Checking in to The Hilbert Hotel

[Chuck]

Everyone is familiar with The Hilbert Hotel, especially since I linked to it, in which room can always be found for more guests, even an infinity of them, and even if every room is already occupied.

But can every room really become occupied? Let's say there are an infinity of guests waiting to check in to the newly built hotel and the front desk can handle one per minute. After an hour there will be 60 guests in rooms and an infinity of people still waiting. After a week there will be 10,080 guests in rooms and an infinity of people still waiting. After a year there will be 525,948 guests in rooms and an infinity of people still waiting. After a millennium there will be 525,948,766 guests in rooms and an infinity of people still waiting. It seems that there will only ever be a finite number of guests in rooms and there will always be an infinity of people still waiting.

But are any of the waiting people dissatisfied? Let's say they're immortal and, living in an infinite population, they're used to waiting in long lines. A person who will eventually get a room, even after a long wait, is satisfied. Anyone who will never get a room is dissatisfied.

Are there any dissatisfied people waiting in line?

[Quailman]

[Scurra]

[L'lanmal]

Upon arriving at the Hilbert Hotel, those that noticed the infinite "take a number" display and grabbed one will be satisfied. Those that were less attentive may be out of luck.

[Chuck]

That's sort of what I was thinking. If the customers take a number or stand in an orderly line then each one of them will eventually get a room even though they'll never all get in.

[Perpentach]

I always got the impression that there were two types of guests in the Hilbert Hotel. The first type have been in the hotel forever. They are the ones who move around when space has to be made for the second type.
The second type are the visitors listed in the problems the Hilbert Hotel is used to explain. The clerk/clerks/whatever finds rooms for the second type of guest, who become the first type once they enter their room. Therefore you don't get a problem with infinite lines to fill up all the rooms, because all of the rooms were filled when the hotel was built.

[Lepton*]

Guests at the Hilbert Hotel, used to discomfort as they are, might be employed to help with the backlog. Consider the following system, which allows guests a 50% discount on their room if they volunteer a bit of time to help run things:

The process of checking in a customer is highly synergistic. In other words, two clerks will be able to check in a twice as many people, twice as quickly as one clerk. Division of labor, right?

During the first 20 minutes, the clerk checks in the first customer. During the next 10 minutes, with the help of the first customer, the clerk is able to check in two more people. Next, the clerk and the first four customers can check in the next four customers in 5 minutes. These 8 check in 8 in 2.5 minutes. And so forth.

So, by continually halving the time, we get closer and closer to 40 minutes, and we process more and more people. Informally, when we've gotten to 40 minutes, an infinite number of people have checked in and they can all go swim in the pool.

Next Challenge: The pool, which occupies the basement of the hotel, is both infinitely long and infinitely wide. Is there enough space in the pool for everyone to take a dip at the same time? Next, everyone decides to play a game, and they all hold hands, make a very long line: do they fill up the pool now?

[Perpentach]

So the first problem is the Cantor Diagonal Argument. The answer is yes. Infinity times Infinity is equal to Infinity. I personally heard that problem as an infinite number of buses with an infinite number of people on each one.
_________________
All questions must be stated in the form of an answer.

[itisally*]

I have always wanted to argue that it is possible to have a "larger" infnity.
There are an infinate number of integers, yet just between 0 and 1 there are an infinate number of rational numbers so therefore there must be more rational numbers than intergers, right?

I know it isn't true because of the definition of infinity. This is where logic and mathmatical terminology diverge.

[jesternl]

There is a 'larger' infinity, it's called Aleph 1.

On different kinds of infinity

[Chuck]

[itisally]

Jesternl,

That is the best thing ever! I have to tell you I feel vindicated in so many ways. I will have to send the link to my old college buddies.

As for the challenge, I wonder if you would concider the pool to be always full as infinity = infinity

infinite space with infinite people would be full no matter how they are arranged, but its ok we can always add more.
_________________
I could agree with you, but then we would both be wrong.

[Perpentach]

Technically with Aleph-0 the pool is full and nobody is left out. With Aleph-1 the pool is full and there are still people outside of it. An example of Aleph-1 is the number of decimal expansions, both rational and irrational.
_________________
All questions must be stated in the form of an answer.

[itisally]

ah yes, I see
_________________
I could agree with you, but then we would both be wrong.

[Jack_Ian]

Shirley ¹ the pool would only need to be infinitely wide and 50m long (after all there are bound to be an infinite number of guests training for the Olympics).
But wait! If 0.001% of guests are training for the Olympics, then we need an infinite number of lanes for them. What about the other guests?
Ahh! That's no problem, just make sure everyone gets in the lane with the same number as their room.
Huh? My head hurts.
Well That's not surprising. You only have a finite number of brain-cells.

^{1. Just watched Airplane again last night. Still funny. It was in a collection of 80's movies for €20, couldn't resist. Ferris Bueller's Day Off is next.}

[jesternl]

Airplane, also known as Flying High (which IMHO is a much better title)
(off to sniff some glue)

[BraveHat]

Are there landing platforms the guests are lowered onto to walk down steps into the pool or does everyone dive in?
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax