# The Infinite Hotel Paradox - Jeff Dekofsky

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**Published: 16 January 2014**- Sign up for our newsletter and never miss an animation: bit.ly/TEDEdNewsletter

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The Infinite Hotel, a thought experiment created by German mathematician David Hilbert, is a hotel with an infinite number of rooms. Easy to comprehend, right? Wrong. What if it's completely booked but one person wants to check in? What about 40? Or an infinitely full bus of people? Jeff Dekofsky solves these heady lodging issues using Hilbert's paradox.

Lesson by Jeff Dekofsky, animation by The Moving Company Animation Studio.

That's easy, infinity is a notion and not a number. You can't add, subtract, multiply, n+-... with it. Point.

3 is a prime number you skipped that .

All I can hear in this video is

"Infinite...infinite...infinite..."

Think of all infinite loads in that hotel. And i dont mean laundry

what

I can’t get pass :60 of this video. If the hotel is “full” (infinity) how is there a room to move the guest over to the next room? And if the hotel has a room to move the guest into why not just put the new guest in the “last” room? I’m sooo confused.

Alright alright keep your secrets!

The real question is who cares bruh

*HotelHell Season ♾ Episode ♾*

sI have so many question so little understanding

Wait, what?

all i understand is 'infinite'

This made me think of that last time I tripped out on LSD ;)

1:27 seems legit.

This is honestly the stupidest paradox theory ever thought of.

dude, why didnt he just do the even/odd separation process again?

Solution is simple there is no such thing as infinite

how can he finish the first bus if there’s an infinite amount of people in the bus

I can't handle so many handle's 😕

Okay , so the last line was scarier than a horror movie

A hotel with an infinite number of rooms would occupy all the space in its universe. The buses couldn’t come from anywhere outside the hotel as there would be no space where the hotel doesn’t exist.

In fact the hotel *is* the universe.

If there was infinite rooms then why did the people have to move rooms every time someone else came, just give the new people new rooms, no one has to move, problem solved

If the hotel has infinite rooms available, how can it be full? 🤔

There's no escape.

Were there infinite rooms

Mean is it equivalent to the wrld population?

ok ted ed i have a question for you

I GIVE YOU INFINITY FENCE POSTS

USE ALL THE POSTS

how do you do it?

When I scrolled down I expected the videos in the middle to keep going for some reason, infinitely. I was surprised when I saw the comments 😂 I guess the concept of infinity got to meh head

Why the heck you even need to move the current guests?!?! Dafak. Just put the infinite incoming guests to the infinite unbooked rooms that your infinite hotel have. Don't mess with your current guests, because exactly THAT you will start losing some of them and loose money.

The 90% of the time of this video is waste, just needed to talk about the Aleph Zero.

Mind blown

So the moral of the story is to ask for Room 6?

0:23 0:23 0:23

Ted used confusion.

It's super effective!

The viewers are now confused and start questioning about their life.

Why doesn't he alot the first guy infinity+1 in first place

With brains like that why is the night manager a night manager 😂😂

God What is this madman doing

Somebody stop him or he be countin forever

Me:Dad where is the Terrance of this hotel

Dad: son your grandfather is on Terrance

Me : you said that he is in the heaven

Dad: what I said is the same as I said before

There is a fire in the hotel and the people say hey the elevators aren't working .

Me:let's jump out of the window

Me:3 2 1

Manager sir you have to move to the room no. 57 billion me

Me : throws the manager out and check what happens

*Wait, that’s illegal*

Why need to jump from 3 to 3 to the power of 7 or whatever. .why not just let him jump to room 4 , and 4 to 5 and 5 to 6 and so on.. 😡😡😡

The goal is to create a mapping that assigns each guest to a single specified room. The scenario here is that infinite guests are arriving. If we just move each guest up a room when a new one arrives, then each guest has to make that n+1 trip infinitely many times. So, after you've finished unloading the infinite bus, where is guest three? They're not in room four, because more than one guest has arrived. They're not in room ten, because more than seven guests have arrived.

In fact, for each room, guest three is not inside it. And, on top of that, this is true for literally every guest, even the ones that were inside the bus. Thus, you need a different mapping. Otherwise, instead of accommodating all the guests, you'll end up effectively accommodating none of them.

Why does this manager keep worrying about making room in a hotel with infinite rooms?

My mine exploted.

Manager: Sir u have to move from room 1 to infinity.

Me: hold my luggage🧳🚛🚚🚚🚚

you see good sir this paradox reveals another paradox within your paradox. if the people moved from n to n+1 you would be asking an "infinite" amount of people to move to n+1 one by one which would take until your day of death.

You don't necessarily have to ask them to move one by one though. Tell everyone to go at some specified time, and the entire thing will take a finite length of time.

I want my 6 minutes back

Me at 3am

I think I vaguely get it? Infinity+1 isn’t an actual number, so you can’t check someone into the next room because you don’t know what that room is. However, you can ensure that everyone gets a room through the processes explained in the video. By putting one new guest in the first room and forcing its current occupant into the next one, everyone will eventually have a room, but there will always be two people switching rooms at any given moment. The process of switching rooms lasts forever.

I think that’s it but I’m bad at explaining

The actual procedure of moving the guests is really irrelevant. Suppose all guests leave their room; now the hotel is empty. Suppose then every guest moves to a room one higher from where they came from; now each guest has a room, and each room except the first has exactly one guest. The first room is empty and can be occupied by a new guest. (Mathematically speaking, the set of all natural numbers can be mapped one-to-one with its strict superset or subset. And in fact, a set can be mapped one-to-one with its strict superset or subset, if and only if it has a countably infinite subset. Assuming axiom of choice, this is true for all infinite sets.)

Manager? Good

Rooms? Infinite

Hotel? Trivago

every time he says "infinite", i swear this video goes 2.0x faster

You don't need to know any math to become hotel manager they said. You just need to finish the studies of hotel management they said