A hotel has an infinite number of rooms, which are all fully booked, but that's no problem as the guests are flexible. They are accustomed to being moved to another room, up to once a day, to make more space, as long as they are clearly told where to go.
On the first day, a thousand new people show up and need rooms. This is easy for the manager, everyone is given a new room by moving from room n to room n + 1,000, which vacates rooms 1 .. 1,000 for the new arrivals.
On the second day, an infinite number of new people arrive. How are they given rooms?
On the third day, an infinite number of buses arrive, each with an infinite number of people. How does everyone now get a room?
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Originally posted by talzamirNo idea... ask Hilbert, I'm sure he knows.
A hotel has an infinite number of rooms, which are all fully booked, but that's no problem as the guests are flexible. They are accustomed to being moved to another room, up to once a day, to make more space, as long as they are clearly told where to go.
On the first day, a thousand new people show up and need rooms. This is easy for the manager, everyon ...[text shortened]... mber of buses arrive, each with an infinite number of people. How does everyone now get a room?
Richard
If infinity and hotels together feel silly, this could of course be mathified, e.g.
Is it possible to pair up with positive integers...
a. all natural numbers?
b. all integers?
c. all rational numbers?
d. all real numbers?
where cases a..c are just about the same as the cases of adding a finite number of new guests, an infinite number of new quests, and infinity squared number of new guests. Correct answers given already by readers on those cases.
Originally posted by talzamira) yes
If infinity and hotels together feel silly, this could of course be mathified, e.g.
Is it possible to pair up with positive integers...
a. all natural numbers?
b. all integers?
c. all rational numbers?
d. all real numbers?
where cases a..c are just about the same as the cases of adding a finite number of new guests, an infinite number of new ques ...[text shortened]... infinity squared number of new guests. Correct answers given already by readers on those cases.
b) yes
c) yes
d) no
Therefore, case a,b,c are all adding the same number of guests 🙂
Originally posted by talzamirOn the third day the buses are numbered as they arrive and the occupants go to the floor indicated by their bus number.
A hotel has an infinite number of rooms, which are all fully booked, but that's no problem as the guests are flexible. They are accustomed to being moved to another room, up to once a day, to make more space, as long as they are clearly told where to go.
On the first day, a thousand new people show up and need rooms. This is easy for the manager, everyon ...[text shortened]... mber of buses arrive, each with an infinite number of people. How does everyone now get a room?
Each floor has an infinite number of rooms to accomodate the infinite number of passengers on each bus.
EDIT: The current guests all remain on the ground floor.
Originally posted by PalynkaJust because you take take infinitely many guests and map a room to each guest doesn't mean you can fill a hotel with infinitely many rooms. There will always be more rooms. That's the definition of infinite.
It's a countably infinite number, so bijection says otherwise.
And...here...we...go!
You put 2 numbered balls in a bag and...
Originally posted by forkedknightIf there is a bijection, then there are the same amount of rooms as guests. By definition.
Just because you take take infinitely many guests and map a room to each guest doesn't mean you can fill a hotel with infinitely many rooms. There will always be more rooms. That's the definition of infinite.
Originally posted by forkedknightA finite number of elements can always be added, but you need to change the mapping else there would be no available rooms.
And yet you can add 1000 more guests, and they all still have rooms.
Using the concept of infinity as though it is a number doesn't make any sense.
What doesn't make any sense is your blatant disregard for set theory and cardinality. If two sets are linked by a bijection, they have the same cardinality, and sets with the same cardinality have the same number of elements.