Hilbert's paradox of the Grand Hotel is a really amusing one. The extensions are also amazing. But first, let us look up what Hilbert's paradox actually is...
One can see Hilbert's paradox as an attempt to define infinity or to make someone realize how infinity can be thought of. You see, infinity is termed as 'not defined' and hence, defining it can be taken as a paradox in itself.
Consider a Hotel with infinite number of rooms. Then, infinite number of guests can be accommodated Now, since there is no definition of infinity, we can't see how we will accommodate infinite guests in infinite rooms. So think of it like this. There are right now 'n' rooms occupied. A new guest comes and asks for accommodation. Now, what the room manager does is shift the guest in room number 1 to room number 2, shift the guest in room number 2 to room number 3 ans so on such that at last, guest in room number n occupies room number (n+1). The room number 1 is now empty and the new guest can occupy that room.
If r number of news guests arrive, we can move each guest into (n+r) room where n is the room number presently occupied by them.
Sounds easy, doesn't it? There always exist a room with number (n+1) for every room with number n. So, whats the problem? What is the paradox?
The paradox is (as i understand it) that the Hotel should be able to accommodate infinite number of guests. So, suppose infinite number of guests arrive at the hotel. We have infinite rooms as described above. But how do we describe infinite number of guests? The infinite number of rooms were described by using a situation where finite number of guests arrived. So, we were able to see how there can be infinite number of rooms. But there is no way we can see how can there be infinite number of guests. Since we are not able to see how can there be infinite number of guests, how will the hotel give them rooms? So, the hotel is not able to accommodate infinite number of guests despite having infinite number of rooms.
Now it doesn't sound that easy.. right?
One way to describe infinite number of guests And this, can be taken as an extension to the paradox.
This is the famous Grand Hotel cigar mystery... (it is used for something in induction but i think it can be used here too)
suppose, no one is allowed to smoke a cigar in the Hotel (same hotel as above). Now, suppose, guest in room number 1 wants to smoke a cigar and goes to guest in room number 2 to ask for one. Now, since the guest in room 2 will also not have any cigar, he goes to guest in room number 3 to ask for two cigars, one for himself and the other for the guest in room 1. This can go one as far as we encounter a guest in say room number n having n cigars. This guest gives n-1 cigars to the guest in room n-1 and smokes one himself. But, this situation never occurs since no one has the cigars. So, we see that infinite guests can be thought of there a guest in room n goes to guest in room n+1 to ask for n cigars, smokes one and pass the rest to the guest in room n-1.
So, we have infinite rooms and infinite guests. But, what about the cigars? They are not described as to be in infinite numbers. How do we do that for a cigar? We may do it with something like we did it for the rooms and the guests. But, in the end, we will be left with an object for which we have not defined infinite quantity.
I hope you got bored.... :-P
PS : All of this is how I understood the concept.. it may be wrong, please do tell me where i went wrong.. :-)
One can see Hilbert's paradox as an attempt to define infinity or to make someone realize how infinity can be thought of. You see, infinity is termed as 'not defined' and hence, defining it can be taken as a paradox in itself.
Consider a Hotel with infinite number of rooms. Then, infinite number of guests can be accommodated Now, since there is no definition of infinity, we can't see how we will accommodate infinite guests in infinite rooms. So think of it like this. There are right now 'n' rooms occupied. A new guest comes and asks for accommodation. Now, what the room manager does is shift the guest in room number 1 to room number 2, shift the guest in room number 2 to room number 3 ans so on such that at last, guest in room number n occupies room number (n+1). The room number 1 is now empty and the new guest can occupy that room.
If r number of news guests arrive, we can move each guest into (n+r) room where n is the room number presently occupied by them.
Sounds easy, doesn't it? There always exist a room with number (n+1) for every room with number n. So, whats the problem? What is the paradox?
The paradox is (as i understand it) that the Hotel should be able to accommodate infinite number of guests. So, suppose infinite number of guests arrive at the hotel. We have infinite rooms as described above. But how do we describe infinite number of guests? The infinite number of rooms were described by using a situation where finite number of guests arrived. So, we were able to see how there can be infinite number of rooms. But there is no way we can see how can there be infinite number of guests. Since we are not able to see how can there be infinite number of guests, how will the hotel give them rooms? So, the hotel is not able to accommodate infinite number of guests despite having infinite number of rooms.
Now it doesn't sound that easy.. right?
One way to describe infinite number of guests And this, can be taken as an extension to the paradox.
This is the famous Grand Hotel cigar mystery... (it is used for something in induction but i think it can be used here too)
suppose, no one is allowed to smoke a cigar in the Hotel (same hotel as above). Now, suppose, guest in room number 1 wants to smoke a cigar and goes to guest in room number 2 to ask for one. Now, since the guest in room 2 will also not have any cigar, he goes to guest in room number 3 to ask for two cigars, one for himself and the other for the guest in room 1. This can go one as far as we encounter a guest in say room number n having n cigars. This guest gives n-1 cigars to the guest in room n-1 and smokes one himself. But, this situation never occurs since no one has the cigars. So, we see that infinite guests can be thought of there a guest in room n goes to guest in room n+1 to ask for n cigars, smokes one and pass the rest to the guest in room n-1.
So, we have infinite rooms and infinite guests. But, what about the cigars? They are not described as to be in infinite numbers. How do we do that for a cigar? We may do it with something like we did it for the rooms and the guests. But, in the end, we will be left with an object for which we have not defined infinite quantity.
I hope you got bored.... :-P
PS : All of this is how I understood the concept.. it may be wrong, please do tell me where i went wrong.. :-)
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