The Infinite Hotel Paradox – Jeff Dekofsky


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.

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96 thoughts on “The Infinite Hotel Paradox – Jeff Dekofsky

  1. It is not the same thing that Rick from Rick and Morty everytime said that all is happening in an Infinite Universe, in Infinite Time, in Infinite Versions from all of this?

  2. dude if the hotel is infinte and the number of people that come in is infinite then the hotel is still infinite so why dont the people that come in just take a new roomwhy does everyone have to move rooms ? every new customer should be able to just go to their room, proof is that if there is room for all the customers that had to move to a new room then that already proofs that the new customers coulve just taken those rooms so what is this for its useless.

  3. So if a person is in room 1000, then he has to move to room 2000. So he gets his morning walk done. Right?😂🤣😅

  4. Or just add the new guests to the next un used room since it is infinite

    If there are 1000 rooms being used and someone walks in just put him in room 1001 and so on

  5. This is all fundamentally flawed due to math and our natural nature of trying to order and conceptualize things. An example would be dividing infinity by infinity. It wouldn't be one in this case but infinity. The numbers never end so it just infinitely divides itself for infinity so instead of one you would just be left with infinity÷infinity= (infinity/infinity)
    Or infinity÷infinity=infinity÷infinity which would be infinity.

  6. You can stop at 0:31 when the narrator says the infinite hotel is completely full. Infinity complex problems solved.

  7. Infinity is an ill conceived human construct to qualify our over blown sense of intelligence , remove the other cancer of time and it all makes sense , there is no place in this cosmos for the less than pathetic human whimsy on problems born from our irrelevant fear of end and mortality .

  8. Wait imagine this, if there are room for people to change room so why don’t the staffs just move the new people to those room

    Also it would literally takes forever to do

  9. Like universe which is infinite born from big bang, the universe bursts and spread so far that it keeps spreading without end.. my question is, what is outside of big bang or outside of the spreading universe???

  10. Imagine how long it would take to get to your room if it was room number 123648172647263856282719782649265992018620173926190472837487374659103847299902929757638278102947491903749263477241469423041256178235276527412741264231407646141674614661721076815615660561006150601576666366222600625376162730522

  11. If you really want to try to wrap your head around a problem: what happens when an infinite number of people flush their toilet at the same time?

  12. This is impossible to be comprehended by a human brain. It’s time we see the world through another species’ point of view.

  13. or..he could just put all the new guest in the room next to room "n" instead of moving everyone the occupants of room 1 to the next room (room 2)

  14. this mistake is already at 0:24 – an hotel with an infinite number of rooms will never be full – not even with infinitive numbers of guests – there will always be at least +1 free room as otherwise nobody could check in

  15. What if a infinite amount of infinite amounts of infinite buses with infinite passengers comes?

    ( ͡° ͜ʖ ͡°)
    If you figured that out,
    An infinite amount of infinite amounts of infinite amounts of infinite buses with infinite passengers

  16. Here's a puzzle for everyone. Count the number of times he says "Infinite" and "Infinity". The person who answers correctly will get to stay in the infinite hotel! All the best!

  17. The whole paradox is putting a max capacity to the term infinity.
    You can't but its human nature to as we utilize the term 'infinite' to simplify it in the first place.

  18. Funny how you didn't mention Raúl the builder of the infinite rooms who was born with a curse of immortality henceforth why the rooms are infinite because he can keep on building when the occupants keep on coming 😎

  19. Why can't you just assign new guests to new rooms? Infinite rooms would mean you don't need to move the guests, right? If you can move rooms for old guests, why not put new guests in those rooms? Plus, if they wake me up at 5:00 (Cause ya know I'mma be awake at 3:00) I'm gonna get mad. Am I just over thinking this? My brain hurts, someone help!

  20. I'm thinking instead of asking them to move, why don't just put the new customers to the new rooms. Since apparently, hotel being full doesnt mean that there aren't any rooms left since there is an infinite amount of rooms

  21. Oh man, even the video has so many colourful and intelligible illustration , math is still the most
    hard-understanding subject

  22. I don't understand! how can an infinite number of rooms ever be full??? I didn't think one infinity could match/exceed another

  23. if it happens what he described in sequence,then room no. 1 is empty forever.i think the night manager stays in room no.1

  24. If everyone moves up one room number, why not just skip that part of the process and send the new guests to first available rooms? Same concept

  25. but if there is an infinite amount of infinite buses that have an infinite amount of infinite passengers, wouldn't that mean, that the first infinite bus of infinite passengers doesn't end? So, how would you move from the first infinite bus to the second infinite bus?

  26. Khách sạn vô tận phòng thì sẽ không thể có khái niệm FULL. Bài toán đã bị sai ngay từ ban đầu. Không bao giờ đổ đầy được một chiếc bình có đáy vô tận.

  27. "Excuse me sir, we're going to have to ask you to move from room number 24 to room number… 27,571,377,035,722,319,940,389,970,188,357,803. Thank you for your cooperation."

  28. If a bus is infinite and there are multiple buses. More buses are useless because he only needs one.

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