Talk:Hilbert's paradox of the Grand Hotel

Moving Guests to occupied rooms???

-'Because the hotel has infinite rooms, we can move any guest occupying any room n to room n+1, then fit the newcomer into room 1'. No, we can't. Room n+1 is already occupied, as are all rooms by definition. Any room in the hotel has 'occupied' as descriptor, as per condition of the problem (' a hypothetical hotel with a countably infinite number of rooms, all of which are occupied '). So, whatever number we call any particular room (be it n+1, n+2, n+3 or any other), it does not change the room's 'occupied' status. Therefore, we can't move a guest from an occupied room to an occupied room to make room for more guests :) Shorr (talk) 12:05, 24 December 2014 (UTC)

Let's start with a hotel with N rooms all occupied where N is a finite integer. If we ask the occupant in Room i to move to Room i+1 (for i=1, N), it can be done, because the occupant in Room i+1 is moving to Room i+2 so that the occupant in Room i can move into Room i+1. Of course, in this case, the last occupant in the last room, Room N, has no room to move in, and this is a problem. Note the keyword here is "last". Comes in the mind-boggling idea of infinity. If the hotel has an infinite number of rooms numbered 1, 2, 3, ... and each and every room is occupied, the above operation can be performed without any problem. The reason: There is no last room or last occupant. One's mind is tethered to the idea of "last" and it is hard to break with it. --Roland (talk) 05:06, 3 March 2017 (UTC)

I think there is a problem in terms of the infinity of rooms, guests and new arrivals. I would assume if the hotel had infinite rooms, no new guests would arrive since everyone would already be in rooms. However, there seems to be the assumption that there is not only an infinite number of guests but also a possibly infinite number of new guests arriving. Ghormax (talk) 02:01, 19 February 2024 (UTC)

(Nick here) Roland, your explanation doesn't make sense to me. There being no "last" room/occupant doesn't make it make sense for all current guests to be accommodated even after a new guest is given a room. It just pushes the problematic part (there being no room for one of the guests) infinitely out of view. I think this is a general problem with a lot of so-called paradoxes that involve infinity-- you follow a process that makes sense, and results in a paradox, but it has a problem. Then that problem just disappears over the horizon because of infinity, and we're supposed to accept that 'the impossible can become possible'. There not being a "last" guest *does not explain* how you can start with nothing but full rooms, add a guest, shuffle the original guests around, and somehow not have any room be double-booked or anyone without a room. Ultimately, what's really happening here is that logic *appears* to dictate that you can always accommodate an additional guest, but since we know there ain't no fuckin' way that could be true, there must be something wrong with the application of the logic. Yes, mathematically, it looks right. But that doesn't mean that it's a proper application and would obtain in the real world if we could somehow test it. What seems much more likely to me is that *the very idea of a hotel with an infinite number of rooms breaks the rules of reality already, so you cannot then go doing thought experiments on it and expect results to be fully logically consistent.* What say you? --Nick (2017-12-21 sorry I don't know how to format)

Hi Nick. This space is not for this sort of discussion. See WP:TPG.

Feel free to ask a question at WP:RD/Math, the mathematics reference desk. People will be happy to help you work through the questions there. --Trovatore (talk) 22:14, 21 December 2017 (UTC)

Here's how it works. The Hilbert Hotel is full. New guests arrive. Desk guy pushes the P.A. button, and announces, to all of the existing guests that there are X new arrivals. If we posit that a countably infinite number of rooms exist, and they are all occupied somehow, we sidestep the issue that an infinite amount of distance needed to be walked to set this up in the first place. If this is somehow true, then surely the global P.A. system would exist too. Everyone gathers all their stuff, and steps out of their rooms at the same time. now every room is unoccupied. Every person now has an unoccupied room to the side that is X higher, where X is the number of new guests. Everyone walks down x rooms, then enters their new room, freeing up x rooms for the new guests, which may then move in. Every one of the infinite number of current guests travels the same finite distance, and the new guests are moved in in a finite and reasonable timeframe. This starts to break down a bit with infinite numbers of new guests, as then the issue of infinite travel time is restored. 198.153.92.254 (talk) 16:12, 13 June 2021 (UTC)

Deletion of edits

I'm very new to contributing, so I was hoping for some further clarification on why my edit from March 8 was deleted. The comment from the user who deleted it marked it as non-notable. How is that determined? I added a reference to the principle in a published work of fiction, which feels like a notable reference. Was it because I failed to provide a citation? The Hitchhikers Guide to Existentialism (talk) 15:16, 31 October 2023 (UTC)

That's...pretty late to bring it up. I can't speak for the person who reverted you (you should ask them specifically), but "notable" refers to topics for articles themselves, not pieces of information within them. Maybe "noteworthy" is a better word here. I agree that it shouldn't be mentioned, which might be confusing given the laundry list already there. But I'd also say the solution is to remove them all, which I've just done, not add more. See WP:IPC, an essay about stuff like this. 35.139.154.158 (talk) 15:49, 31 October 2023 (UTC)

I wasn't notified it was removed until I opened the article again today. The Hitchhikers Guide to Existentialism (talk) 16:25, 31 October 2023 (UTC)

Title Change

I suggest that the title of this article should change to "Hilbert's grand hotel paradox" or "Hilbert's hotel paradox". It sticks out like a sore thumb when compared to other pages of paradoxes (e.g. https://en.wikipedia.org/wiki/List_of_paradoxes). I looked into how to change the title and it seems I cannot find this illusive move button or I do not have the permissions. Lucian Chauvin (talk) 04:20, 4 January 2024 (UTC)

Given that this article has been at this title for many years, I would recommend that you not try to move it unilaterally, even if you have the permissions, as that is supposed to be for "uncontroversial moves" and I predict controversy for this one. You can nominate the article to be moved and try to get consensus. The procedure is explained at WP:RSPM. --Trovatore (talk) 07:14, 4 January 2024 (UTC)

1924 or 1925?

The date of January 1924 here is cited to this paper, which is a bit confusing to read. It explicitly says "January 1924" on a couple of occasions including the abstract. However, it later refers to "the winter semester 1924-1925", "A few months later ... on 4 June 1925". It has a section titled "3. January 1925: The birth of Hilbert's hotel" with the comment "This is what Hilbert had to say about his hotel in January 1924". One of those years must be a typo.

I do not have access to the 2013 book of Hilbert's lectures that is cited, but the preview pages seem to support the 1924/25 dates, which would strongly support this being January 1925.

I'll change the article but given the cited paper has the wrong date up-front, thought it best to leave a note here explaining. Andrew Gray (talk) 18:33, 18 January 2024 (UTC)

Article has gotten repetitive

I guess I hadn't looked here in a while. It strikes me that there's a large amount of material that, while presumably correct (I haven't checked it in detail) doesn't really add much of value. I'm worried that readers will "lose the plot" as it were.

To be specific, I'm talking about (possibly among other things) most of the section titled "Infinitely many coaches with infinitely many guests each". Is it really necessary to enumerate five different "methods" of accommodating the guests, each with its own separate section?

All this really comes down to is that any countable collection of new guests can be accommodated, which can be shown in a sentence or two. All the rest of the verbiage seems to be about methods to show that certain sorts of sets are countable, which I think is beyond the scope of an article on this particular "paradox".

I think a massive trimming is in order. The ideal length for this article is probably at most half its current length. --Trovatore (talk) 06:48, 2 September 2024 (UTC)

But I do think casual readers need to gain the intuition that the Cartesian product or (even disjoint) union of countably many countable sets is still countable, which makes the paradox even more paradoxical at first glance. I would say we should trim down to a single pairing function and note that it's not the only possible one.--Jasper Deng (talk) 07:05, 2 September 2024 (UTC)

Um, what? The Cartesian product of countably many countable sets is not in general countable. I haven't drilled down into what part of the text you're applying this to, so I'm not sure whether you're making any actual error in context, but that statement is wrong. --Trovatore (talk) 20:53, 2 September 2024 (UTC)

I must've only been thinking of a disjoint union or finitely many, not countably many, countable sets, since my claim immediately fails to Cantor's diagonal argument. Or I was thinking that given a sequence of countable sets, the set of finite prefixes of all possible corresponding sequences of members is countable. But my argument for taking out the "infinite layers" section should still hold.--Jasper Deng (talk) 01:29, 3 September 2024 (UTC)