Question about hilberts hotel.

[cragar]

So if I have an infinite amount of rooms and an infintie amount of people can I pack the hotel.
And both quantities are countable. Ok so I could just put all the people in the odd numbered rooms and have the even rooms open. But it seems like I could pack the hotel if I wanted too.
Can I constucrt both situations if I want too.

 

[DonAntonio]

So if I have an infinite amount of rooms and an infintie amount of people can I pack the hotel.
And both quantities are countable. Ok so I could just put all the people in the odd numbered rooms and have the even rooms open. But it seems like I could pack the hotel if I wanted too.
Can I constucrt both situations if I want too.

Yes...and after packed, you can still free an infinite (countable) number of rooms to host an infinite number of new guests.

DonAntonio

 

[cragar]

so you are saying I can have both situations if I want to.

 

[Mark44]

"Packed" for Hilbert's Hotel doesn't have the same meaning it would for a normal hotel. Even if all the rooms are assigned, there is still room for another countably infinite number of guests. So in some sense, it is never full.

 

[HallsofIvy]

What do you mean by "packed"? If you mean "every room is occupied", yes, you can do that. If you mean you "cannot add another person", no you cannot do that.
For a "hotel with countably infinite rooms", "every room is occupied" and "cannot add another person" are NOT the same.

 

[cragar]

so I can pack the hotel, but I can still add people. Thanks for your responses by the way.