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Hilbert's paradox of the Grand Hotel colloquial : Infinite Hotel Paradox or Hilbert's Hotel is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that source fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often.
Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not dating someone with roommates able to accommodate any roommatws arriving guests, as would be the case with a finite number of rooms, where the pigeonhole principle would apply.
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Suppose a new guest arrives and wishes to be accommodated in the hotel. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests. It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2 n 2 times nand all the odd-numbered rooms which are countably infinite will be free for the new guests. It is possible to accommodate countably infinitely many coachloads of countably infinite dating someone with roommates each, by several different methods.
Most methods depend on the seats in the coaches being already numbered or use the axiom of countable choice. In general any pairing function can be used to solve this problem.
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This solution leaves certain rooms empty which may or may not be useful to the hotel ; specifically, all odd numbers that are not prime powerssuch as 15 orwill no longer be occupied. So, strictly speaking, this shows that the number of arrivals dating someone with roommates less than or equal to the number of vacancies created. It is easier to show, by an independent means, that the number of arrivals is also greater than or equal to the number of vacancies, and thus that they are read morethan to modify the algorithm to an exact fit. Wtih every number has a unique prime factorizationit's easy to see all people will have a room, while no two people will end up in the same room.
Like the prime powers method, this solution leaves certain rooms empty. This method can also easily be expanded for infinite nights, infinite entrances, etc. Treat each hotel resident as being in coach 0.
If either number is shorter, add leading zeroes to it until both values have the same number of digits. Interleave the digits to produce a room number: its digits will be [first digit of coach number]-[first digit of seat number]-[second digit of coach number]-[second digit of seat number]-etc. The hotel coach 0 guest in room number moves to room i.]
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