Talk:Hilbert's paradox of the Grand Hotel

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[edit] Really?

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)

[edit] 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.

[edit] Old comments

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)

[edit] 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)

[edit] 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)

[edit] 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
–adjective
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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] (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)

[edit] 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, \textstyle \left | \mathbb{N} \right | \notin \mathbb{N} . Maelin (Talk | Contribs) 07:58, 11 February 2007 (UTC)

[edit] "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 (talk • contribs) 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)