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.

100 thoughts on “The Infinite Hotel Paradox – Jeff Dekofsky”

  1. Honestly lost me at moving occupant. If someone tries to move me out of my hotel room the only place is to a different hotel.

  2. By reading the comments, I agree that the finite minds of us humans can not wrap our minds around the infinite concept.

    But then again, it's us humans who came up with this in the first place.

  3. Its not a good explenation because it gives you a feeling that there is some logic in infinity. Infinity is 0 (nothing) and infinity is everything, every single number,sitauation, memory etc

  4. "Never any vacancy, but always room for more"………. As much as this video tries to be clever, it makes no sense. Firstly, to claim that the hotel is full at the beginning implies a finite amount of space in the hotel. You cannot be full if you have a countably infinite amount of rooms. You can never run out of rooms so therefore you can never be full. Even more absurd and pointless, if you have rooms to slide the guests already staying there into to accommodate new guests, then just put the new guests in those higher-up rooms instead. What is the point of moving the original guests into rooms that you can just put the new guests into, for the sake of sounding clever? This video just seems to try to solve a problem that doesn't exist.

  5. What if someone in the hotel doesn’t understand that math? What if someone goes to the wrong room? Wouldn’t that cause problems?

  6. So this is a very poorly explained paradox, mostly because how you portrary to problem is applying infinite to apply to finite tasks in order to get to your paradox; when there are numerous paradoxes already present in your attempt to explain the situation (Which I guess is the point of this, but it comes across as less of a paradoxical riddle and more of gibberish.)

    For one, we are talking about a hotel with infinite rooms, and the first obvious error is that this can be full. By the definition of infinity, this doesn't make any sense. Sure you could argue that for every room there is an occupant, but then you can't move these people as that breaks the analogy.
    Furthermore, there is the concept of time.
    Because for each person of the infinite people, they have to get to their rooms individually. It would never end, because there is an infinite number of people checking in. Adding more to this pile comes the continual subsets of infinity of new infinite buses, where you couldn't even finish one infinite bus because an infinite bus cannot end it's arrival of infinite people, and infinte check ins. At best, in order for this system to work, it would have to be applied as a continuous, exponential system, but then by definition that system would never end, you'd never get a full hotel, and you'd never stop having people check in.

    Forget the negative numbers and fractions, the entire premise of an infinite hotel with numerous subsets of infinite people itself cannot be resolved.

  7. When does infinity end? It has to end eventually. It can't go on FOREVER. Infinity must end. It reminds me of sleep. We never remember falling asleep, even if we try to wait. As well as how do we sleep 10 hours when it seems like 1 minute?

  8. Easier way for infinite buses:
    Existing: r -> 2^r
    First bus: s -> 2^s*3
    Second bus: s -> 2^s*3*3
    Third bus: s-> 2^s*3*3*3
    And so on!
    And also, it only puts people in 3-smooth rooms! Those are very smooth, I've heard.

  9. Worker: Sir, you have to go to room 1,605,546,491,106,999,545,694,136,742,623,645,786,434,205,316,1405,000,000,000,000.

    Me: say sike right now

  10. Imagine if the elevator broke, will the people on it from the Infinite floor be fall and stuck forever in that elevator ?

  11. When your room number is 84700000000 and the sign near the elevator says "Elevator under repair, please use the stairs..
    Sorry for inconvinience"

  12. If hotel have infinite number of rooms and we have infinite number of people for all rooms it means infinite = infinite so all infinite people have infinite rooms. So problem is.

    What is my problem all persons are comfortable manager have infinite money so what am I doing here so what is the aim of this video now I am very very confused. Oooh

  13. "A hotel with an INFINITE number of rooms" would never run out of vacancies, period. The rest of it, seems like an unnecessary, chaotic charade.

  14. Imagined riding an elevator to room 183838281828384858827191003748382
    That would take years to get to their room and more years to leave the hotel
    That hotel would have killed tons of people in the elevator

  15. But it is wrong…
    If I have infinite candies and infinite people eat one at each the jar of candies is now empty!
    In my opinion doesn't exist a BIGGER infinite and a smaller infinite! All infinites are equal!
    If I do:
    infinite×infinite=infinite<2
    It no have sense because in this case infinite <2=infinite:because it is the same do 0×0! It is the same!
    When an infinite is empty anyone number can full it but an other infinite can do it!

  16. Well… The next person who comes to stay in the hotel could just go to the next room since its infinite. No one would need to move to another room

  17. Hey mom dont worry my hotel number is room number 100000029392928392938383920384897888990992846577929388888273 so you coming over christmas?

  18. these paradoxes come from a quantitative thinking, its humans nature to measure things but a that human forgots that universe isnt for measuring and understanding it can be achieved better in a qualitative thinking proccess

  19. The moment you said infinite is full. You spoilt everything. Infinite cant be full. Full is a term used for only finite number of items. So all this logic doesnt make sense

  20. Infinity is never full so of there's an.infinite number of rooms then the hotel can't be full. You could just keep everyone in their rooms and put the new people in the infinite rooms that aren't full

  21. I would hate to stay here, I gotta move from room 7 to room 928574939014024292997 cause somebody had to pull up in an infinitely large bus smh

  22. So its impossible for our finite minds to grasp something as big as infinity, yet its possible for those same finite minds to grasp the concept of creation? Or I guess its just better to shut off our brains and hang our hopes on a theory (just a theory) or two when this vast universe with all its might exists outside of its limitations? Kind of sad to see people think that way. Why not try reading about how great some creations are, or how vast this universe is in Islam prophet's hadiths (NOT as evidence or proof, but merely to get a peek about what may be out there) and start seriously looking for the true religion in midst of all the false ones? Its not like death waits for anyone's permission. Nor is it like all these planets with all of their contents appeared by "coincidence".

    Judaism was going good and all until they deviated, christianity after until they did same. Then Islam came, here to stay until the end of time with God himself looking after it to keep it from deviating like the previous two (not to mention their prophecies of Islam's coming). Nice summary eh?

    You're still breathing, you still got a chance
    You got a brain, use it. Question, read, seek and learn
    Wake up already, enough messing around
    Time's ticking, the second that passes never comes back
    Stop letting others think in your shoes like some cattle,
    have some alone time and seriously think about all of this
    Death is but once, no second chances after it occurs.

  23. You won't need an infinite bedrooms because theres only a few billion persons living in this planet.
    If there is an infinite number of rooms then the person that just came in can just take the next consecutive number. Since numbers are infinite. No need for the guest to transfer to different rooms.😂

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