The Infinite Hotel, also known as Hilbert’s Grand Hotel, is a thought experiment conceived by mathematician David Hilbert in 1924. It is used to understand the unintuitive effects when working with infinite sets.
The crux of the thought experiment is this:
Imagine that you have a hotel with infinite rooms that are filled with infinite guests, exactly one guest per room.
A new guest arrives at the hotel, but every room at the hotel is occupied.
However, you the new guest can still be accommodated if you have the guest in Room #1 (Guest #1) move to Room #2, displacing the guest in Room #2 (Guest #2) , who goes to Room #3. Guest #3 then goes to Room #4, Guest #4 to Room #5 and so on, each guest simply moving to the next numbered room.
Suddenly, Room #1 is empty and ready for the new guest to fill.
Just then, a bus full of infinite passengers arrives all at once, and needs to be accommodated right away. You could try to move them in as before, but no matter how many times you have the first passenger of the bus move into Room #1, there is still an infinite number of passengers disembarking the bus.
To solve this conundrum, you can have the guest in Room #1 move to Room #2 as before, but then move Guest #2 to Room #4. Guest #3 can then take Room #6, and Guest #4 who had been displaced by Guest #2 then goes to Room #8, displacing that guest, and so on. Instead of each guest going to Room N+1, each guest instead goes to Room N*2, leaving an infinite number of odd-numbered rooms for the bus passengers to move into.
Similar methods can be used to accommodate infinite ferries carrying infinite busses of infinite people, or any various other hypothetical combinations of vehicles.
So next time you go to a hotel, be sure to tell them that you’re not going to switch rooms into the N+1th or N*2th room, regardless of how mathematically sound it would be.
Have a great Night Thread!