Olutions to Hilbert Hotel's Infinite Room Challenge

[Shing]

Homework Statement

There are infinite rooms in Hilbert Hotel, room
number is natural number 0, 1, 2,

Story: AhQ comes into Hilbert Hotel, but
the waiter Kong Yiji tells him that all rooms
are booked up. AhQ is disappointed. If you
were the waiter, what would you do?

2. If there are n persons q1, q2,
· · · , qn, what
will you do?

3. If there are infinite persons q1, q2,
· · · , qn, · · · ,
is there any solution?


**Basically, I understood 1 and 2, but I am confused about #3.
Please help me about #3! thanks!

Homework Equations

N.A.

The Attempt at a Solution

1.The persons in the i-th to (i+n)th rooms moves into the
(i + 1)-th to (i+n+1)th room as follows:
0123
q1q2,q3...,qn,0,1,2,3...

2.h to (i+n)th rooms moves into the
(i + 1)-th to (i+n+1)th room as follows:
0123
q1q2,q3...,qn,0,1,2,3...

 

[cristo]

For your first solution, I would say move each person "up" a room; i.e. the ith room moves to the i+1 th room. For the second, move each person up n rooms, so the person in the ith room moves to the i+n th room-- we can do this since n is finite. (I think this may have been what you meant in your solution)

For the third you need to find a way to free infinitely many rooms.

[Hint: how many odd natural numbers are there? how many even natural numbers are there?]

 

[Shing]

Basically, I met this question when I was trying to self-study set theory. So I really don't know about that...btw I am still in high school.

So could you tell me more please?

 

[cristo]

Well, for the third problem, you want to free up infinitely many rooms. One possible way of doing this is to move each person in the hotel to the room with twice the number of the current room-- so the nth person will move to the 2nth room. Thus the even numbers, 2,4,6,... are now taken. Then allocate each new customer an odd room (1,3,5,...) of which there are infinitely many.

 

[Shing]

So if there are two times infinite persons q1, q2...
I just need to free up two times infinitely many rooms,
like that: n1->n3, n2->n6, n3->n9
right?

 

[cristo]

No, infinity doesn't work like that, as infinity is not a number. Think of infinity as being "larger than the largest number." Therefore it makes no sense to talk about "two times infinity" since this would just be infinity.

I hope that makes sense-- perhaps someone else could come and give a better explanation!

 

[turdferguson]

Along the same lines, its important to remember that there are just as many prime numbers and integers as there are natural numbers. Intuitively, youd think there would be more natural numbers, but the cardinality of all 3 sets is aleph0