Basic Set Theory: Can You Accommodate Countable Guests?

[saadsarfraz]

I found this question in a book.

Q-Suppose you own a hotel with a countable number of rooms. One night a
traveler wishes to stay in your hotel, but all the rooms are occupied. Can
you give him a room without kicking anybody out of the hotel? What if
a tour bus shows up with countably many passengers, all wanting a room?
(Assume each room only accommodates one person.)


I would assume the answer is no since the hotel is completely occupied but something tells me that's not right.

 

[John Creighto]

I found this question in a book.

Q-Suppose you own a hotel with a countable number of rooms. One night a
traveler wishes to stay in your hotel, but all the rooms are occupied. Can
you give him a room without kicking anybody out of the hotel? What if
a tour bus shows up with countably many passengers, all wanting a room?
(Assume each room only accommodates one person.)


I would assume the answer is no since the hotel is completely occupied but something tells me that's not right.

Say you indexed the rooms with the natural numbers. Simply give room n+1 to the person who is in room n. Then room one becomes free.

 

[saadsarfraz]

but it says all the rooms are occupied? giving room n+1 to the nth person would mean that one room was already empty isn't it?

 

[bpet]

In a similar paradox, there is a comic book where every 4 or 5 issues they reprint an old issue - if they continue indefinitely will every issue be reprinted?

Most people say no, because of the increasing gap between the number of new issues and reprints.

 

[HallsofIvy]

but it says all the rooms are occupied? giving room n+1 to the nth person would mean that one room was already empty isn't it?

Which room would that be? You can move the person in room 1 to room 2, leaving room 1 empty. And room 2 is open because the person in room 2 has moved to room 3. Which is itself open since the person in room 3 has moved to room 4, etc.
IF there were only a finite number of rooms that would eventually terminate- but there are an infinite number of rooms.

In fact, when that bus with countably many passengers shows up, we can create room for all of them by moving the person in room n to room 2n, leaving all the odd numbered rooms empty.

 

[Tobias Funke]

I would assume the answer is no since the hotel is completely occupied but something tells me that's not right.

That's because you're using your intuition for an object that doesn't exist. This is a math problem where phrasing it in terms of a hotel may help people visualize what's going on easier. On the other hand, real world, common sense notions may not make sense for an impossible object.