Math question on Hilbert's Hotel...

[mjr]

Question regarding Hilbert's Hotel.

https://en.wikipedia.org/wiki/Hilber...he_Grand_Hotel

I get the general idea of what it's talking about, but I don't understand why you'd have to move guests.

So if you have an infinite hotel, with an infinite number of guests (i.e. the hotel is "fully occupied"), and another guest (or guests) shows up, why do you have to move the infinite number of guests to different rooms? Why not just put the new guests in "new" rooms?

The only thing I can think of is that it has to do with moving X spaces. If one guest shows up, and they're put in the infinity-ith room, they'd never make it (putting aside how all the other guests got to their rooms to begin with).

So, what say you?

[Crai]

The thought process is there is no end to Infinity, so how do you decided what number the newly arrived guest would enter? Are they the last person in infinity? What room number would be infinity?

[mjr]

The thought process is there is no end to Infinity, so how do you decided what number the newly arrived guest would enter? Are they the last person in infinity? What room number would be infinity?

Hmmm...I hadn't considered it from that perspective. "Go into the infinity-ith room" doesn't make sense. "Everybody shift one room" makes sense in that respect.

[dalesys]

It's PHBs all the way down.


Oh. I thought you said "Dilbert's" hotel.

[wolfie]

SC interpretation of this math problem: "No matter how many people are already at a hotel, the desk clerk can always pull another room out of his ass, so there's no need to make a reservation even if there's a big event in town. If you're a walk-in and he tells you every place in town is booked solid, he's lying.".

[Crai]

Truth there Wolfie lol Truth!!!

[Geek King]

Hmmm...I hadn't considered it from that perspective. "Go into the infinity-ith room" doesn't make sense. "Everybody shift one room" makes sense in that respect.

However, it seems like you could also say, "Send the new person to the last unoccupied room", since a new person creates a new room, assuming the thought construct of infinite rooms and infinite guests.

Although a philosophical argument could also be applied, since true infinite guests would include everyone already, that there could be no new guests.

All I know for sure is that I bet the wi-fi signal sucks. Infinite guest use up all the bandwidth

[Nunavut Pants]

But if there were an infinite number of people who could decide to become guests...

The thought experiment is all about different types of infinity. Specifically, countable versus uncountable.

[mjr]

But if there were an infinite number of people who could decide to become guests...

The thought experiment is all about different types of infinity. Specifically, countable versus uncountable.

Yes, but when does a countable infinity become an uncountable one? When we reach beyond the Aleph numbers?

https://en.wikipedia.org/wiki/Aleph_number

And that still doesn't explain why everyone has to shift a room. It really shouldn't matter if you put the new guest in room #1, or the room "next to" infinity...unless I'm missing something.

[dalesys]

...And that still doesn't explain why everyone has to shift a room. It really shouldn't matter if you put the new guest in room #1, or the room "next to" infinity...unless I'm missing something.

My take is that we have no address for the last room beyond "Out There!", so the only definable algorasm is for the little one to say "Roll Over" and see who falls out.

[sstabeler]

It's more supposed to be a proof that infinity + anything = infinity- if you don't move the guests, nothing is actually proven. (basically, the guests moving to their new rooms means you can definitively prove the new guests' rooms exist.)