with an infinite number of guests.
一天晚上，无限酒店满房了，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。
Rather than turn him down, the night manager decides to make room for him. How?
夜班经理并没有拒绝他，而是决定给他一个房间。怎么可能？
Easy, he asks the guest in room number 1 to move to room 2,
很简单，他让1号客房的客人搬到了2号客房，
the guest in room 2 to move to room 3, and so on.
2号客房的客人搬到3号客房，以此类推。
Every guest moves from room number 'n' to room number 'n+1'.
每个客人从“n”号房间搬入“n+1”号房间。
Since there are an infinite number of rooms, there is a new room for each existing guest.
因为那里有一个无限个房间，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 for any finite number of new guests.
这个过程可以被重复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，有40人下车要找房间，
then every existing guest just moves from room number 'n' to room number 'n+40', thus, opening up the first 40 rooms.
那么每个已在的客人只要从“n”号房间搬到“n+40”号房间，因此，就能打开新的40个房间。
But now an infinitely large bus with a countably infinite number of passengers pulls up to rent rooms.
但是现在有一个无限大的巴士拉了可数的无限多的乘客来租房间。
Countably infinite is the key.
可数的无限是关键。
Now, the infinite bus of infinite passengers perplexes the night manager at first,
现在，无限大的巴士的无限的乘客让夜班经理一开始为了难，
but he realizes there's a way to place each new person.
但是他意识到有一个方法来安置每一个新人。
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 to move to room 4, the guest in room 3 to move to room 6, and so on.
他然后让2号房间的客人搬到了4号房间，3号房间的客人搬到6号房间，以此类推。
Each current guest moves from room number 'n' to room number '2n' -- filling up only the infinite even-numbered rooms.
每一个当前的客人从“n”号房间搬到了“2n”号房间，填补了只有无限的偶数号房间。
By doing this, he has now emptied all of the infinitely many odd-numbered rooms,
通过这么做，他现在清空了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。
Everyone's happy and the hotel's business is booming more than ever.
每个人都很开心，酒店的生意达到了从未有过的兴隆。
Well, actually, it is booming exactly the same amount as ever, banking an infinite number of dollars a night.
好吧，事实上，它只是和以前一样，一直在兴隆，在一夜之间把无数的美元存入银行。
Word spreads about this incredible hotel. People pour in from far and wide.
关于这家惊人的酒店的消息传开了。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 and sees an infinite line of infinitely large buses,
夜班经理看了看外面，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. What can he do?
每个大巴上都有一个可数的无限多的客人。他能做些什么？
If he cannot find rooms for them, the hotel will lose out on an infinite amount of money, and he will surely lose his job.
如果他不能给他们找到房间，这个酒店可能会失去一大笔无数的钱，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 of prime numbers.
欧几里得证明了质数的一个无穷量。
So, to accomplish this seemingly impossible task of finding infinite beds for infinite buses of infinite weary travelers,
所以，为了完成这个看上去不可能的任务，找到无限的床给无限的大巴上的无限的疲倦的旅客们，
the night manager assigns every current guest to the first prime number, 2, raised to the power of their current room number.
夜店经理安排给每个当前的客人第一个质数，2，幂指数则为他们当前的房间号。
So, the current occupant of room number 7 goes to room number 2^7, which is room 128.
因此，当前所居住房间号为7，那么就要住到房间号为2的7次方的房间里，也就是128号房。
The night manager then takes the people on the first of the infinite buses
夜班经理然后带着第一个无限大的大巴们上的人们，
and assigns them to the room number of the next prime, 3, raised to the power of their seat number on the bus.
并且安排了房间号给他们，下一个质数，3，幂指数为他们在大巴的座位号。
So, the person in seat number 7 on the first bus goes to room number 3^7 or room number 2,187.
因此，座位号为7的第一辆大巴上的人到房间号为3的七次方，即2187号房间去。
This continues for all of the first bus.
第一辆大巴上的所有人都按照这个流程进行。
The passengers on the second bus are assigned powers of the next prime, 5. The following bus, powers of 7.
第二辆大巴上的乘客们被安排到了下一个质数，5的幂。接下的大巴，7的幂。
Each bus follows: powers of 11, powers of 13, powers of 17, etc.
每辆大巴如下：11的幂，13的幂，17的幂，等等。
Since each of these numbers only has 1 and the natural number powers of their prime number base as factors,
因为这些数字每一个都只有1和它们的幂本身作为因数，
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 using unique room-assignment schemes based on unique prime numbers.
所有的客人便遵循这从质数发展出来的独特房间安排方式，各自散开进入他们的房间。
In this way, the night manager can accommodate every passenger on every bus.
这样一来，夜班经理就能够安排每辆大巴的每位乘客入住了。
Although, there will be many rooms that go unfilled, like room 6, since 6 is not a power of any prime number.
尽管，还有许多房间是空的，像是6号房，因为6不是任何质数的幂。
Luckily, his bosses weren't very good in math, so his job is safe.
幸运的是，他的老板数学不是很棒，所以他的工作是安全的。
The night manager's strategies are only possible
这位夜班经理所想出的策略之所以可行，
because while the Infinite Hotel is certainly a logistical nightmare,
是因为当这间无限旅馆虽然从后勤上来讲像是一场噩梦，
it only deals with the lowest level of infinity,
却也只处理无限的最低层级，
mainly, the countable infinity of the natural numbers, 1, 2, 3, 4, and so on.
也就是可数的无限，像是自然数1、2、3、4，等等。
Georg Cantor called this level of infinity aleph-zero.
数学家格奥尔格·康托尔称这种程度的无限为阿列夫零。
We use natural numbers for the room numbers as well as the seat numbers on the buses.
旅馆的房号和巴士上的座号都是使用的自然数。
If we were dealing with higher orders of infinity, such as that of the real numbers,
假如我们今天所面对的是更高等的无限，例如实数的程度，
these structured strategies would no longer be possible as we have no way to systematically include every number.
此类架构之下的策略便不再可行，因为我们将无法有系统的包含每一个数字。
The Real Number Infinite Hotel has negative number rooms in the basement, fractional rooms,
实数无限旅馆将会有负数房号在地下楼层和几分之几的房号，
so the guy in room 1/2 always suspects he has less room than the guy in room 1.
于是住在1/2号房的人便总会怀疑自己的房间大小不如1号房。
Square root rooms, like room radical 2, and room pi, where the guests expect free dessert.
平方根号的房号，像是√2号房，以及数学常数π号房，入住此房的客人也许会期待被招待免费点心。
What self-respecting night manager would ever want to work there even for an infinite salary?
即便薪水是无限高，又会有哪一个有自尊心的夜班经理想在这里工作？
But over at Hilbert's Infinite Hotel, where there's never any vacancy and always room for more,
但在希尔伯特的无限旅馆住房率总是百分之百，却总还是有空房可供入住，
the scenarios faced by the ever-diligent and maybe too hospitable night manager
这名勤奋并可能过于好客的夜班经理所面对的各种状况，
serve to remind us of just how hard it is for our relatively finite minds to grasp a concept as large as infinity.
可以提醒我们，理解无限这样的概念对人类相对有限的头脑来说是多么的困难。
Maybe you can help tackle these problems after a good night's sleep.
或许在好好睡一顿觉后，你会有办法解开这些难题。
But honestly, we might need you to change rooms at 2 a.m.
但老实说，你可能会在半夜两点时被通知要换房间。
