Talk:Hilbert's paradox of the Grand Hotel

message: 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)
thread_title:  infinity + 4 = ? 

message: 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example Hilbert gave in a lecture, claiming in turn to have got this story from an R. Courant who was working on a then unpublished book to be called "The Complete Collection of Hilbert Stories".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)
replies: text: Possibly references at Wolfram's Mathworld article will be helpful...  See http://mathworld.wolfram.com/HilbertHotel.htmlCiaPan 16:18, 17 July 2006 (UTC)
thread_title: Source

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Subtracting infinities 

message: "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)
replies: text: 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)
replies: text: 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)
thread_title:  Infinite sets 

message: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title:  The Cosmological argument 

message: 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)
replies: text: Fixed by creating a disambiguating link to coaches. --kostmo 01:48, 11 September 2007 (UTC)
thread_title: "Coach" meaning "bus" is a UK-specific idiom

message: 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)
thread_title:  axiom of choice 

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

message: I added a link here, 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)
thread_title:  External link removed 

message: 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:
# 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.
# 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".
# 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.
# 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."
# 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.
If this is indeed how the Grand Hotel "paradox" was put forth, then the "paradox" is specious.  It is disingenuous to claim that this is a "paradox" when it is predicated on conditions that no reasonable person would agree to, not to mention that they are unstated.  It's probably also illegal in most places to rent an already occupied room to a new guest.  If this is really the complete "paradox", I would argue that this is not a true paradox, but actually a half-truth.  The "paradox" requires that one assumes that the "the expectations of what it means to stay at a real world hotel are the same for the Grand Hotel" when they very clearly are not.  Compare the Grand Hotel Paradox to the Birthday Paradox, which is not a "true" paradox, but in my opinion, is certainly more of a "paradox" than the Grand Hotel.  To me, the fact that the paradox requires that guests never get an unoccupied room, and the hotel can demand that you move to a new room whenever they want, and as often as they want, makes it pretty hard to claim that it is a "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)
thread_title: This article is obviously flawed

message: 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.
This article claims:
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 (talk • contribs) 10:34, 21 March 2010 (UTC)
replies: text: See my response to the above part. The hotel is not necessarily full all the time. Awychong (talk) 05:34, 27 April 2010 (UTC)
thread_title:  It's inconsistent, not a paradox 

message: 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)
replies: text: 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)
thread_title:  Clarification 

message: 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)
thread_title:  Sources 

message: 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 (talk • contribs) 02:23, 29 October 2011 (UTC)
thread_title:  New Solving for Infinite Buses 

message: 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)
thread_title:  Luggage 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Hilbert doesn't understand the difference between an infinite number of rooms and as many rooms as imaginable 

message: At least two commentaries on this page said "Hilbert didn't understand blah blah blah".   Hilbert was the leader of the the world's mathematical community in his time, and his mathematical reputation was so formidable that, according to Scientific American, Einstein feared Hilbert might work out the General Theory of Relativity before Einstein did. Hilbert was no fool.
Hilbert was trying to demonstrate that the usual rules of arithmetic don't work when applied to infinity  (something pointed out 300 years earlier by Galileo) and he succeeded.  People who don't follow his line of reasoning then claim the problem with the rules is Hilbert's fault.
I heard of the Infinite Hotel Problem in my college math class, AFTER a discussion of Infinite Set Theory, and understood perfectly well what Hilbert was doing. 2001:558:6011:1:11B7:55ED:94CC:AC2C (talk) 21:23, 28 April 2018 (UTC)
thread_title:  Hilbert Knew Perfectly Well What He Was Doing 

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

message: 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 n, and their coach number to be 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)
replies: text: 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)
thread_title:  Infinitely many coaches 

message: 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)
thread_title:  Infinite layers of nesting 

message: 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)
replies: text: That's an interesting connection to the Ross-Littlewood paradox. Do you have any sources? Paradoctor (talk) 03:35, 15 December 2013 (UTC)
thread_title:  The most mysterious paradox is missing... 

message: This analogy is just plain stupid. If you tried to construct such a system of spaceships in R^3 and represent each person as a point, then clearly each person is an isolated point, and you can only have countably many isolated points in R^3, so total number of people is countable. If you think about it, each person correspond to a sequence in 10^omega thats eventually constant, and this subset of 10^omega is clearly countable.  — Preceding unsigned comment added by Money is tight (talk • contribs) 23:28, 3 February 2016 (UTC)
replies: text: Assuming AC, a countable number of countable infinities is indeed countable. All of those infinite layers can indeed be accommodated, but there's no way to give a constructive example of how to do it because you can't prove two to the aleph-null is regular without some form of choice. (You can prove that aleph-two is not the countable union of countably many things, but not aleph-one; and if you're foregoing choice then the cardinality of the continuum need not be directly comparable with any aleph.) That said, you're missing the point of the exercise. This is an entirely fictional mathematical universe. Even the Universe we have access to does not admit infinitely many physical things (the part we are able to access, at any rate, which is essentially our 'Universe'). TricksterWolf (talk) 01:56, 16 March 2017 (UTC)
thread_title:  The infinite layers spaceship analogy is wrong 

message: Pick any paradox (with or without an wikipedia entry)! They all follow the same rules: First you create a seemingly possible situation. Like a hotel. Or a man with a mule. Or a race between Achilles and a tortoise. It must be something that people can imagine. Next, you create a rule or set of rules. Sometimes this is not obvious, but it is always done. For example: If you make changes in your hotel, you have to begin at the first room, not at the last. And also totally absurd rules like: "infinite is a countable number". And you don't have to clear a room first before you put in another guest. And you can take cigarettes from a room before you brought them in. Or for the race: you can't wait until the race is over to see who wins, you must check the state in ever smaller timeframes. If you look closer, all of these rules don't make any sense at all. They are the exact opposite of the seemingly possible situation, as they never appear in real life. Real life problems always have multiple solutions. For example: even if somebody forces you to choose between A and B, you can always choose not to choose. So the paraxdox is created by applying a set of rules, which are not based on logic or experience. Instead they are specially crafted to create the so called paradox. Most talk pages to paradoxes have a lot of arguments to show flaws in the paradox. But whenever a flaw is exposed, somebody will say this is not possible. It is against the rules which were set up. So for me, the paradox is that nobody sees this although it it obvious and a common property of all paradoxes.--TeakHoken79.203.243.199 (talk) 12:21, 25 March 2016 (UTC)
replies: text: The set of rules that you describe is called a model. It is not a mistake, but a way to study some particular aspect of reality or of a formal system (such as a mathematical theory). In the article I've linked above ant in this other one you can read about how models are used and why they are benefitial. Diego (talk) 17:18, 25 March 2016 (UTC)
replies: text:  The law of gravity is not "things fall down" but "matter exerts a force of certain intensity".  When the zeppelin stays up it's not because it's "breaking the rule" but because it's taking advantage of other forces that pull in other directions. 2001:558:6011:1:38F3:171E:261:EBEA (talk) 17:42, 10 July 2017 (UTC)
replies: text:  A bit of nitpicking: those 'other forces' actually push in other direction, not pull. --CiaPan (talk) 16:45, 31 December 2020 (UTC)
thread_title:  It is as wrong as any paradox 

message: Every room is occupied, but since the setup states that (unlike typical hotels) the hotel can rent out occupied rooms to new guests, there is no paradox or contradiction in saying that new guests can be accommodated.  — Preceding unsigned comment added by 153.204.75.136 (talk) 21:03, 11 February 2017 (UTC)
replies: text: This again? The setup does not state that. The entire purpose of moving all the guests around is to make one room unoccupied, which is given to the new guest. 2.25.149.66 (talk) 04:02, 28 February 2017 (UTC)
thread_title:  Simple Solution 

message: In the case where each person is identified with a finite string of digits, the result is countable. Cantor's diagonal argument relies on the ability to produce a counterexample string that has at least one digit per enumerated string. This cannot be done in the case where the addresses are all finite. Put differently, finite strings can be identified with a subset of the rationals (in a suitable base) and are thus countable.--Jasper Deng (talk) 01:04, 26 November 2017 (UTC)
replies: text: "If this variant is modified in certain ways, then the set of people is countable again. For example, suppose there were a largest ship, directly containing a finite (or countably infinite) number of both ships and people, and each of these ships in turn contained both ships and people, and so forth. This time, any given person is a finite number of levels "down" from the top, and thus can be identified with a unique finite address. The set of people is countable again, even if the total number of layers is infinite, because we do not have to consider an "infinitieth layer" in either direction." You could have say 9 ships and people  in larger ship and model it as the reals from 0 to 1. 11cookeaw1 (talk) 01:32, 26 November 2017 (UTC)
replies: text:  No. The number \frac{1}{\pi} corresponds to no person under this scheme. Put more formally, the set of all addresses of a given finite length is finite, and the countable union of countable (in particular, finite) sets is countable. If we relax to countably many objects at each level, the result is still countable: for any finite number of countable sets, their Cartesian product is countable. Since each address is a member of said Cartesian product for a given address length, the set of all addresses for objects at a certain level is countable, and because there are countably many levels, the same argument applies.--Jasper Deng (talk) 01:59, 26 November 2017 (UTC)
thread_title:  Finite-address case 

message: Once the article gets to the Infinitely many coaches with infinitely many guests each section, I think it starts to go off into the weeds a bit.  The key takeaway here is that any countable set of new guests can be accommodated.  I don't think we need to go into endless variations about what sets are countable.  Also, since there are no inline refs, it's not easy to determine which of these arguments come from the sources, and which ones are just someone's personal take (for example, the notion of "layers" of infinity, which is not any standard terminology that I'm familiar with, though what the author means is probably clear enough if you bother to work through it). --Trovatore (talk) 03:12, 4 May 2018 (UTC)
thread_title:  Repetitive exposition 

message: I recently added a section explaining how to incorporate the diagonal argument into the story of the hotel. This isn't original research, I came across the idea in the arxiv paper I cited. However, the edit got unrolled by  the reason given was "Unpublished ArXiv source; may not be considered reliable." While the bulk of the paper itself might or might not be reliable, the reader can check from the text I wrote that indeed the diagonal argument is thus incorporated into the story. The role of the citation is not to provide credibility but merely to give a original source for where the diagonal argument-addition was first seen. Is it wikipedia policy that, unless someone publishes the diagonal argument in this context, it cannot be added to the article? If so, it's a shame, given how concrete it makes the diagonal argument that many people find confusing at first encounter.
— Preceding unsigned comment added by  159.8.170.6 (talk) 13:32, 14 September 2018 (UTC)
replies: text: Wikipedia is an encyclopedia, not a sourcebook. We do not document first occurrences of ideas that may be good or bad. Instead, we report on what the experts have said (reliable secondary sources) about a topic. Unpublished work has not been vetted by the academic community and so can not meet that level of assurance. Wikipedia's definition of original research is very broad. An idea, no matter where it originates from, that is not backed up by a reliable secondary source, can be considered to be original research in this context. So, in answer to your question, yes it has to be published and moreover it has to be commented on in order to be eligible for inclusion in this encyclopedia.--Bill Cherowitzo  (talk) 17:51, 14 September 2018 (UTC)
thread_title:  Incorporating the diagonal argument 

message: I clicked on this Talk page expecting to see references to hallways, and I wasn't disappointed. This is a complete misunderstanding of the procedure involved. Try this:
An infinite number of convention-goers have emailed their preregistration to the Accommodations Chairman, who has emailed them all back with their room number. Then another preregistration comes in. The Chairman assigns this new applicant room #1, then emails all the others informing them that their new room number is now their original room number plus one. No hallways! 108.20.114.62 (talk) 18:35, 4 January 2019 (UTC)
thread_title:  no hallways! 

message: In the first line of the chapter "Prime powers method", it says: "Empty the odd numbered rooms by sending the guest in room i to room 2^i". This is technically correct, of course. But why 2^i instead of just 2i? The goal is to remove guests from all even rooms. And the easiest way to achieve it, is to move the guests to double their current room number. Using 2^i seems totally arbitrary to me. 
After clearing out all even rooms, the "Prime powers method" can be applied to find rooms for all the guests. Epaminaidos (talk) 14:57, 10 June 2020 (UTC)
replies: text: It's not arbitrary because that way, the occupied rooms are the prime powers, which was the apparent point of this.  What could be done is to simply omit the "Empty the odd-numbered rooms" bit if you think it's misleading.  Of course, this is all pretty crufty WP:OR and should probably be removed anyway, but ehh. –Deacon Vorbis (carbon • videos) 15:10, 10 June 2020 (UTC)
thread_title:  prime powers 

message: There is a confusion of language which creates an apparent paradox.
1. being full has no meaning with an infinite number of rooms.
2. having an infinite number of guests leaves no possibility of any new guests unless they are part of a separate infinite set of not-guests. 
3. A not-guest can become a guest if an empty room exists. But all rooms are by definition full and there are no empty rooms to convert a not-guest to a guest.  — Preceding unsigned comment added by Kripan (talk • contribs) 19:01, 7 November 2020 (UTC)
replies: text: Not sure I really follow exactly what you're trying to say, but in any case please review the talk page guidelines.  If you want to make a change to the article, you need reliable sources.  If you just want to argue about it, we're not supposed to do that here. --Trovatore (talk) 20:05, 7 November 2020 (UTC)
thread_title:  Not a false paradox but a non-existent one 

message: You are all missing the point. To get an infinite amount of new guests into their rooms you would need an infinite amount of receptions. And that would really cut into profit margins. I am afraid it's just not feasible.  — Preceding unsigned comment added by 2A02:810B:C63F:DF78:54E9:118B:787A:A150 (talk) 21:22, 1 December 2020 (UTC)
replies: text: I do indeed think that somebody is missing the point. --Trovatore (talk) 01:46, 2 December 2020 (UTC)

text: Lol! :-D -- 92.10.212.241 (talk) 19:54, 23 February 2023 (UTC)
thread_title: Not thought through

message: A. 
Hilbert’s Hotel
Hilbert’s Hotel is full. Guest X arrives. Every guest n moves intro room n+1 simultaneously. Guest X moves into room 1. 
X and all other guests are in a room.
B. 
It does not matter if an infinite amount of moves taken simultaneously or an infinite amount of moves taken one after another.
Hilbert’s Hotel is full. Guest X arrives. X moves intro room 1 and guest 1 moves out of room 1 simultaneously. Guest 1 moves into room 2 and guest 2 to moves out of room 2 simultaneously… After an infinite amount of moves taken, every guest n is in room n+1.
X and all other guests are in a room.
A works like B but one after another.
C. 
Hilbert’s Hotel is full. Guest X arrives. Guest X has a red hat and all guests in the hotel have no hat. X moves intro room 1 and guest 1 moves out of room 1 simultaneously. Guest 1 moves into room 2 and guest 2 to moves out of room 2 simultaneously… After an infinite amount of moves taken, ever guest n is in room n+1.
X with his red hat is in room 1 and all other guests are in a room.
B works like C but X has a red hat.
D. 
Hilbert’s Hotel is full. Guest X arrives. Guest X has a red hat and all guests in the hotel have no hat. X moves intro room 1 and guest 1 moves out of room 1 simultaneously. X gives guest 1 the red hat. Guest 1 moves into room 2 and guest 2 to moves out of room 2 simultaneously. Guest 1 gives guest 2 the red hat… After an infinite amount of moves taken, ever guest n is in room n+1.
X is in a room and all other guests are in a room. There is also no room with a guest, which has the red hat.
C works like D but with a red hat moving from room to room.
What is happening with the red hat? Is it completely unclear, what is happening after an infinite amount of moves taken, with the red hat? If the red hat still exists, then in which room should it be?
For me, Hilbert’s Hotel argument, every guest n moves into room n+1, is as saying E or F:
E.
Hilbert’s Hotel is completely empty. Guest X arrives. Guests X moves from room to room, starting by room 1. After an infinite amount of moves taken from X, every room in Hilbert’s Hotel is empty. There is no room where X could be in, every room n is free.
F.
There is a hotel with only two rooms. In room 1 there is a „uneven“-sign above the door. In room 2 there is a „even“-sign above the door. Both doors have a counter, counting +2 if a guest moves out (so the room number gets higher each time). Guest X moves into room 1, then moves into room 2, then back into room 3 and so on. After an infinite amount of moves taken by X, there is no even or uneven number, where guest X could be in. Is than the hotel empty and there are no more room numbers above the rooms, because there is no even or uneven number left.
*Note how the phrase „after an infinite amount of moves“ can work for B, C and D? In a physical world we can say after an infinite amount of time, every single guest has moved to room n + 1. Or getting to a non physical world we could say guests have no speed limit for moving . Guest 1 moves in 1 minute, guest 2 in 1/2 minute, guest 3 in 1/4, guest 4 in 1/8 minute and so on. After two minutes every guest n moved from room n to n+1. There are an infinite amount of smaller time periods needed to reach the 2 minutes (if there are an infinite amount of smaller time periods between two time periods). But after 2 minutes, an infinite amount of action happened. 
For me Hilbert’s Hotel is an unsolved mathematical topic. What is happening with the red hat, wen guests start to give the hat from one room to another. What is happening, when a guest takes an infinite amount of moves in an empty hotel. If it is not clear, what is happening to this single guest, moving an infinite amount of times in an empty hotel, how can we know what is happening when an infinite amount of guests move from room n to n+1 simultaneously?
Back to Hilbert’s Hotel, every guest n moves into room n+1 simultaneously. Let’s say every guest needs exactly 2 minutes to move. A drone is flying over the moving guests. At the first minute it is flying directly over guest 1 and pointing its laser at the moving guest 1. At the next 30 seconds on guest 2. At the next 15 seconds at guest 3 etc… After 2 minute, the drone would have pointed to each moving guest. There is no guest left, which the drone has not pointed to. But what is happening to the drone after two minutes. Did the drone disappear? How the drone could stop pointing on a guest? The drone would be like the moving guest in an empty Hilbert Hotel. What happens to the guest / drone after two minutes? Would the guest have been disappeared, because there is no possible room the guest could be in? 
  — Preceding unsigned comment added by TaineMccoy (talk • contribs) 20:45, 30 December 2020 (UTC)
thread_title:  Questioning Hilbert’s Hotel as a vertical paradox, where the counter initiative result is provably true 

message: I'd pay to see an edit that applies what about a number of mathematicians and several video essay creators on YouTube did, which is the diagonal method, in which an infinite number of people arrive, using infinitely-long identities, a list is created, the nth number or letter of the label is circled(leading to a diagonal line), the digits constructing a name, only for each one to be switched out for an entirely different digit, revealing one of the many people that might still be without a room, proving that while the hotel would be full of people from that group of people with infinitely-long names, there would still be an infinite amount of people of the same group missing, thus also proving that there are infinities greater than others. There could be a bigger hotel built that could house them via rooms labeled with both natural numbers and real numbers alike. Just ask YouTube username Hotel Infinity.
And it peeves me seeing people getting rid of my edits, especially if I find good citations that meet Wikipedia's standards. Bruh. -_- What ever happened to common sense?
I'd also pay to see a page dedicated to this theory that a YouTube mathematician created that serves as a "sequel" to Hilbert's paradox, and the arguments people, excluding fame hounds, would make in regards to YouTube username keystone's theory that ought to be well-known among mathematicians the world over. Kaden Bayne Vanciel (talk) 04:19, 18 March 2023 (UTC)
replies: text: The first proof you mention is explained at Cantor's diagonal argument. No need to pay! I think you are right that this article needs to explain for that most infinite cardinalities it's not possible to put a set of that size in the Hilbert Hotel. This is something that won't be obvious to the lay reader. As for your final paragraph, I don't know what ideas you're referring to. If they are legitimate mathematical arguments, then they are probably already covered in journal articles and books. If the ideas are only expressed on YouTube, and then only on one channel, then it's dubious that they should be treated as knowledge. MartinPoulter (talk) 16:42, 19 March 2023 (UTC)
replies: text: Here's the video: https://www.youtube.com/watch?v=g02KJ4kxldA Kaden Bayne Vanciel (talk) 02:38, 21 March 2023 (UTC)
replies: text: Well that was 6 minutes of my life I'm never getting back.  Overall, a poor overview with some confused mathematical ideas.  Not that it really matters...material should be added based on reliable sources, not some random dude on Youtube.  There's already a ton of unsourced, iffy stuff in the article, and if I had the stomach for it, I'd take a machete to it, but I'm not sure that I do.  35.139.154.158 (talk) 02:58, 21 March 2023 (UTC)
thread_title:  Applying Some Ideas to the Paradox 

message: I'm very new to contributing, so I was hoping for some further clarification on why my edit from March 8 was deleted. The comment from the user who deleted it marked it as non-notable. How is that determined? I added a reference to the principle in a published work of fiction, which feels like a notable reference. Was it because I failed to provide a citation? The Hitchhikers Guide to Existentialism (talk) 15:16, 31 October 2023 (UTC)
replies: text: That's...pretty late to bring it up.  I can't speak for the person who reverted you (you should ask them specifically), but "notable" refers to topics for articles themselves, not pieces of information within them.  Maybe "noteworthy" is a better word here.  I agree that it shouldn't be mentioned, which might be confusing given the laundry list already there.  But I'd also say the solution is to remove them all, which I've just done, not add more.  See WP:IPC, an essay about stuff like this.  35.139.154.158 (talk) 15:49, 31 October 2023 (UTC)
replies: text: I wasn't notified it was removed until I opened the article again today. The Hitchhikers Guide to Existentialism (talk) 16:25, 31 October 2023 (UTC)
thread_title:  Deletion of edits 

message: I suggest that the title of this article should change to "Hilbert's grand hotel paradox" or "Hilbert's hotel paradox". It sticks out like a sore thumb when compared to other pages of paradoxes (e.g. https://en.wikipedia.org/wiki/List_of_paradoxes). I looked into how to change the title and it seems I cannot find this illusive move button or I do not have the permissions. Lucian Chauvin (talk) 04:20, 4 January 2024 (UTC)
replies: text: Given that this article has been at this title for many years, I would recommend that you not try to move it unilaterally, even if you have the permissions, as that is supposed to be for "uncontroversial moves" and I predict controversy for this one.  You can nominate the article to be moved and try to get consensus.  The procedure is explained at WP:RSPM. --Trovatore (talk) 07:14, 4 January 2024 (UTC)
thread_title:  Title Change 

message: The date of January 1924 here is cited to this paper, which is a bit confusing to read. It explicitly says "January 1924" on a couple of occasions including the abstract. However, it later refers to "the winter semester 1924-1925", "A few months later ... on 4 June 1925". It has a section titled "3. January 1925: The birth of Hilbert's hotel" with the comment "This is what Hilbert had to say about his hotel in January 1924". One of those years must be a typo.
I do not have access to the 2013 book of Hilbert's lectures that is cited, but the preview pages seem to support the 1924/25 dates, which would strongly support this being January 1925.
I'll change the article but given the cited paper has the wrong date up-front, thought it best to leave a note here explaining. Andrew Gray (talk) 18:33, 18 January 2024 (UTC)
thread_title:  1924 or 1925?