Hilbert's Hotel: new Guest arrives (Infinite number of Guests)

[pbuk]

Or Buzz Lightyear "To infinity and beyond!"

Anyways,
How did the hotel fill up in the first place..
There is a convention in town and new The Infinite Hotel is open for business.
Guests arrive and keep on arriving. In fact there is an infinite number of them.
As they arrive in the lobby they check in and are assigned guest / room 1,1 . 2,2 , .. m, n, ... , ...
The hotel management always sees an infinite number of guests in line, and they always have a room available.

Well this is a different question - in the OP you start with the hotel full and have to squeeze in one more guest.

But it's not worth starting a new thread to answer your side question - the hotel can easily fill up if the first guest takes 1 minute to check in, the second guest 30 seconds, then 15, 7.5 etc. After 2 minutes there are no more guests in the queue and every room is occupied.

 

[srfriggen]

Interesting physics question here: The communication of the order to move is of finite speed. Whereas this doesn't seem to be a problem for the first billion rooms, will it work out at infinity?

That's a long game of "telephone." The billionth guest would hear, "Purple baby monkey uncle" and have no idea what to do.

 

[Vanadium 50]

Really, the guests are not numbers

I am not a number! I am a free man!

I like the version: As can easily be seen ...

I like "One can show". One did. His name was probably Gauss.

 

[fresh_42]

I like "One can show". One did. His name was probably Gauss.

Or Euler. I wonder whether it was Fermat who started to write like this.

 

[Buzz Bloom]

I think the OP has a point related to the distinction between a math problem and a real world problem. There is no constraint that axioms related to a math problem must reflect reality. If we make an assumption that there are an infinite number of rooms, say all in a line with a shared corridor, how long does it take for the message to be passed to all of the rooms that each occupant has to move to the room next door? How can it be done in less than an infinite amount of time?

 

[phinds]

I think the OP has a point related to the distinction between a math problem and a real world problem.

I disagree. It seems to me that the OP is trying to force a math problem to look like a real world problem and using that false comparison to confuse himself about what really is just a math problem. His mistake is understandable since it is, unfortunately, POSED as a real-world problem but only a beginner would try to interpret it as actually BEING one.

 

[gmax137]

I think 'the Hilbert Hotel' is a teaching tool to help you understand what "infinity" is about; what it means to have a set with "an infinite number" of members.

Is it weird? yes, but that's the point.

 

[bob012345]

Or Euler. I wonder whether it was Fermat who started to write like this.

All I can say is that it's uncertain if Heisenberg ever checked into the Hilbert Hotel. But Schrödinger was probably spread over several rooms. But all this is getting rather Bohring.

 

[phinds]

All I can say is that it's uncertain if Heisenberg ever checked into the Hilbert Hotel. But Schrödinger was probably spread over several rooms. But all this is getting rather Bohring.

Yes, even aside from the pun, it is, and based on the fact that the whole conversation started based on a misunderstanding by the OP, it seems to me it should be closed, as is normally done here on PF with such threads.

 

[fresh_42]

Yes, even aside from the pun, it is, and based on the fact that the whole conversation started based on a misunderstanding by the OP, it seems to me it should be closed, as is normally done here on PF with such threads.

Guess this is a good idea.

Hilbert's hotel is a heuristic, no mathematical construction.