The Infinite Hotel Paradox – Jeff Dekofsky
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The Infinite Hotel Paradox – Jeff Dekofsky

October 9, 2019

In the 1920’s, the German mathematician David Hilbert devised a famous thought experiment to show us just how hard it is to wrap our minds
around the concept of infinity. Imagine a hotel with an infinite
number of rooms and a very hardworking night manager. One night, the Infinite Hotel
is completely full, totally booked up
with an infinite number of guests. A man walks into the hotel
and asks for a room. Rather than turn him down, the night manager decides
to make room for him. How? Easy, he asks the guest in room number 1 to move to room 2, the guest in room 2 to move to room 3, and so on. Every guest moves from room number “n” to room number “n+1”. Since there are an infinite
number of rooms, there is a new room
for each existing guest. This leaves room 1 open
for the new customer. The process can be repeated for any finite number of new guests. If, say, a tour bus unloads
40 new people looking for rooms, then every existing guest just moves from room number “n” to room number “n+40”, thus, opening up the first 40 rooms. But now an infinitely large bus with a countably infinite
number of passengers pulls up to rent rooms. countably infinite is the key. Now, the infinite bus
of infinite passengers perplexes the night manager at first, but he realizes there’s a way to place each new person. He asks the guest in room 1
to move to room 2. 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. Each current guest moves
from room number “n” to room number “2n” — filling up only the infinite
even-numbered rooms. By doing this, he has now emptied all of the infinitely many
odd-numbered rooms, which are then taken by the people
filing off the infinite bus. Everyone’s happy and the hotel’s business
is booming more than ever. Well, actually, it is booming
exactly the same amount as ever, banking an infinite number
of dollars a night. Word spreads about this incredible hotel. People pour in from far and wide. One night, the unthinkable happens. The night manager looks outside and sees an infinite line
of infinitely large buses, each with a countably infinite
number of passengers. What can he do? 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. Luckily, he remembers
that around the year 300 B.C.E., Euclid proved that there
is an infinite quantity of prime numbers. So, to accomplish this
seemingly impossible task of finding infinite beds
for infinite buses of infinite weary travelers, the night manager assigns
every current guest to the first prime number, 2, raised to the power
of their current room number. So, the current occupant of room number 7 goes to room number 2^7, which is room 128. The night manager then takes the people
on the first of the infinite buses and assigns them to the room number of the next prime, 3, raised to the power of their seat
number on the bus. So, the person in seat
number 7 on the first bus goes to room number 3^7 or room number 2,187. This continues for all of the first bus. The passengers on the second bus are assigned powers of the next prime, 5. The following bus, powers of 7. Each bus follows: powers of 11, powers of 13, powers of 17, etc. Since each of these numbers only has 1 and the natural number powers of their prime number base as factors, there are no overlapping room numbers. All the buses’ passengers
fan out into rooms using unique room-assignment schemes based on unique prime numbers. In this way, the night
manager can accommodate every passenger on every bus. Although, there will be
many rooms that go unfilled, like room 6, since 6 is not a power
of any prime number. Luckily, his bosses
weren’t very good in math, so his job is safe. The night manager’s strategies
are only possible because while the Infinite Hotel
is certainly a logistical nightmare, it only deals with the lowest
level of infinity, mainly, the countable infinity
of the natural numbers, 1, 2, 3, 4, and so on. Georg Cantor called this level
of infinity aleph-zero. We use natural numbers
for the room numbers as well as the seat numbers on the buses. If we were dealing
with higher orders of infinity, such as that of the real numbers, these structured strategies
would no longer be possible as we have no way
to systematically include every number. The Real Number Infinite Hotel has negative number rooms in the basement, fractional rooms, so the guy in room 1/2 always suspects he has less room than the guy in room 1. Square root rooms, like room radical 2, and room pi, where the guests expect free dessert. What self-respecting night manager
would ever want to work there even for an infinite salary? But over at Hilbert’s Infinite Hotel, where there’s never any vacancy and always room for more, the scenarios faced by the ever-diligent and maybe too hospitable night manager serve to remind us of just how hard it is for our relatively finite minds to grasp a concept as large as infinity. Maybe you can help tackle these problems after a good night’s sleep. But honestly, we might need you to change rooms at 2 a.m.

Only registered users can comment.

  1. If the amount of rooms in the hotel is indefinite why can't just the night manager give rooms on the higher floors that are not taken yet, a.k.a add elements at the end of the list, and not make room for them (no pun intended) in another sides of the list?

  2. What happened to "M" or "O" and "P"? This all seems a little partial to me. I dont like this "N", no matter how much power it has.

  3. sir i think u need to just move to room 654857845784985784985785867858675858784587658675876876589876545678987656789876567876567898767898767890987678987678766787656787667876567e+56745565665676567667656787678767876

  4. The initial assumption that the infinite number of rooms are occupied by infinite number of guests is senseless in the first place. İnfinity divided infinity is an uncertainity

  5. I would like to book room number (square root of complex pi) to the power of (integral of the derivative of sine 90 degrees) for avocado's constant number of people, of which plank's constant are adults and absolute 0K are children.

  6. I missed something, I can comprehend infinite hotel being full, I get lost at infinite hotel getting 1 more guests though. That would be like the universe getting energy out of nowhere, which while not impossible, infinity would no longer apply in the same way.

  7. 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  8. … that's like the story of the university student who cuts his instant noodles brick in half every day so he'd have infinite amount of food to eat…

  9. Ok, I'm not the smartest guy… Maybe someone can help explain the answer to my question. I understand the example presented is hypothetical, however how could you fully book a hotel with infinite rooms? I know it was used for arguments sake just to demonstrate the paradox, but if infinity doesn't end, how could you book all of the rooms? The paradox crumbles right away under the pretenses it has set upon itself. Thoughts?

  10. Well, I'm a layman, not a mathematician so maybe have the wrong terminology. But it seems to he is trying to explain the infinite with finite numbers and examples. What's the point? I don't find the concept of infinity complicated. Probably most people don't. This seems kind of childish to me, like kind of a mind fuk. Doesn't accomplish anything , teach anything, further understanding in any way, or maybe I am missing it all.

  11. But technically he can't infinitely tell infinite people to move rooms without doing it infinitely so technically he would of got stuck straight away

  12. Some of the customers died while travelling to their next room.
    Imagine from room 9 to room 89
    thousand six hundred eighty seven.

  13. No rooms go unfilled right? Because the hotel is already full as the premise of each thought exercise? So room 6 already has a guest.
    Is there any practical application for the concept of infinity?

  14. Wouldn’t it take an infinite amount of time to tackle the first infinite bus? If it’s infinite it never stops

  15. I did not understand the last paradox, the infinite buses of infinite passengers, can anyone explain it in simple terms please?

  16. Недавно видела более понятное объяснение для этого явления)

  17. What if there is an infinite amount of visitors that are complaining and want their money back, is he making plus or minus at the end?

  18. YouTube recommendations:
    1. One night the infinite hotel is full
    2. Poor kitten hisses at me
    3. I learnt how to open an apple with my hands
    Me at 3 am: duuuude, that’s interesting!

  19. If its infinite, then how does it become full; and how does asking everyone to move a room over correct this? Why wouldn’t the same technique not also work for the infinite bus? Ask people to move a room over infinite, amirite?

  20. Glad he was able to make all the way up to an infinite amount of money instead of only an infinite amount of money

  21. Why deal with a lot of maths when he could've just assigned the new guests to the last rooms…sSssS so they don't have to move rooms

  22. There's people here saying it being full doesn't make sense but that's because when they think of infinite rooms they think of it like the rooms are being constantly made infinitely but no the rooms are there already there is no infinite growth of rooms just an infinite number of rooms, in this situation when they say the rooms are full it's because there are ininfinite amount of guest, and there's always one guest in every possible finite number of that infinity currently so yes it can be full, and therefore no more guest can enter the hotel apart from the infinite ones that are already their, which is why the formula of moving people up works because there is no caps on how far you move them up creating vacant rooms infinitely for the next batch of infinite guest

  23. “Excuse me Sir go to room 1.63937636843777393749429406726205826306936259593524959263504826295726249582730593625057272055829204847320396740193735”
    “Can you repeat that again sir?”


  24. This fails from the very beginning. For the manager to ask the guests to move 1 room down, there had to be an empty room at the end. By definition, there had to be an empty room at the end. And therefore, the hotel is not full and there is an empty room at the end.

  25. Did you forget that the infinite hotel was already full before any additional guests arrived? If the hotel is full, it is full, infinite or not.

  26. Someone once explained infinity to me using pi.

    Pi is random, and infinitely long. So, if you worked out a code that converted the numbers of pi into letters, eventually pi would tell the story of every person that ever lived. Perfectly. If you came across it and it told your story and just before the end there was a misspelling, you could continue reading until you found your life story again, spelled correctly this time.

    That's how large infinity is.

  27. The guests cannot be moved up since each room in this series of infinite rooms is already occupied. Infinite rooms are full. This is obviously true. If they already have an infinite number of guests, they can't accommodate, they're full to infinity. Also,there is no top room. Since there is no top room, you couldn't make it there in the first place (nor even come anywhere close), thus the solution to ask the guests in rooms 1 and 2, etc. to move is absurd and impossible, thus the hotel is not capable of accepting more guests. Even if the hotel employee (and the new arrivals) had infinite lifetimes, the new arrivals would be waiting for infinity for the hotel employee to ever find a guest to speak with, which would be a vain pursuit, the employee searching endlesslessly for all eternity and the new arrivals probably would choose to leave and find a hotel that could accommodate.

    Edit: Nevermind all of the above, yes I see now there would always be another room available, the hotel would just always have some number of guests walking the hallways (every guest walking to the next room times the number of new guests accepted).

  28. No need to change rooms as this hotel can never be full,just fill up the next room each time.
    If it had reached the n room there is always an n+1 room available,n+2 etc.
    ((n)^n)^n and many more n rooms.
    If it was full wouldn't be infinity but limited

  29. I haven't understood something: Can't the butler also give the room n+m to the new people? Because if there are infininetly rooms, there is always n+1=m and then m+1=k ,….
    Or have I missed something?

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