# Talk:Hilbert's paradox of the Grand Hotel

## Really?

"Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests." This ignores a significant point; in this case, the process of switching guests to new rooms is infinite, so although the new guest is settled in room 1, the infinite hallway between the rooms always contains a person switching rooms with the next guest in the line. This means, though each room has a new occupant, Template:Fontcolor. Whereas before, there was an infinite number of occupied rooms and no one in the hallway, now there is the same infinite number of occupied rooms and one person in the hallway- Template:Fontcolor, because there are always new rooms to cycle into, and new guests to move out. An analogous operation would be to move the person out of room 1, and send him to the end of the infinite hallway of occupied rooms to move into his new room- he will never make it, because there is no end to the process. Besides, there is truly no such thing as an infinite number of anything, because no matter how many rooms and guests there are, that number is not infinity. You can't have an infinite number of occupied rooms, even conceptually- the concept simply doesn't allow it. The fact that mathematicians accept this so called paradox seems to make it clear that they really don't fully understand the character of infinity. Perhaps my wiki-talk will change all that. I won't be holding my breath, though- my lungs are not of infinite capacity.

Perhaps a better explanation of the paradox would work like so, "Infinity is a paradox because it never stops; yet we classify it as the end which is never-ending." It's an inherently circular concept. This is the very definition of a paradox. --Xtraeme (talk) 22:29, 1 July 2010 (UTC)

Excellent post.
Just granpa (talk) 15:11, 10 January 2011 (UTC)

...............

I just try to use simple logic, and this "paradox" does not work out in the end.

"every room is occupied"

In a hotel with finite rooms and a finite number of guests, that is entirely possible. However, this new hotel has infinite rooms but yet it still has a finite number of guests. I say the guest number remains finite because we can distinguish that 4 guests are staying at the hotel at the time. Clearly, 4 is a finite number. The addition and subtraction of guests also supports this because you can't physically subtract or add a finite amount to an infinite number. 4 + infinity = ?

We have a hotel with infinite rooms and finite guests. An infinite number can not equal a finite number, so we can never actually fill the hotel.

I doubt I'll check up on this any time soon, so email me here if you want to reply to me personally.

I would just note in general that a question that has fascinated specialists in any field for many years probably does not have an obvious yet overlooked solution that can be spotted by a non-specialist in that field. Nareek 05:47, 30 November 2006 (UTC)

The hotel is a thought experiment. It should not really be called a paradox, it's more of a little idea designed to illustrate some of the interesting properties of infinite quantities. Worrying about the practicalities like filling the hotel in the first place is missing the point. Obviously we can't build infinite hotels in real life, and obviously even if we could, we couldn't fill them up one-by-one. That isn't the point of the exercise. The point is to see that infinite quantities don't behave the way finite ones do. Maelin (Talk | Contribs) 12:14, 30 November 2006 (UTC)

People are getting confused, attempting $4 + \infty$ to see how many rooms are needed, or $\infty - \infty$ to see if the hotel is full. Perhaps the article should explicitly mention Hume's principle and the further develpments of Cantor. The bijection is more fundamental than the number. To see if two sets are the same size, you don't count through each set and compare the resulting numbers; you test if their elements can be placed in exact correspondance. Endomorphic 23:56, 7 December 2006 (UTC)

Is there a name for that rule? I've seen it stated in response to claims that the theory of evolution is a tautology, as well; it seems to be rather well-applicable. grendel|khan 19:56, 3 January 2007 (UTC)

I agree, this paradox is moronic, it simply doesn't work logically.

RootBeerFanatic 03:34, 1 August 2007 (UTC)

## infinity + 4 = ?

Since all infinities are the same size I would argue that 4 + infinity is equal to infinity.

All infinities are not the same size, though. Take the set of Natural Numbers and the set of Real Numbers. It has been proven that the Set of Real Numbers are larger than the set of Natural Numbers.

You are right. If you take an infinite set and add four elements, the number of elements in the set does not change, so "infinity + 4 = infinity". The same is true when you add an infinite number of elemens (e. g. take the naturals and add the rationals), so "infinity + infinity = infinity". Going from naturals to reals is a different story, check "aleph numbers" for details. -- wr 87.139.81.19 (talk) 12:42, 9 January 2009 (UTC)

I am not sure that the article is correct when it says "that such a movement of guests would constitute a supertask". What is generally regarded as a supertask is an infinite series of actions occurring sequentially throughout a finite interval of time. Here the actions must occur simultaneously otherwise some of the guests would be moving to already full rooms. I have included Hilberts Hotel on the supertasks page which I am currently maintaining, but I'd like some consensus before I change anything. --NoizHed 11:19, 27 August 2005 (UTC)

This is quite interesting, but doesn't deal with the different "kinds of infinity"

If a bus load of people mapping on to the Real numbers turned up, they would not fit in a Grand Hotel with an infinite number of rooms each having an integer as a room number.

I think I'm right in this - but I don't know what Hilbert actually said. David Martland 09:48 Dec 11, 2002 (UTC)

It is indeed correct due to the fact that the cardinality of the reals is greater than the cardinality of the naturals. A more interesting question would be whether if the hotel rooms *and* the people mapped on to the reals you could fit everyone, but more importantly what method would you use. Asking everfone to move up to double their room number wouldn't make a difference in this case, each room would still be full. Asking everyone to move up their room number plus one would work because the rooms with numbers in the interval [0, 1) would then be free, and of course this is enough space for all the new guests. --NoizHed 11:19, 27 August 2005 (UTC)

but there is no paradox! if hotel is full rooms n and n+1 are full too! you can't move guests to make free room, because there aren't any free rooms to move guests to. porneL, icq:145671338. Oct 19, 2003

You get them to move up simultaneously. It is for this reason I don't believe that to do this involves a supertask, for with what we ordinarily classify as supertasks the actions occur sequentially. --NoizHed 11:19, 27 August 2005 (UTC)

I agree. This paradox assumes since it is possible for there to be a hotel with infinite capacity, it is therefore possible for such a hotel to be full, but obviously, it cannot. You can't say, "The hotel has infinity rooms, and it currently is hosting infinity guests, therefore, if you subtract the number of guests from the number of rooms, you find we have zero rooms available." It seems painfully obvious, but since infinity isn't a finite number, it therefore cannot be treated as such. - RealGrouchy 19:49, 15 Jul 2004 (UTC)

• Claim 1: The hotel is full (it can't contain any more guests)
• Claim 2: The hotel has an infinite number of rooms (it can contain an infinite number of guests)}
• Claim 3: We can still add more guest
Is it just me or is there a contradiction between claim 1 and 3? I think the paradox just proves that the hotel can't be full.PoiZaN 10:42, 16 May 2006 (UTC)
The first claim should be rephrased: "The hotel is full (every room contains a guest)". This doesn't contradict claim 3, although claims 1 and 3 would be incompatible in a finite hotel. 213.249.135.36 19:52, 13 June 2006 (UTC)
Or in more formal terms, the hotel is full when the set of guests and the set of rooms have the same cardinality. The fact that an infinite hotel may be full but still able to accept new guests is at the heart of the seeming paradox, because it clashes with our common-sense intuitions about finite-sized hotels.

---

Well, you can just fill all the even numbered rooms. Since there's an infinite amount of rooms, half the hotel will be forever free. Or not. --199.67.140.76 03:05, 21 Aug 2004 (UTC)

I have a strong urge to replace these occurences of "infinite (countable)" with 'indefinite'. I think the major flaw in this 'paradox' is that it confuses infinite with indefinite; infinite is not measureable, indefinite is -- so to speak, indefinite is "always one more". If you truly had an infinite number of (empty) rooms in the Hotel, you could never possibly reach a point where they were all full; contrastingly, if you had an indefinite number of (empty) rooms in the hotel, the problem would be unsolvable and thereby a paradox: since the number of rooms is not even defined, in any state (an integer, a decimal, or infinite), having just a single occupant would fill the entire Hotel. In this scenario, you could add rooms, but the would be automagically occupied from the infinite number of guests.

I think ;)

Don't change anything - this isn't how the terms "indefinite" and "infinite" are used. You seem to be referring to the actual/potential infinite distinction, but that doesn't appear to be relevant here. --NoizHed 11:23, 27 August 2005 (UTC)

## Source

No source is given for the claim that this story is due to Hilbert. AFAIK the earliest source for this problem is:

• George Gamow, 1947. One, two, three... Infinity. New York: Dover.

where on p17 he claims it is an example Hilbert gave in a lecture, claiming in turn to have got this story from an R. Courant who was working on a then unpublished book to be called "The Complete Collection of Hilbert Stories". I'm guessing this has to be Richard Courant, who would be a good source for this attribution, but as it stands I see only an attribuition of an attribution... Does anyone have a better source, or know anymore about the manuscript Gamow talked about? --- Charles Stewart 17:50, 6 December 2005 (UTC)

Possibly references at Wolfram's Mathworld article will be helpful...
See http://mathworld.wolfram.com/HilbertHotel.html
CiaPan 16:18, 17 July 2006 (UTC)

## Subtracting infinities

Not sure that the last edit is very helpful. Yes, you can subtract an infinite quantity from an infinite quantity and still have an infinite quantity--that's why the number of whole numbers is equal to the number of even numbers. It's one of the interesting qualities of infinity, but I don't think it ought to be posed as a question. I'll leave it to someone else to revert if they agree. Nareek 04:53, 22 January 2006 (UTC)

Is the cigar story really considered a valid thought experiment? While one can imagine an infinite number of hotel guests changing rooms, it's difficult to imagine a hotel guest handing over a billion cigars--or a trillion cigars--or a million billion trillion--all of which would be some of the smaller numbers of cigars dealt with in this hypothetical. Also, unlike the room transfers, which can be imagined being done simultaneosly, the cigar handovers would have to be done sequentially, which means the guest in the first room would be dead an infinite number of years before he got his cigar. Is the cigar thing really part of the canonical paradox? Nareek 17:24, 5 February 2006 (UTC)

A friend thought of this; Since there are an infinite number of guests staying at the hotel, then the newcomer would count as 0% of the total guests, therefore being nonexistant? could this be right? there's probably something I'm missing since I don't know too much about these things..

aznshorty67 04:02, 17 February 2006 (UTC)

The size of the universe is infinite and there are an infinite number of molecules in the universe. You consists of a finite number of molecules and so you count as 0% of the universe, and are therefore nonexistent. How do you feel about being nonexistent then? Isn't it a marvellous feeling? The Hitchhiker's Guide to the Galaxy has an entry about infinity somewhat stating the same. PoiZaN 20:05, 16 May 2006 (UTC)
Um. Nobody knows whether the universe contains an infinite amount of matter, or even if it is infinite in size. Regardless, being a finite proportion of an infinite quantity, or an infinitesimal proportion of a finite quantity, does not mean that one is nonexistent. No infinite or infinitesimal quantities are known to exist, and it's not really meaningful to make percentage comparisons of such hypothetical things. Maelin (Talk | Contribs) 09:04, 8 November 2006 (UTC)

## Somethings bothering me

If the hotel consists of an infinite amount of rooms why would they fill up? They wouldn't.They suggest that you keep moving one room to the next, but why do this? They could just check into the infinite amount of rooms available.Also, you couldn't take up any rooms; because if one group of 4 people check into a room when there is infinite rooms available, they obviously take up a room, so how many are available now? There is an unlimited amount of rooms left. Leading to make me think thats there's actually no such thing as infinity.

--I agree... it would take a supertask to make the hotel full in the first place. Either that or an infinite number of people check in at the same time(talk about an opening-day rush). Incidentally, this infinite number of guests happen to be the infinite construction crew that built the darn place, now on vacation after completing it.

The concept of the Grand Hotel is meant to be an educational tool for developing an understanding of countable infinities. It's not supposed to be an actual thing you could build, since even the amount of matter in the universe is finite, so worrying about issues of logistics is not constructive. Don't worry about how an infinite hotel got full in the first place, that's not the point. The point is to realise that even when it is full, you can still fit in as many more people as you need. Maelin (Talk | Contribs) 06:14, 2 November 2006 (UTC)
That doesn't sit well with me. If there's room for more, this is not a full hotel.
full
1. completely filled; containing all that can be held; filled to utmost capacity
If there are always more rooms in the hotel, no matter what, it can't be full, even if there are always more people in the hotel, because for every one of those 'more people' who comes along, there's a room for them. If it were full, it would be 'containing all that can be held', but if the number of rooms is infinite, there is always room for the hotel to hold more. --Bertieismyho 02:01, 8 November 2006 (UTC)
Well, that's just a quibble on the word "full". It seems to me pretty natural to say the hotel is "full" if there's someone in every room, but if you don't like the word in that context, feel free to think of another word. --Trovatore 02:22, 8 November 2006 (UTC)
I edited the first paragraph to make it clear that in an infinite hotel, one must be careful about what one means by "the hotel is full". The point of the article is really that an infinite hotel is never 'full' in the sense that no more people can be accomodated. I hope it's clearer now. Maelin (Talk | Contribs) 08:59, 8 November 2006 (UTC)

## Arguing with itself

The article as it is now is having a debate with itself--which isn't good. It's possible (indeed necessary) to provide more than one point of view without having the paragraphs bickering. Nareek 22:32, 5 July 2006 (UTC)

I removed the bickering paragraphs, which seem to be rather pointless anyway--Hilbert's hotel paradox is about the nature of infinity, rather than about the practicalities of cosmic hotel management. It's rather like responding to Einstein's thought experiments by saying, "Well, if you were traveling that fast, how could you see the clock?"
If someone feels a need to put this material back in, please try to do it in a way that reflects the various points of view rather than starting the internal argument again.Nareek 16:16, 12 September 2006 (UTC)

## Infinite sets

"In fact, infinite sets are characterized as sets that have proper subsets of the same cardinality."

This is clearly false.

You'll have to say more. The characterisation seems fine to me. You might like to provide a pair of counterexamples: 1/ A finite set having a proper subset of the same cardinality 2/ An infinite set having no proper subset of the same cardinality. One would really be enough, but I give you two goals, in the hope that in failing to achieve one you may strive for the other, to eventually realise why the characterisation holds.

I think the problem is the wording, while I understood what was meant, at first I read it as "if a set A has two proper subsets B and C, such that B has the same cardinality as C, then A is infinite", which is clearly not true. Would someone less tired than me like to come up with a wording that is unambiguous and not too messy? (By the way, I am not the original poster, in case the lack of signatures and dates makes this unclear.) Bistromathic 14:16, 8 June 2007 (UTC)
How about "In fact, an infinite set is characterized as a set which has a proper subset of the same cardinality"? I don't see how it could be much simpler. grendel|khan 17:46, 8 June 2007 (UTC)
Still with the ambiguity. You want to say "A set X is infinite if and only if X contains a proper subset Y of the same cardinality as X". Endomorphic 22:40, 10 June 2007 (UTC)

## The Cosmological argument

I'm confused by this section. It appears that the people using Hilbert to argue against the possibility of physical "real-life" infinities are trying to use that as proof that God DOES exist? Isn't that self-defeating though? Or is the God they're arguing in favor of actually a finite one? Though I recognize this isn't the main focus of the article, a fuller/clearer explanation of how this objection is resolved (as I'm assuming they can't possibly be unaware of it) would be very helpful. --Arvedui 09:05, 2 October 2006 (UTC)

Guessing a little here -- could it be along the lines of: Infinities exist, physical infinities don't exist, therefore something non-physical exists, therefore God exists? Obviously this would not be the whole argument, but it could conceivably be the outline, with lots of reasons to be filled in for the various assertions and therefores. --Trovatore 20:04, 2 October 2006 (UTC)
The argument is anyway really weak, with obvious fallacies. Since when did something being counter-intuitive prove anything? I can't be bothered to go dig up a notable refutation of this argument, and I doubt anyone else can either. I vote we just chop the whole section. --61.214.155.14 (talk) 05:53, 7 March 2008 (UTC)

Craig's argument was an argument to prove logically the universe had a beginning because an infinite past would require an actual countably infinite quantity. While this argument is largely unnecessary today, since we've proven the big bang with scientific experimentation and confirmation, this method is used to argue for a beginning for the universe strictly from mathematics and logic. It was far more relevant before the big bang was discovered. Lionside (talk) 22:27, 7 August 2009 (UTC)

Frankly this section has no place here. It is a plant for christian apology in an article that should be about mathematics. It could perhaps be moved to pages that cover christian apologetics. I'll remove the section. People can revert and discuss if there is a real case for keeping it here. 99.118.116.9 (talk) —Preceding undated comment added 22:50, 12 June 2010 (UTC).

## (Comment by 76.0.30.157 moved)

This comment moved here from the top of the talk page - Maelin (Talk | Contribs) 14:28, 27 December 2006 (UTC)

This is not a mathematical paradox. It's an example of flawed use of language. For a hotel to be "full" presupposes (by definition) a limited capacity, and the problem explicitly states that the capacity is infinite in this case (i.e. without limit).76.0.30.157

The problem with the word "full" is that, in normal usage for finite hotels, there is no need to distinguish between "every room contains a person" and "there is no space for more people", since they are synonymous. In the case of infinite hotels, that synonymity no longer holds and we need to be careful with what we mean by "full". It was carefully stated in the first main paragraph of the article that when the hotel is described as "full", this should be taken to mean "there is a person in every room". Beyond that, the point of Hilbert's Hotel is to demonstrate the interesting properties of infinity, not to illustrate ambiguity in the interpretation of the word "full". Try not to get caught up on the minor details, they aren't the important part. Maelin (Talk | Contribs) 14:34, 27 December 2006 (UTC)
The paradox article defines paradox as "A paradox can be an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition", so this does count as a paradox in the weaker sense. I don't find it satisfactory personally, since this is simply a case where infinities result in a situation contrary to intuition. If you've done much mathematics, then you'll realise that this is completely normal.
It appears that Hilbert's Hotel is a thought experiment designed to help people who haven't had exposure to this sort of thinking get up to speed. I don't like the description as a "mathematic paradox" which confuses it with real mathematical paradoxes like the Banach–Tarski paradox. --61.214.155.14 (talk) 05:49, 7 March 2008 (UTC)
What's the distinction between this and Banach–Tarski? In Quine's terminology, they're both veridical paradoxes. Algebraist 23:21, 30 May 2008 (UTC)

## Confused

However, in an infinite hotel, the situations "every room is occupied" and "no more guests can be accommodated" do not turn out to be equivalent

It is never explained why this is the case. This needs to be explained.

An even stranger story regarding this hotel shows that mathematical induction only works in one direction. No cigars may be brought into the hotel. Yet each of the guests (all rooms had guests at the time) got a cigar while in the hotel. How is this? The guest in Room 1 got a cigar from the guest in Room 2. The guest in Room 2 had previously received two cigars from the guest in Room 3. The guest in Room 3 had previously received three cigars from the guest in Room 4, etc. Each guest kept one cigar and passed the remainder to the guest in the next-lower-numbered room.

Is this written correctly? Seems to me the infinitieth guest had to have brought an infinity of cigars into the hotel for this to work. --24.57.157.81 00:46, 9 February 2007 (UTC)

"However, in an infinite hotel, the situations "every room is occupied" and "no more guests can be accommodated" do not turn out to be equivalent" It is never explained why this is the case. This needs to be explained.
Of course it is explained why this is the case. That's what the whole article is about.
Seems to me the infinitieth guest had to have brought an infinity of cigars into the hotel for this to work.
For one thing, there is no infinitieth guest. The very nature of an infinite number of guests is that every guest has another guest in the next room along. Still, your mistrust in the story's validity is well-founded, for the example is intended to demonstrate that mathematical induction can only work by starting from the base case and working up, and not the other way around. Maelin (Talk | Contribs) 09:09, 9 February 2007 (UTC)
Oh I get it. The article neglects to mention that there are also an infinite amount of guests who want rooms, right? In an hotel with an infinite number of rooms (empty rooms, to be precise), there will always be another empty room available. Sounds reasonable. A hotel with an infinite number of empty rooms is never full. If an infinite number of guests arrive, they cannot all be accomdated because there is always an unoccupied empty room. That way the hotel is never full, but cannot accomodate an infinite number of guests. See, I don't think this is explained very well in the article, it just needs some reorganization I think.
As for the cigars, I understand what it's trying to show (working backwards can give you any result you want--deriving a formula from a pattern is the same sort of concept), but I don't think it works out the way it is written. It should explain how the cigars got into the hotel, or why this doesn't matter, etc. --24.57.157.81 23:17, 9 February 2007 (UTC)
You might be right about the cigar thing, it's not very clear. Regardless, I didn't quite follow your comments about the infinite number of guests. One of the things about a hotel with an infinite number of empty rooms is that, indeed, if you put guests in one by one, you will always have empty rooms left over, so filling the hotel with an infinite number of guests in the first place is a supertask. But the important part of the Hotel thought experiment isn't about filling the hotel up, it's about discovering that, even if you have somehow managed to fill -every- room with a guest, there is still space for a finite number of extra guests, even an infinite number of extra guests, and even an infinite number of groups of infinite numbers of extra guests. Maelin (Talk | Contribs) 00:03, 10 February 2007 (UTC)
I'm wrong again I think, having read about it somewhere else: The hotel starts off full. It has an infinite number of occupied rooms, not empty rooms. The hotel can create an unoccupied room by moving the guest from room 1 to room 2, and room 2 to room 3, and so on. Eventually*, the guest in room infinity will have to move to room infinity + 1. Since infinity + 1 = infinity then it does not matter. I think it does matter, but I only want to get the idea straight, regardless of whether or not I think it works out.
And to clarify my previous comment, it's irreleveant but if you're interested, it seems I made up something like the opposite of the situation. (The hotel is never full, can accept a finite number of new guests, but can never accomodate an infinite number of guests).
.*I don't think the fact that this is a supertask enters into it, I think someone just added that in because it is interesting.-- 24.57.157.81 19:37, 10 February 2007 (UTC)
You've got the right idea now, except for your comment about the guest in Room ∞. There is no infinitieth room, at all. Every room has a room number that is a finite, positive whole number, called a natural number. That's why they can always move up one, because there are infinitely many natural numbers. This is a bit confusing because when we are dealing with counting finite quantities (like, say, a set of five apples), we can number them off (apple 1, apple 2, ...) and the number we get on the last one is exactly the number of apples there are. In contrast, every hotel room in Hilbert's Hotel has a room number that is finite, but the total number of rooms is not finite. Put succinctly, ${\displaystyle \textstyle \left|\mathbb {N} \right|\notin \mathbb {N} }$. Maelin (Talk | Contribs) 07:58, 11 February 2007 (UTC)

## "Coach" meaning "bus" is a UK-specific idiom

Very few English speakers in the US refer to a bus as a "coach", the word should be changed for clarity. —The preceding unsigned comment was added by 155.212.242.2 (talkcontribs) 16:18, 19 June 2007 (UTC)

Well, does anything really change if what pulls up to the hotel is an infinitely capacious stagecoach, presumably drawn by infinitely many horses? --Trovatore 22:05, 19 June 2007 (UTC)
Fixed by creating a disambiguating link to coaches. --kostmo 01:48, 11 September 2007 (UTC)

## The Hallway

With the infinite hotel, I don't understand. Since the hotel is infinite the hallway is infinite. Either you can't add a new guest until all guests are in their new rooms, which would be a infinite amount of time, hence never, or there would be one guest propagating in the hallway for each new guest you add to the hotel. —Preceding unsigned comment added by Websaber (talkcontribs) 03:53, 13 October 2008 (UTC)

You're getting hung up on irrelevant detail here. You might as well ask how the hotel got a building permit. Just start with the assumption -- yes, I understand it's impossible, but that doesn't matter -- that the hotel exists, the guests are all in their rooms, and the signal can be sent to them all simultaneously to pick up and move to the next room. What logically follows? That's the content of the paradox. --Trovatore (talk) 05:59, 13 October 2008 (UTC)
That does clear it up thanks —Preceding unsigned comment added by Websaber (talkcontribs) 05:32, 5 November 2008 (UTC)

Your (Trovatore) solution doesn't address the fundamental problem with the "paradox". People are finite. Rooms are occupied by people.
• We assume that at some point in the past, there was the "first person".
• For the sake of argument, let's assume that you're the "first person", and the "second person" has yet to be born.
• Again, for the sake of argument, since you're the "first person", no one else has yet to check in to the hotel.
Now, once you check in, and you get to your room, who is in the room?
• You're the only person.
• You're the first person, EVER. No one else has checked in to the hotel
• You're pretty sure you're not in the room.
• There's no one else that can be in the room since no one else has been born. To be clear, no one has checked in to the hotel since you checked in.
• There must be someone in the room, or else the room is unoccupied.
There's two choices:
1. Explicitly name the person who is in the room that isn't me, isn't someone yet to be born, or otherwise require a "paradoxical" person (since this would resolve the Grand Hotel paradox by shifting the paradox problem somewhere else).
2. Accept that there is no paradox because the "paradox" is logically inconsistent and requires impossible things to be true (other than the infinite number of rooms, which we're accepting as possible for the sake of argument).
I can prove all sorts of things true if you allow impossible things to be true. Johne23 (talk) 12:45, 21 March 2010 (UTC)

## axiom of choice

The argument with accomondating infinitely many coaches actually uses the axiom of choice. This should be clarified at some point. Katzmik (talk) 13:20, 23 October 2008 (UTC)

## Hilbert doesn't get it

The idea of the Grand Hotel fails to mention the space required. For this to work then guest (1) would have to share room (a) with guest (2) until guest (2) leaves for room (b) and so on. This implies that the infinity of the surrounding space would have to be larger than the infinty of the number of guests. It makes more sense to imagine an infinite space occupied by an infinite number of particles. Particle (1) cannot occupy the next space until particle (2) moves, but it can't because the next space is always occupied. The new particle we are trying to fit into the space couldn't exist. There would never be a new guest because all the guests in the universe are already in the hotel.

You are missing the point entirely. The Grand Hotel thought experiment is intended to demonstrate certain unintuitive properties of countably infinite quantities. The logistics and engineering considerations of an infinite hotel are irrelevent, as are problems in having an infinite number of people, and so forth. Maelin (Talk | Contribs) 02:35, 11 May 2009 (UTC)
Maelin, the paradox is fundamentally flawed and logically inconsistent. Having an "infinite number of rooms" hides the fundamental problem and just confuses people. Contrary to everyones claim here, the "magic" of the paradox does not come from having an infinite number of rooms. The real problem is revealed when you start to ask "Who is in the room?"
• Let P be the population of people, which we represent with the set of natural numbers, N, where N > 0.
• We define "the room is occupied" as containing at least one member from P. No cheating by trying to substitute the empty set in a room.  :)
• I am the first person, that is to say, me = { 1 }. For simplicity, we all agree that we will not use a member from P that is > 1 for the moment (i.e., any member of P that we discuss will satisfy both x > 0 and x < 2).
• I check in to the Grand Hotel. You can pick any door you want, but you guarantee that the room is occupied "by a member of P".
• We open the door. In order for you to claim that the room is occupied, you must show me someone from P that is > 1 (not me) and, simultaneously, is < 2 (not any of "children", so to speak).
• To be clear, there is no member from P that satisfies both > 1 and < 2 at the same time.
• One way out of the paradox is to axiomatically define the statement "The member occupying the room is both > 1 and < 2 and is a member of P" as "True".
• By this route, we have created a genuine paradox- by definition, you have created a condition that is true that can not be true. For most people, though, this does not make it "provably true".
• Another way to "solve the paradox" is for me to be in the room. This is perfectly reasonable from a mathematical perspective- there is only one '1', yet I have used it multiple times. However, since we're discussing people, it's fair to say that we are implicitly applying constraints that would only allow '1' to be in one place at a time.
This is the fundamental paradox of the Grand Hotel. You can toss the infinite number of rooms out- it just postpones having to deal with the above case. Since this is a case that will eventually happen, and in fact must happen, you have to come up with a logically consistent solution to it. The infinite rooms is just a red herring, it allows you to defer dealing with this case, which is the real problem. Since we are dealing with countably infinite set, I can demand an explanation for "what happens the very first time", even though there are an infinite number of numbers after me. Johne23 (talk) 04:17, 22 March 2010 (UTC)

I am sorry, but I just don't get what the "paradox" is. 117.97.26.35 (talk) 15:56, 24 October 2009 (UTC)

I added a link here[1], a page that talks about the infinite hotel, but it was automatically removed. Is it because it's in a different language? In that case, what are the polytics of the english wikipedia in linking? —Preceding unsigned comment added by 84.78.246.14 (talk) 02:24, 12 January 2010 (UTC)

This is the first time I've heard of "Hilberts paradox of the Grand Hotel", but the content of this article is obviously flawed. Personally, I suspect it is because it has been transcribed incorrectly, or who ever wrote the original content did not appreciate the nuances of how the original "paradox" was stated.

The line stating the paradox contains a fundamental flaw:

Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.

The flaw is the underlined text. This statement, at least the way it is phrased, means there is a paradox if, and only if, one assumes the following conditions:

1. The Grand Hotel can rent out a room more than once. That is to say, it can rent a room that is already occupied by a guest to a newly arriving guest.
2. A newly arriving guest can only get a room on the condition that the room they get is currently occupied. The management assures the newly arriving guest that they will immediately re-locate the guest currently occupying the room to another room "right away".
3. A newly arriving guest must agree to be relocated to another room at any time. The room that they will be relocated to will be occupied by another guest, which the management assures that they will immediately re-locate to another room.
4. A newly arriving guest must agree and guarantee, forever, that they will not do anything to cause the room they are in to become "unoccupied."
5. A newly arriving guest must take a room. They can not decide that they don't like the terms and would rather go somewhere else.

Point #5 might seem odd, but it is probably required due to the wording of the paradox. It is required because if any guest is given the opportunity to refuse a room at the Grand Hotel, there exists the possibility that every guest before you has refused a room, and you are "the first guest". This would require the hotel to have a room occupied by a guest before you, the very first guest, because whatever room you get, it will already be occupied. The "paradox", as stated, only works if you can make some kind of hard guarantee that you are not the first guest at the hotel.

Consider the "paradox" when the offending line is removed:

Consider a hypothetical hotel with countably infinitely many rooms. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.

Once the qualification that "all the rooms are occupied" is dropped, it no longer seems like a "paradox". On the other hand, if you want to keep the "all the rooms are occupied" qualifier, it doesn't really seem like much of a "paradox" when you read the "fine print of the contract".

The article continues with:

Finitely many new guests - Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.

IMHO, this is not actually true as it is worded since it glosses over the fact that you can't ever cause the state of a room to become "unoccupied", though it is not clearly defined what qualifies as "unoccupied". It also hand waves the fact that it's possible your entire stay at the hotel will be 100% filled with moving from room to room, something most people would probably object to.

Another small, minor detail not really mentioned is the fact that checking out of the hotel would create an unoccupied room... or at least glosses over some of those "messy details" like requiring a room to instantaneously "vanish" once it becomes "unoccupied"... and the fact that you're pretty much screwed if you forget something important in the room and need to go back for it...

These cases demonstrate the 'paradox', by which we mean not that it is contradictory, but rather that a counter-intuitive result is provably true: The situations "every room is occupied" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.

Again, the "paradox" relies on the fact that most people reasonably assume that "to be accommodated" means that they will be given an unoccupied room, and once they've been accommodated, they're not going to be kicked out of their current room and relocated to another room at any time. The "paradox" also requires that a guest can be relocated as many times as the Grand Hotel wants to relocate them. In fact, they may spend their entire stay at the Grand Hotel relocating from room to room, literally closing the door only to have it the hotel knocking on it to relocate you again. Most guests subjected to the "worst case scenario" for the entire duration of their stay at the Grand Hotel would not consider themselves to have been "accommodated" by the Grand Hotel. This is probably yet another unstated caveat in order to make the "paradox" logically consistent: Every guest that stays at the Grand Hotel is "happy with the service provided" and can't claim that they weren't accommodated.

The Grand Hotel Cigar Mystery section also has a number of fatal flaws, at least as it is stated in the article:

Another story regarding the Grand Hotel can be used to show that mathematical induction only works from an induction basis.

Suppose that the Grand Hotel does not allow smoking, and no cigars may be taken into the Hotel1. Despite this, the guest in room 1 goes to the guest in room 2 to get a cigar2. The guest in room 2 goes to room 3 to get two cigars3 - one for himself and one for the guest in room 1. In general, the guest in room N goes to room (N+1) to get N cigars4. They each return, smoke one cigar5 and give the rest to the guest from room (N-1). Thus despite the fact no cigars have been brought into the hotel, each guest can smoke a cigar inside the property6.

The fallacy of this story derives from the fact that there is no inductive point (base-case) from which the induction can derive. Although it is shown that if the guest from room N has (N+1) cigars then both he and all guests in lower-numbered rooms can smoke,7 it is never proved that any of the guests actually have cigars. The fact that the story mentions that cigars are not allowed into the hotel is designed to highlight the fallacy.8 However, unless it is shown that in the limit there is a guest with infinitely many cigars, the proof is flawed regardless of whether or not cigars are allowed in the hotel9.

This one is even worst than the original "paradox"- it puts forth a "mathematical induction thought experiment" that can be trivially disproven.

• 1 - Establishes how cigars can make it in to the hotel- specifically that they can't. The thought experiment does not specify any other way for a cigar to enter the hotel, nor does it specify where the cigar "comes from" that a guest eventually smokes.
• In general, all the conditions need to be stated up front. Anything that depends on pulling something out of thin air is generally considered to be logically flawed. Anything can be proven if you're allowed to make up rules at any point that happen to benefit your argument.
• Since the thought experiment provides no other way for cigars to enter the hotel (i.e., a room comes stocked with a random number of cigars), and a condition of the thought experiment is that guests can't bring cigars in to the hotel, we can prove that there are no cigars in the hotel.
• If a cigar does manage to turn up, we can prove that it "magically appeared". In order for a "proof" to be legitimate, it can not depending on any "magically appearing cigars".
• It is unstated, but assumed, that the only way a guest can get a room is to "enter the hotel as a new guest", and that all of the guests occupying a room in the hotel were initially a new guest at some point in the past.
• We make this unstated, but completely reasonable assumption explicit to prevent "magically appearing cigar" loopholes.
• 2 - As stated, this creates a number of problems.
• In the context of the Grand Hotel "paradox", all rooms are "occupied", but what "occupied" means is not well defined. Does a guest leaving a room to go to the room of another guest make the room unoccupied?
• In the context of the thought experiment, it creates a condition where a guest goes from their room to another guests room, leaving the original room unoccupied — if someone knocks on the first guests door at that point, no one will be there to respond.
• This seemingly trivial detail probably dooms the entire thought experiment just by itself.
• 3 - This really should be cleaned up. As worded, it allows for the possibility that "room 3" is unoccupied and happens to contain a cache of cigars. The thought experiment should clearly state what happens when a room in unoccupied.
• It is assumed that there are no magical caches of cigars anywhere since the thought experiment never mentions such an important detail.
• By most standards, if a room is unoccupied, it can't be entered.
• What happens when a guest comes across an unoccupied room should be clearly stated. Is the guest required to keep trying rooms until an occupied one is found? This seems to be implied and it is assumed.
• 4 - This establishes that an infinite number of rooms can be checked. This is analogous to "recursion".
• 5 - This makes two critical and fatal assumptions-
• 1) a guest "guaranteed" to return with cigars.
• Although we can make the reasonable claim that "there are no cigars in the hotel", the thought experiment requires that when a guest doesn't have the required number of cigars (which they are guaranteed not to since they can not have any cigars by definition), they need to "go to the next guests room" and ask that guest for cigars. No limit is specified on how many times this can take place.
• 2) a guest is "guaranteed" to return.
• Because the thought experiment deals with an infinite number of rooms, and no "bounding conditions" are given, there is no guarantee that a guest will ever actually return. Fundamentally, this is known as the halting problem.
• 3) Because of 2, and the fact that the thought experiment deals with an infinite number of rooms, there is no guarantee that when a guest is asked for cigars by another guest, and that guest leaves their room to go knocking on doors, that they will ever find another occupied room- the possibility exists that a guest is infinitely unlucky.
• This has important ramifications- you can even change the conditions so that all the other guests have an infinite number of cigars, it still wouldn't guarantee that a guest is actually going to find an occupied room.
• 6 - This implies that having an infinite number of rooms is all that is required to create the possibility that cigars will eventually be found, even though no guest is permitted to bring cigars in to the hotel.
• Not only is this absurdly ridiculous — how, exactly, does having an infinite number of rooms create the possibility that there are cigars in the hotel? — the halting problem proves that there is no deterministic guarantee that even if there were cigars to be found, that a guest would ever return with some cigars in a finite amount of time.
• 7 - The only thing that is "shown" is that if, and only if, the condition of a guest having n+1 cigars ever "happens", then all the guests at n and below can have a cigar.
• This is not even remotely close the same thing as proving that this condition is ever going to happen. You can not claim to prove that guests are smoking in the Grand Hotel if you can not even prove that a single guest can smoke a cigar.
• Even if the example was restated to address the fundamental flaw of where the cigars come from, it still wouldn't prove that guests can smoke cigars in the hotel since a guest may have to wait an infinitely long time for another guest to return with cigars. The halting problem proved long ago that it is impossible guarantee that a guest would only have to wait a finite amount of time for their cigars. A proof that guests can smoke a cigar requires a guarantee that they only need to wait a finite amount of time- the fact that the cigar is being smoked requires that some event in the future has come to pass, and it can not come to pass if it never arrives.
• Just about anything can be proven true if you're allowed to wait an infinite amount time — if you don't like the result, you only need to claim that if you wait a little bit longer, the desired result will happen. It is not uncommon for "proofs" about what will happen in the future to be predicated on unprovable axiomatic statements: Gödel's incompleteness theorems
• 8 - The fact that it is never proven that any of the guests have a cigar is sufficient to prove that none of them do.
• Technically, if we accept the proof, the proof proves that a guest has a cigar — in order for the guests to be smoking a cigar, there has to be one, and by induction, a valid proof that a guest is smoking a cigar proves that at least one guest has a cigar (the one being smoked).
• My guess is that the author meant to say that it is never proven that a guest brought a cigar in to the hotel — just because you're smoking a cigar doesn't necessarily mean that a "guest broke rule" and brought one in.
• Yet the thought experiment provides no other way for guests to obtain cigars.
• Since only a single way is provided for cigars to enter the hotel, and it is specifically forbidden for any guest to bring a cigar by the way described... if you want to claim that guests are smoking cigars, you must accept the fact that guest(s) brought cigars in to the hotel, which for most purposes proves that guest(s) brought cigar(s) in to the hotel (i.e., you can't get there from here without going through the forbidden zone).
• It is unstated, but strongly implied, that having an infinite number of rooms is all that is required for cigars to exist in the hotel.
• This is tantamount to "pulling cigars out of thin air by magic."
• The fact that where the cigars come from is ambiguous, assuming one accepts that no guest brought a cigar in to the hotel, means that the "proof" depends on everyone accepting the fact that cigars appear out of thin air, like magic.
• There is no requirement on anyones part to accept the fact that cigars just magically appear. As a general rule, most people will not assume that cigars magically appear. The proof does not say that I'm required to accept that cigars magically appear.
• Not only does this means that nothing was proven, it disproves the entire hypothesis, or at least demonstrates that it is fundamentally flawed.
• I choose to assume that cigars can not magically appear out of thin air since I'm not required to. Now there are no cigars for any of the guests to smoke no matter how many rooms and other guests they check with.
• 9 - This would seem to offer evidence that the proof is flawed- the thought experiment provides no plausible way for any guest to get even a single cigar, which by definition would rule out the possibility of a guest having an infinite number of cigars.

Johne23 (talk) 20:03, 19 March 2010 (UTC)

What you have stated contains several major misunderstandings.

Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.

The statement of the paradox is an assumption in that at the current point in time every room in the hotel is occupied. You seem to be claiming that the statement of the paradox means that every room is always occupied, regardless of what the guests do. This is not what the paradox is about, and it is the main point in your misunderstanding.

The Grand Hotel Paradox is not supposed to be a "true" paradox. The point is that if the hotel was finite, then one could not accommodate more guests when every room is occupied, however this can be done when the hotel is infinite. You also seem to be suggesting that the hypothetical hotel has some relation to a real hotel, that just because the hotel can move guests at any time means it cannot be a hotel. While real hotels cannot do this, this is a hypothetical hotel and does not fall under the rules of real hotels. Similarly, the guests are hypothetical and just because no reasonable person would agree to the conditions does not mean that these hypothetical people won't. The entire paradox is hypothetical.

IMHO, this is not actually true as it is worded since it glosses over the fact that you can't ever cause the state of a room to become "unoccupied", though it is not clearly defined what qualifies as "unoccupied". It also hand waves the fact that it's possible your entire stay at the hotel will be 100% filled with moving from room to room, something most people would probably object to.

Another small, minor detail not really mentioned is the fact that checking out of the hotel would create an unoccupied room... or at least glosses over some of those "messy details" like requiring a room to instantaneously "vanish" once it becomes "unoccupied"... and the fact that you're pretty much screwed if you forget something important in the room and need to go back for it...

This has been covered by my earlier comments, as are all the sections up to the one quoted below.

The Grand Hotel Cigar Mystery section also has a number of fatal flaws, at least as it is stated in the article: ... This one is even worst than the original "paradox"- it puts forth a "mathematical induction thought experiment" that can be trivially disproven

The thought experiment is supposed to be disproven. You seem to have missed the point here as well.

In conclusion, your arguments have no basis as they are based on misunderstandings of what is intended. Awychong (talk) 05:26, 27 April 2010 (UTC)

Personally, AWchong, I think you're full of crap. The fundamental declaration is this - the hotel contains an infinite number of rooms, and each room is occupied. We are dealing with exactly one entity: an occupied room. The paradox does not claim "an infinite number of rooms, and an infinite number of guests" (two discrete entities), it claims "an infinite number of occupied rooms". If it claimed "an infinite number of rooms, and an infinite number of guests" then adding additional guest would be no problem, since infinity is undefined and adding to the set of infinity remains undefined. However, that the rooms are qualified as 'occupied', means there are no unoccupied rooms, so no more guests can be accommodated.

Also, I can't see where your getting the idea that "it doesn't say every room is always occupied". Actually, it does. It says quite clearly "countably infinitely many rooms, all of which are occupied". No where does it even imply that there is a time constraint to this condition.

This is basic set theory, it's actually pretty simple. —Preceding unsigned comment added by Zencycle (talkcontribs) 19:01, 19 November 2010 (UTC)

Actually, "all of which are occupied" does not mean "all of which are always occupied". I think you're running into confusion here because you're thinking of a countably infinite number of guests arriving one by one, when in this thought experiment all moves are made simultaneously.

The heart of your misunderstanding, though, is with countably infinite sets. I believe you are thinking of the infinity from calculus (a number bigger than any finite number), or the infinity from theology (an incomprehensible, never-ending hugeness), instead of that of discrete mathematics (a set of objects where every object corresponds to one and just one unique positive integer). Finite sets, like you're used to working with, just behave so much differently than infinite sets. It'd be like trying to comprehend some of the weirder quantum physics theories using only Newtonian physics.

173.23.82.190 (talk) 06:55, 22 September 2013 (UTC)

I have removed the following sentence at the end of the cigar mystery section

"However, unless it is shown that in the limit there is a guest with infinitely many cigars, the proof is flawed regardless of whether or not cigars are allowed in the hotel."

This is meaningless as there is no such thing as a "guest in the limit": a limit is a topological concept and here it is used out of place. The best thing I can think of while trying to inject some sense into this sentence is to consider a countable poset of guests of order type (omega + 1) - that is, the set of natural numbers extended with a sigle element (called "omega") greater than every natural number. It still would not follow that every guest can smoke a cigar, no matter how abundantly cigar-laden guest omega is because nobody is ever going to ask him for a cigar, since guest N only asks guest N+1. I think. — Preceding unsigned comment added by 93.65.38.28 (talk) 15:00, 25 August 2011 (UTC)

## It's inconsistent, not a paradox

After some further thinking on this, this isn't a paradox at all. Whether or not there is a paradox depends critically on how one defines the Grand Hotel. I'm of the opinion that any formulation of the Grand Hotel is either logically inconsistent or simply "shifts the paradox" somewhere else, and therefore not a paradox at all.

For example, a slightly more formal definition of the hotel might be something like:

Let P be the set that contains people. The set P contains a finite number of members. The empty set is not a member of P.
We define "hotel room occupied" as a room that contains a single member from the set P. To be clear, a hotel room can not be occupied by the empty set.
The hotel contains an infinite number of rooms.

This is the Grand Hotel, where there are an infinite number of rooms, and each room is occupied, but the hotel can still take in new guests!

Now, let us say that the set P is {1}, that is to say- you're the very first person ever, and the only one.
You check in to the Grand Hotel.
As promised, the room you check in to is occupied, and the current occupant is shuffled off to another room, since there's an infinite number of them.

The obvious question is... who was the occupant in the room that got shuffled off? The answer is: it was you. No one ever explicitly said it couldn't be you, after all. Oh wait, is it a problem for there to be two of you when there is only one of you? The Grand Hotel is not responsible for paradoxes you create yourself.

If you restate the problem so that the occupant of the room isn't you, you will most likely create a logically inconsistent model that, for all practical purposes, boils down to:

• All true statements are false.
• All false statements are true.

These cases demonstrate the 'paradox', by which we mean not that it is contradictory, but rather that a counter-intuitive result is provably true: The situations "every room is occupied" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.

Anyone who accepts that the Grand Hotel is provably true also accepts that it is provably false because all true statements are false. In general, though, most people don't put a lot of weight behind provably true statements that are the result of inconsistent logic. —Preceding unsigned comment added by Johne23 (talkcontribs) 10:34, 21 March 2010 (UTC)

See my response to the above part. The hotel is not necessarily full all the time. Awychong (talk) 05:34, 27 April 2010 (UTC)

## Errors in the article

Some find this state of affairs profoundly counterintuitive. The properties of infinite "collections of things" are quite different from those of finite "collections of things". In an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is as many as the total quantity of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets, this cardinality is called ${\displaystyle \aleph _{0}}$ (aleph-null).

Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers - those numbers which can be written as a quotient of integers - contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: There is a bijection from the naturals to the rationals.

This wrong, very wrong. The set of natural numbers, ${\displaystyle \mathbb {N} }$, is countable, and contains either all the numbers > 0 or > 1 (depending on your definition), such as { 1, 2, 3, ... }. The set of natural numbers continues on in to infinity. The set of integers, ${\displaystyle \mathbb {Z} }$, is countable, and contains all of the numbers in ${\displaystyle \mathbb {N} }$, along with all the "negative integers", such as { ..., -3, -2, -1, 0, 1, 2, 3, ... }. The set of integers "continues backwards infinitely and forwards infinitely from 0". Even though both ${\displaystyle \mathbb {N} }$ and ${\displaystyle \mathbb {Z} }$ contain an infinite number of numbers, ${\displaystyle \mathbb {Z} }$ contains "even more". The set of natural numbers, ${\displaystyle \mathbb {N} }$, is a subset of ${\displaystyle \mathbb {Z} }$, which is to say, for any number in ${\displaystyle \mathbb {N} }$, it also exists in ${\displaystyle \mathbb {Z} }$. There reverse is not true: -1 does not exist in ${\displaystyle \mathbb {N} }$, but does exist in ${\displaystyle \mathbb {Z} }$.

These two sets of numbers have the following property: Given two numbers, say 2 and 3, there is no number in either set that is both > 2 and < 3. This is one of the properties that "makes them countable" is that you can provably distinguish between any two of them.

The set of rational numbers, ${\displaystyle \mathbb {R} }$, is uncountably infinite. The article incorrectly states that the set of rational numbers is countable. The set of integers, ${\displaystyle \mathbb {Z} }$, is a subset of the set of rationals, ${\displaystyle \mathbb {R} }$. For every number that exists in ${\displaystyle \mathbb {Z} }$, it also exists in ${\displaystyle \mathbb {R} }$. But, just like there were numbers in ${\displaystyle \mathbb {Z} }$ that were not in ${\displaystyle \mathbb {N} }$, there are numbers in ${\displaystyle \mathbb {R} }$ that do not exist in ${\displaystyle \mathbb {Z} }$.

Using our example of 2 and 3 from before, one of the properties of the set of rational numbers is "Are there any numbers that are > 2 and < 3?" Yes, 2.5 for example. And one can keep finding smaller and smaller subdivisions between any two numbers- in fact, between the numbers 2 and 3 there are an infinite number of numbers. This is why the set of rational numbers is uncountable- Starting at 0, and counting to 1, you can never actually reach 1.

Whether or not this paradox "makes provably true statements" depends entirely on how one defines the problem. If we limit ourselves to just considering the set of natural numbers, ${\displaystyle \mathbb {N} }$, (which seems perfectly reasonable with all the discussion of "countably infinite" and the fact that people are finite, with an undeniable "first person" at some point in the past), it is trivial to prove the Grand Hotels claims that every room is occupied, but it can still accommodate guests because it has an infinite number of rooms, are impossible, and therefore false. The short proof is this: Using the set of natural numbers, ${\displaystyle \mathbb {N} }$, where N > 0, I can provably claim to be the first person in the hotel: 1. My proof is absolute and irrefutable, because by definition I am the first person to check in: No one has checked in before me, and no one is allowed to check in until after I check in. I check in to the hotel, they show me to my room, which they claim is occupied. The hotel has an infinite number of rooms to choose from, and they are free to choose any one they want. When we open the door, the room must be occupied by someone > 1 and < 2. Since we can irrefutably prove that there are no numbers between 1 and 2, the room can not be occupied. In fact, we'll even allow the room to contain "0", but this would be tantamount to proof of an unoccupied room that can not be repudiated, otherwise I would have non-repudiable evidence that there exists a number between > 1 and < 2, even though no such number could possibly exist.

Any definition of the Grand Hotel which makes Provably True statements is going to depend on some kind of "loop-hole" in order to be true. For example, it's perfectly valid from a mathematical perspective for the hotel to populate it's room with numbers from the set of integers, ${\displaystyle \mathbb {Z} }$. However, once you "allow" this, you sort of concede to "not playing fair". In fact, once you go this route, there's no need what-so-ever to have an "infinite" number of rooms. You can have The Humble Hotel, with just one room, and even when the guest currently using the room checks out, the hotel can still take a new guest, even though it's "provably" occupied.

Even under "looser" definitions of the hotel, I can still demand that the only room that I will accept as "being occupied" is the room that is occupied only by me, the countable version of me. With an infinite number of rooms, there must exist a room that meets my standard. I can "demand" the paradox be resolved, and whatever the outcome is, it is non-repudiable from that point on (we'll define it as axiomatically true from that point on- up until that point in time, no one could prove it one way or another, but after that point, someone has Proof that settles the question). Evidence in my favor would be: the hotel was full and had to turn me away, or the room I was assigned was unoccupied, or the hotel was found to be "cheating", etc . The hotels proof would be showing me the room that I am in, even when I'm not in it. From that point forward, the hotel can't claim "If I'd only waited another minute..." or "Our contract says that even though we have an infinite number of rooms, that one specific room that you wanted, you can't ask for it, but... that doesn't mean we're full! You just have to accept one of our other rooms, and there's an infinite amount to choose from, you just can't have that one room." —Preceding unsigned comment added by Johne23 (talkcontribs) 17:46, 23 March 2010 (UTC)

You clearly do not understand the concept of "countable". A set is countably infinite if there exists a one-to-one correspondence between the natural numbers and the members of the set. See Countable_set. The correspondence does not have to be in ascending order, as you seem to be suggesting. In fact, this paradox is supposed to demonstrate properties of countably infinite sets.
In addition, you claim that the rational numbers are not countable. The article I linked to earlier demonstrates a proof that they are. As an aside, you seem to be using the wrong symbol for the rational numbers - ${\displaystyle \mathbb {R} }$ is the set of real numbers, while ${\displaystyle \mathbb {Q} }$ is the set of rational numbers.
For my response to the rest of this section, see my response to "This article is obviously flawed" earlier on this talk page. Awychong (talk) 05:41, 27 April 2010 (UTC)

## Clarification

To be clear, it is only possible to accommodate a new occupant in the hotel if all the guests move to the next room correct? Just sending a new guest into the hallway to find a room for himself would prove a futile task since he would wander the halls forever and never find an unoccupied room.Ziiv (talk) 19:24, 7 April 2010 (UTC)

The management are able to specify an exact room number for each guest to move to. (We ignore the congestion on the corridors, and provide in-corridor transport at near-infinite speed for those who would have to walk a long distance.) Dbfirs 12:15, 15 December 2012 (UTC)

## Cardinality Cleaners

I read about this from somewhere, can't remember where, and wonder whether it could be added in the article. It's a related thought game, where the company that cleans the Grand Hotel is introduced. The Cleaners also demonstrate that there are infinities of different sizes.

The Cardinality cleaners are responsible for cleaning the Grand Hotel. The Cardinality Cleaners have an infinite amount of employees, each having a distinct set of rooms to clean so that for each set of rooms in the Hotel, there is exactly one Cardinality Cleaner. Every day the Grand Hotel's administration calls the Cleaners requesting a set of rooms to be cleaned - the worker responsible for that particular set is sent to work.

The Cleaner's boss is satisfied with his workers and decides to book them a weekend stay at the Grand Hotel. Unfortunately, the Grand Hotel is too small for the Cleaners, as one-to-one pairing can't be established between counting numbers (rooms) and sets of counting numbers (cleaners). They have to check in to a bigger hotel (infinite, with each real number used as room number), which in turn is cleaned by a bigger cleaning company, which needs an even more humongous hotel... 88.112.51.212 (talk) 17:54, 7 January 2011 (UTC)

I don't see any great point in adding this to the article. The point of the Hilbert Hotel is to connect some of these concepts from basic set theory with intuition. The learner needs to figure out to ignore some stuff, like "where would you put a hotel that big" or "how can a guest move from room 10^100 to 2*10^100 in the same time another guest moves from room 1 to 2".
In the case of the cleaners, though, I think the stuff you have to ignore has gotten distracting. For example, why exactly would you assign, to each set of rooms, a single worker to clean them, when that means that each room will end up getting cleaned ${\displaystyle 2^{\aleph _{0}}}$ times? I'm just not sure the story is worth the trouble any more. That wouldn't matter if this were a standard follow-on to the original story, but I kind of doubt that it is. --Trovatore (talk) 18:37, 7 January 2011 (UTC)
Sorry if I was confusing, but every day only one cardinality cleaner is sent to work - if rooms 4, 16 and 200157 need cleaning, the cleaner specializing in that set goes to work, and the others slack off. Nevertheless, this part of the story is mainly flavor text. The very idea is to demonstrate that there are different sizes of infinities, as, no matter how the Cleaners are stuffed in, it's always possible to find at least one who hasn't got a personal room. However I'm not insisting that this should be added, it's up to your judgment. 88.112.51.212 (talk) 09:09, 8 January 2011 (UTC)
Oh, I see. That's sort of cute, I guess. If it can be found in the sources it's at least plausible that it might be mentioned. Still not sure though — might be just more confusing. --Trovatore (talk) 06:14, 9 January 2011 (UTC)
I don't see why Cardinality Cleaners need to employ an infinite number of cleaners for each room in the hotel! Why must their cleaners be uncountable? ... (later) ... If the company wanted to book a separate training room for each set of cleaners, then I agree that the Hilbert Hotel could not accommodate the (${\displaystyle \aleph _{1}}$)≠?) ${\displaystyle 2^{\aleph _{0}}}$ sets, but this is not the same as being too small to book one cleaner into each room. Dbfirs 18:55, 26 November 2012 (UTC)
No one has mentioned ${\displaystyle \aleph _{1}}$.
Apparently there are ${\displaystyle 2^{\aleph _{0}}}$ cleaning employees, each of whom specializes in a single subset. Why they so massively overstaff is not explained; they just do. There's nothing mathematically wrong with the story — the question is whether it's worth including in the article. My feeling right now is that it's probably not, even if sourceable, because it's too hard to explain and doesn't add enough value. But if it were a standard follow-on to the original paradox, we'd probably include it. --Trovatore (talk) 19:41, 11 February 2013 (UTC)
Sorry, ${\displaystyle \aleph _{1}}$ might (or might not) be an over-estimate, but I would just line them up and fire them one by one! Dbfirs 20:54, 11 February 2013 (UTC)
"Might (or might not) be an under-estimate". I assume that's what you meant to say. --Trovatore (talk) 20:59, 11 February 2013 (UTC)
Yes, under. I agree that the article is tidier without Cardinality Cleaners. Dbfirs 21:33, 11 February 2013 (UTC)

## Sources

There appears to be a lack of sources, despite the abundance of external links. To call attention to the fact, I have added the unreferenced tag to the page. --159.91.127.30 (talk) 05:02, 12 September 2011 (UTC) (forgot to sign in on the first post, but quickly edited it in afterward)

## New Solving for Infinite Buses

I have added a new piece on what to do when an infifnite number of coaches arrives, which I believes works better than the others, because it easily includes every number, and every person, only once, but still includes every room. I did not see this anywhere, but rather developed it from the existing proof that all of the rationals can be expressed as the integers. In a few steps, I drew this conclusion out. Does this conflict with the "new material" rule of wikipedia? Please tell me if it does, and I will do something about it. — Preceding unsigned comment added by 5^.5*.5+.5=phi (talkcontribs) 02:23, 29 October 2011 (UTC)

I think something close to this is said on the page about pairing functions. On the graph, you can see all of the numbers displayed. The first row is the triangluar numbers. The next row adds one. The next row adds two and starts with the second number. In this way, every row represents a coach of people, and the first one represents the guests already in the hotel. 5^.5*.5+.5=phi (talk) 19:09, 1 November 2011 (UTC)

## Luggage

But what if all the luggage was really heavy. Would the servants build an infinite amount of muscle? — Preceding unsigned comment added by 85.229.199.220 (talk) 09:18, 11 August 2012 (UTC)

## Hilbert doesn't understand the difference between an infinite number of rooms and as many rooms as imaginable

For example, there are an infinite amount of numbers between 1 and 2 yet a set of numbers containing 1-2 and 4 (first set) contains more numbers than the set of numbers simply containing the numbers 1-2 (second set). This is a fact since we know that the first set contains all the numbers of the second set and the second set does not contain all of the numbers of the first set. So let's say the hotel has an infinite number of filled rooms numbered with all the numbers between 1 and 2, it would not be possible to hold the one extra person even though the hotel has an infinite amount of rooms since the number of people exceed the number of rooms even though they are both infinite value. — Preceding unsigned comment added by 192.245.87.10 (talk) 11:33, 22 October 2012 (UTC)

This article needs some serious rewriting, if only to make it abundantly clear that "paradox" is not used in the usual sense (as it appears from the talk page, it does not stress the point enough), and to clear up misconceptions like this and the ones exemplified by Johne23's comments above. The sets {x|x∈R}∪{4} and {x|x∈R} do have the same cardinality, even though they may appear that the first has "more" than the other (they both have the cardinality of the continuum. Your argument (that one does not contain all the numbers of the other) is flawed; the even numbers do not contain all the natural numbers, yet the two sets have the same cardinality. You cannot number the rooms of the hotel with reals between 1 and 2, because they are uncountable, as opposed to the rooms. I don't know what you mean by "as many rooms as imaginable", though. -Anagogist (talk) 17:31, 21 November 2012 (UTC)
The word paradox is in fact used in the usual sense. The usual sense of paradox is "apparent contradiction", which is exactly what this is (of course once you're used to it it no longer appears like a contradiction, but to many naive observers, it does). It's true that the word is also used sometimes to mean "genuine contradiction", but if you claim that's the usual sense, I disagree. --Trovatore (talk) 20:55, 21 November 2012 (UTC)
After taking a look at List of paradoxes#Logic and Category:Paradoxes, I have to retract my statement and agree with you. The article, then, should not make it abundantly clear that "paradox" "is not used in the usual sense" as I said above, but rather underline that it is not the contradictory type of paradox. I edited the intro to reflect this, and I hope it makes the distinction clear. -Anagogist (talk) 14:25, 22 November 2012 (UTC)

## Hype vs. payoff

The entire cigar mystery is too much hype (the title), too many words, and too trivial to be worth the investment for all the people who'll end up reading it. It's not sponge-worthy. Let's get rid of it or abridge it.

173.25.54.191 (talk) 02:57, 1 December 2012 (UTC)

I think it's a useful warning about the dangers in mathematical induction. Where's the "hype"? Dbfirs 09:05, 1 December 2012 (UTC)

## Infinitely many coaches

In the section about infinitely many coaches, there is this sentence: For each of these methods, consider a passenger's seat number on a coach to be ${\displaystyle n}$, and their coach number to be ${\displaystyle c}$. The hotel may be considered "Coach #0." It isn't at all clear to me what this sentence is telling us. There is no explanation of what to do with the numbers n and c. TomS TDotO (talk) 14:03, 7 August 2013 (UTC)

The subsections use those numbers, though it may be less clear in the first subsection (the prime powers one). ± Lenoxus (" *** ") 01:06, 14 August 2013 (UTC) Wait, I see you've fixed it up, thanks! ± Lenoxus (" *** ") 01:08, 14 August 2013 (UTC)

## Infinite layers of nesting

This section is very confusing and/or misleading. Suppose that there are infinitely many ships as described in this section. Label each ship with a digit 0 - 9, in a way such that the subships of any given ship are all labeled with different numbers. Then each passenger can be labeled with an infinite sequence of digits, namely the labels of the ships the passenger is in. BUT, for any two passengers, these sequences will agree for all but finitely many digits. This is because, living in physical space, they are presumably a finite distance apart. Since the set of sequences which agree with a given sequence except for finitely many digits is countable, the passengers would be able to fit in Hilbert's hotel.

I see what the section is trying to go for, but it doesn't seem to be doing a very good job. I think it would be better if the nesting were downwards rather than upwards, so that each ship has ten subships, each of those has ten subships, etc., and in the intersection of each sequence of ships is a passenger. This would be closer to how uncountable sets (e.g. the Cantor set) are usually talked about in a mathematical setting. Unfortunately, it runs into the problem that each passenger is infinitely small. It seems that this problem is unavoidable, because if each passenger had positive volume, it would contradict the fact that the uncountable union of positive measure sets cannot be sigma-finite. David9550 (talk) 19:24, 12 September 2013 (UTC)

## The most mysterious paradox is missing...

When I learned about this paradox in the Technion, it had an interesting punch-line, which for some reason does not appear here. Here is the version I learned:

• At 23:00, a countable number of guests arrive and fill the rooms.
• At 23:30, another countable number of guests arrive. The existing guests move to the even-numbered rooms (1->2, 2->4, etc.), and the new guests fill the odd-numbered rooms.
• At 23:45, another countable number of guests arrive, and again the existing guests are moved to even rooms and the new guests are put in the odd rooms.
• In a similar manner, whenever the time to midnight decreases by half, a new group with a countable number of guests arrive and fill the odd rooms, while the existing guests are moved to the even rooms. Up until midnight, a countable number of such groups arrive.
• At midnight, the manager of the hotel decides to see how the guests in his hotel are doing. So he goes to room number #1, and... the room is empty! Why? Because any guest that has been in room #1 at time T before midnight, has moved to room #2 at time T/2 before midnight.
• From a similar argument, room #2 is also empty, and so are room #3, room #4, and so on.
• It turns out that, by midnight, an infinite number of groups with an infinite number of guests has arrived to the hotel, but, all rooms are empty!

This is not only a paradox - it is a mystery. A very scary mystery. --Erel Segal (talk) 21:02, 14 December 2013 (UTC)

That's an interesting connection to the Ross-Littlewood paradox. Do you have any sources? Paradoctor (talk) 03:35, 15 December 2013 (UTC)