# 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. Swapnilganesh Pathak says:

The concept of infinity should under
Infinity not an number as so many
It's Sunya = infinity

But the bus was infinite so it never left at all so every kept moving forever and forever and ever

3. Sandra Jones says:

He would never get past the infinite bus, because they would be checking in infinitely.

4. Saggitaire Auger says:

But infinite cannot be full, so the whole thing isn't even needed to be thought.

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.

6. Fazbear Gaming says:

Infintite rooms? Then Just give The New People Higher Number Rooms?

7. Cathy Jagson says:

Nope 1 gogolplex floors

8. Ivan Cazarez says:

Good luck triying to get out of the hotel

9. Jason Juneau says:

Plot Twist : Quit job immediately, and sleep in one of the buses

10. MALLIKARJUN PATIL says:

where is the paradox in this

11. Virus Alpha says:

12. Alexis Papageorgiou says:

This is amazing. I now find myself wishing the power of 25 enters your room

13. Shelley Jackson says:

Oh lordy lordy lord

14. Shelley Jackson says:

enters elevator in 2019
Person: yawns Its been 3 hours!
*reaches floor in the year 3567

15. satvik pardhan says:

A math teacher has got /infinite/ goosebumps

16. satvik pardhan says:

I''

17. OishikG says:

So that manager was a mathematician ?

18. BlackCent says:

Welp I officially hate math…

19. indigomage says:

Its an infinite hotel so it accomidate an infinite nuber of people . Its like x:x ratio .

20. Boris Gleichmann says:

I wonder how many Karens (entitled moms) are in the hotel

21. Wilson Kowira says:

Instead of moving the quest, why didnt you put the quest in the room thats not occupied?

22. Jasmine Angie says:

Yeah but what about the day manager?

23. Zoe Macdonald says:

Orrrrrr you could put them all in the next room. There are an infinite number of rooms, after all.

24. Chooong7 says:

This reminds me of how much I hated math 😂

25. Ardian Sallauka says:

This is probably the worst video on INTERNET!

26. Magical Mahnoor says:

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.

27. k2111 6 says:

RIP JEFF BEZOZ

28. Pani Beata says:

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

29. Exolette says:

At the end of this hotel… is the start… of me leaving this video cause of brain cell damage.

30. Alfred Naccarato says:

"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.

31. Matt Brown says:

This supports the cosmological argument

32. Niamh Hobson says:

i feel smart that i lowkey understood this

33. Andres Rodriguez says:

Because logistics doesn't matter

34. Juan Cabrera says:

I thought hotels didn't have a 13th floor though.

35. Juan Cabrera says:

Send them to imaginary rooms! 😆

36. EvanMc07 1 says:

Or you know put them in the extra rooms remember infinite is the same as infinite with infinity members

37. somazwolf says:

One night, the Infinite Hotel is full

Error
Success!

38. dank says:

Or you could just put the person at the end

39. Paulo 98 says:

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

40. Nodnarb says:

Fantastic

41. Alex Cav says:

Wait hold on hold on in the 1920s there was no bus 😐

42. rodrigo cosmo says:

AAAAAAAAAAAAAAAAAAAAH

43. acajudi100 says:

https://youtu.be/kdJ3YwbjldM

44. Shockwave herp says:

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.

45. Camryn Schnoor says:

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?

46. KUMA says:

Burn the infinite hotel, get infinite heat, cause infinite smoke, inhale infinite co2, infinite global warming and so on

47. Michael Darrow says:

infinity times infinity is infinity

48. Michael Darrow says:

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.

49. ananasie1211 says:

take a shot everytime he says infinite

50. Corue says:

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

51. Nirmala Sahu says:

why im watching this

52. Alex Kharatishvili says:

And in the elevator there would be only 1 person because of entering and leaving on each floor

53. jsusj nsjjs says:

"Hotel Del Luna"
In real life

54. I Want People to be Whooshed says:

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

55. I Want People to be Whooshed says:

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

56. Lunatic Ranger says:

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

57. Kose926 says:

Why not just make a room on the top of the hote- o h w a i t

58. Jingkai Lim says:

Wait,if there is an elevator,how many buttons are in it?

59. Gaming knight says:

That’s how many times he said infinity
👇🏻

60. Yoshua Ho says:

The nightmanager should be a mathematician

61. ben bundidsilp says:

give this nibba a raise oh wait infinity X 2 + infinity but cooler

62. magicmushroom says:

"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.

actually, there's one hotel named Infinite Hotel here
imma check it now

64. Axo Drago says:

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

65. aanya verma says:

just keep adding floors, cuz its infinte
duh

66. 1000 subs by only commenting challenge says:

I lost you after you said: ”Infinite”

67. Ceckinus 2017 says:

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!

68. THE ULTIMATE MEME says:

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

69. August Artist says:

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

70. Huy2007Chuck says:

BEG BRAIN!!

71. iloveflowers says:

wait what… ?

72. albionmeraj says:

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

73. Dr. 47 says:

This makes no sense

74. Loquacious Noggin says:

Does the night manager sleeps?

75. Leon Au says:

76. Tiffany Lee says:

I would hate to be the cleaner though

77. Dhylan Gamez says:

Hotel-TRIVAGO!

78. Khalifa Ashiru Danliman says:

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

79. Aminta Seay says:

Infinity times infinity equals infinity

YOLO

80. ipi ipi says:

a infinite amount of room being full make it no longer infinite.

81. Rohan Kodibagkar says:

The people on the infiniteth bus will have to wait forever to get off
… feel sorry for them …

82. oa aika says:

if I had to change rooms so much, i wouldn't stay here lol

83. Beast Master says:

what

84. tbhvivian says:

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

85. Alpha Banger says:

If there are infinite rooms the hotel should not have problems with infinite busses

86. Helen Calcagni says:

G

87. Pooch the dog says:

Why not just put the new guest to a new room in the first place and not waste your time moving everyone?

88. GOD소다 says:

DOES ANYONE NOTICE THE POOR MAID????????

89. Renai Support says:

Infiny but FULL
Full which mean a.. Failed of infinite Task

90. HotBurnBaby says:

Can't he just say: get the last room number. Ye the guests would be walking forever but whatever

91. iwanttodie says:

It's super easy wtf

92. John Corcoran says:

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

93. Matthieu BAILLY says:

So basically this video is saying infinity is infinite

94. kosayen2 says:

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.

95. renz says:

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.😂

96. Vinicius Lima says:

"Infinite Hotel is full"

Euclide: "Am I a joke to you?"

97. Polar Bear says:

technically, since we are dealing with infinity, he should be stuck at 1:17 forever

98. Thomas Pastor says:

I think you need to make an infinite number of videos explaining this video.

99. Regulus Altus says:

Room Pi, where the guests expects free dessert.
– Ahhh! Something i can understand

100. Yo Lo Man says:

0 |” |_| € |< \$ GIVEN