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.

4:56 it's Kim Jong Hyun

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?

Hilbert: you know what, what if there was a hotel..

Wife: Oh No, not this again?!

I infinitely dont understand this. Infinite…

Why can’t the new guest go into the empty rooms instead of making people move every 5 mins?

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.

If the room number 1 went to 2 then so on, couldn’t the guest go to the room that the highest person went to?

The maid is super pissed at the hotel manager lol

This was kind of a pointless watch

Night manager : Ma’am, I need you to move to room 749749

Me: Say sike rn

Infinity hotel Full? BS!

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

HOTEL? – TRIVAGO

INFINITE HOTEL? – TRIVAGO

trivago does the work for mathematics.

And how many lifts are there?

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

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.

that moment when an infinite number of guests are staying there but room 6 is empty

It look all like an infinite circle.. No idea from it

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

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 .

So how was the hotel built?

But if there are an infinite amount of buses, how on earth did he actually sort all of them?

Imagine a guy living in Room Number -91 million… Cause I cant.

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

Whats the electricity billl

my brain hurts

If the hotel is infinite, it can't be full, and the guests can just stay in the infinite available rooms.

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???

My head expolded

Infinight holet : full

Population : infinight

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

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?

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

Welcome to the Hotel Cortez! 😂😂😂

This is not a paradox. This is just a math riddle

a hotel with infinite rooms, what would be the profit of the hotel? lol

stonks

The hotel is always full

Yeah, this didn't change anything.

Infinite hotel cannot be full?

Hotel : Trivago !!

Was over it when he said "countably infinite." Hold up, what. Haha

The true mystery of this video is…

How fast are the HECKIN elevators

Why can't he just assign to the next vacant room tho?

This was so confusing

شن الوهم هو

4:55 BIG BRAIN TIME

That doesnt make sense…. hotel w infinite rooms cant be full

That’s a lot of moving

some one wants to leave at floor 1 thousand any ideas how to help that person? and he goes 1 step a day

just make a 2×2 room and your good to go.

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)

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

Working at that place must be even worse then heck

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

Can’t you just make him go in the room all the way in the end (room infinity)

we get it the manager is making alot of money

let's get 10 million 👉SUBSCRIBERS 👈

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!

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.

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 😎

Türkçe altyazı nerede amk

But why move guests to new rooms instead of assigning new guests to unoccupied rooms?

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!

I’d hate to be in a fire while staying here.

4:56 oh yeah it’s big brain time

The next season of Infinity Train seems interesting

that manager needs to be fired

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

Oh man, even the video has so many colourful and intelligible illustration , math is still the most

hard-understanding subject

Ma'am Here's your space suit, Please move to room 18 vigintillion, Thank You

Uhhhhhh…..

What did I just see

Is this even real 👀

a lot of things in maths you just have to accept

how could a infinite hotel be full

His voice is so boring it feels like a robot

Man it’s creepy watching this at 1am

한국어는 번역을 하다 말어

1:19 how does someone make such a big bus?

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

Someone grab the manager a nobel prize or something

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

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

Hmmm why isn’t this video infinite like the 24/7 lofi music channels🤔

he is giving keys to the infinite costumers

Everyone gangster till the Elevator stops workinghi sir sorry to trouble you but i have moved your room to room 3.141…

sir : oh FU-

The maid would get like 1000 dollars

I feel bad for that ONE maid clening by herself.. oof

Infinite hotel? I would be too lazy to ride the elevator.

Oh my 🥧 !

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?

2:42

Did he say….

Euclid?

CONTAINMENT NOISE INTENSIFYS

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.

Excellent; what did he say?🙄

"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."

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

How do you even sleep