Question about hilberts hotel.

  • Thread starter cragar
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In summary, the conversation discusses the concept of Hilbert's Hotel where an infinite number of rooms and people can be accommodated. It is possible to pack the hotel by assigning all rooms, but there is still room for an infinite number of new guests. The term "packed" has a different meaning in this context as compared to a normal hotel.
  • #1
cragar
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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.
 
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  • #2
cragar said:
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
 
  • #3
so you are saying I can have both situations if I want to.
 
  • #4
"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.
 
  • #5
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.
 
  • #6
so I can pack the hotel, but I can still add people. Thanks for your responses by the way.
 

Related to Question about hilberts hotel.

1.

What is Hilbert's Hotel and why is it significant in mathematics?

Hilbert's Hotel is a thought experiment created by mathematician David Hilbert to illustrate the concept of infinity. In this hypothetical hotel, there are an infinite number of rooms, each occupied by a guest. However, if a new guest arrives, the hotel manager can still accommodate them by moving each existing guest to the next room number, effectively creating an infinite number of vacancies. This paradoxical scenario helps to demonstrate the counterintuitive nature of infinity in mathematics.

2.

How does Hilbert's Hotel relate to the concept of infinity?

Hilbert's Hotel demonstrates the concept of infinity by showing that even if there are an infinite number of guests, there can still be room for more. This challenges our understanding of infinity, as we often think of it as a fixed and unchangeable quantity. However, in mathematics, infinity is a concept that can be manipulated and understood in different ways, as demonstrated by Hilbert's Hotel.

3.

What is the paradox of Hilbert's Hotel?

The paradox of Hilbert's Hotel is that it can accommodate an infinite number of guests, even though it appears to be full. This is because infinity in mathematics is not a finite quantity, but rather a concept that allows for manipulation and expansion.

4.

How does Hilbert's Hotel challenge our understanding of the physical world?

Hilbert's Hotel challenges our understanding of the physical world by demonstrating that our intuition and everyday experiences may not apply to abstract mathematical concepts, such as infinity. In the real world, a hotel with an infinite number of rooms is impossible, as it would require an infinite amount of physical space. However, in mathematics, this concept can exist and be explored.

5.

What are the real-world applications of Hilbert's Hotel?

While Hilbert's Hotel is a thought experiment, it has real-world applications in fields such as physics and computer science. The concept of infinity is used in calculus to solve problems of motion and change, and in computer science, the idea of infinity is essential for understanding and developing algorithms and data structures. Additionally, the paradox of Hilbert's Hotel can help us think critically and creatively when approaching complex problems in various fields.

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