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The following two questions are on an assignment of mine:

a) Suppose that Hilbert’s Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel.

b) Show that a countably infinite number of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.

All the information I've been told about Hilbert's Grand Hotel is:

1) Hilbert’s Grand Hotel is a paradox. The Grand Hotel has a countably infinite number of rooms, each occupied by a guest. A new guest is accommodated by moving each current guest in a room with number n to room number n + 1 and lodge the new guest in room 1.

I am confused by the question and what exactly it is asking. My understanding of the Hotel is that, the rooms are functionally infinite. Therefore, don't both of the questions answer themselves?

  • Even if you close half of all the rooms, theres still an infinite number of them, so you will not need to send any guests away.

  • Because theres an infinite number of rooms, how can the hotel ever be considered fully occupied? Wont there always be a room n+1?

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    $\begingroup$ Rather than phrasing the questions in terms of the context of a hotel... it may help to see the questions written purely in "mathspeak." The question is asking you to prove that there exists a bijective function between $2\Bbb N +1= \{2n+1~:~n\in \Bbb N\}$ and $\Bbb N$ as well as a bijective function between $X\cup \Bbb N$ and $\Bbb N$ where $X$ is any countably infinite set thereby showing that they are all equinumerous and of the same cardinality. $\endgroup$ – JMoravitz Mar 26 at 23:10
  • $\begingroup$ "Because theres an infinite number of rooms, how can the hotel ever be considered fully occupied? Wont there always be a room $n+1$?" For any finite number of guests, sure there will always be another room available. But, there are (countably) infinitely many guests, so it is certainly feasible that no room is unoccupied which would imply that every room is occupied. $\endgroup$ – JMoravitz Mar 26 at 23:14
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Hilbert's hotel is usually brought up as a prelude to a discussion of cardinality, where one of the lessons to be learned is that it is not a sufficient argument to throw up your hands and shout "infinity!" as if that automatically settles everything.

In (a), what you're asked to do is (to stay with the analogy) to construct concrete instructions for each guest: Where should he go, exactly? The instructions should lead to every guest being the only person in a room, and no guest in the rooms that close for renovations. Consider for example

moving each current guest in a room with number $n$ to room number $n + 1$ and lodge the new guest in room 1

That tells everyone exactly where to go, at least after they check the number of the room they're currently in and do a bit of easy arithmetic. What are the similar instructions that will let you shut down half of the rooms?

As for (b), "fully occupied" means there is a guest in each room. That requires an infinite number of guests, but if we're in a universe that has space and building materials for infinitely many hotel rooms, finding people to rent them all out to is a simple matter of marketing.

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  • $\begingroup$ So, the answer for a) would be something along the lines of "you would move each current guest in a room with room number n to room number n+2"? As far as I can see that would account for all the even rooms being closed As for b), I'm not quite sure how that applies to answering the question. Is this just a matter of showing that the overall cardinality of the set of total rooms is the same as the set of infinite guests? $\endgroup$ – Anerdson Mar 26 at 23:30
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    $\begingroup$ Nice answer. I worried for a while about whether all the room changes in (b) also require the hotel residents to be immortal, but it should work out OK if the hotel management isn't too fussy about getting rid of the bodies. $\endgroup$ – Rob Arthan Mar 26 at 23:32
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    $\begingroup$ @Anerdson: The guest currently in room $42$ would end up in room $44$ which is to be closed, so that won't work. (But it's the right kind of answer). $\endgroup$ – Henning Makholm Mar 26 at 23:32
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    $\begingroup$ @RobArthan: The hotel has a very meticulously planned network of internal corridors, stairs, and lifts that somehow makes it possible for everyone to reach their new room before dinner time. Visiting architecture students find it difficult to believe it exists in three dimensions. $\endgroup$ – Henning Makholm Mar 26 at 23:35
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    $\begingroup$ @Anerdson: That would make the to-be-closed rooms vacant, but suddenly the guests formerly in rooms $41$ and $42$ need to share room $43$. All the rooms are single beds, so these guests are not happy. $\endgroup$ – Henning Makholm Mar 26 at 23:37

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