-3

I am learning the fundamentals of mathematics.

A bit background: This article says that "The mathematical paradox about infinite sets" envisages Hilbert's Grand Hotel:

"...a hotel with a countable infinity of rooms, that is, rooms that can be placed in a one-to-one correspondence with the natural numbers. All rooms in the hotel are occupied. Now suppose that a new guest arrives – will it be possible to find a free room for him or her? Surprisingly, the answer is yes."

My issue is around this two statements:

  1. "All rooms in the hotel are occupied"
  2. "a new guest arrives"

From an excellent answer here, I gather that 1. is taken to mean that the hotel is hosting an infinite set of guests and that 2. means things have changed, we now have to reassign every room again to accommodate a new infinite set of guests (eg: the ones before + 1).

I saw other threads and answers. But the "new" set is just the same old set. Why not simply tell them all to momentarily leave and then to come back in to their new room? that is, remap the new set every time as the first set, and have no need to come up with fancy algorithms to reassign every new addition to the set guests to host tonight.

Finally, How is it useful to say things like:

  • The "hotel is completely full"?
  • Each and every room is occupied (or all rooms are)?

*[I have changed this question many times. Trying to keep it in one single question.]

Prudencio
  • 11
  • 1
    $\infty$ actually is NOT a number and this experiment is , as you pointed out correctly , just a thought experiment. A set allowing a bijection to the natural numbers is called countable. Even the algebraic numbers form a countable set ! Quite counterintuitive , but it is the case. Of course the Hilbert Hotel cannot be full , since then it would be impossible to put another guest into it. – Peter Aug 13 '23 at 16:42
  • 2
    "And there is no such thing as all when we talk about infinity ...". 99.9999% of mathematicians will disagree with you on that point. – JonathanZ Aug 13 '23 at 16:51
  • 3
    Maybe the idea of an infinite physical hotel is not convincing to you, but it's just a visual aid for thinking about abstract mathematical objects. When we talk about filling the hotel by assigning people to rooms, we're really just talking about choosing a function from $\Bbb N$ to $\Bbb N$. Do you disagree with the existence of such functions? – Karl Aug 13 '23 at 16:59
  • Note however that ZFC can precisely capture the set of the integers (or the set of the natural numbers) and this set contains actually ALL integers (natural numbers). There are "bigger" sets that have still the same cardinality , not possible of course for finite sets. – Peter Aug 13 '23 at 16:59
  • 2
    Does a sentence like "all integers have a unique factorization" or even "all natural numbers are either 0 or 1 greater than another natural number" make sense to you? – Alex K Aug 13 '23 at 17:08
  • Is it ok to downvote the question without reason? It's a perfectly valid question, and I thank those who take the time to comment. -Peter, I also think the hotel cannot be full. And that saying such things as: "every room is taken" treats infinity as bounded. -JonathanZ, I expect some of them to help me clarify this. -@Karl all I say is that is already mapped, all possible guests have a room. It's not full, and there is no new guests coming. -AlexK, how could I say that better? I still see "all rooms are taken" as odd. – Prudencio Aug 13 '23 at 21:02
  • @JonathanZ yes, put so crudely. I guess I can't say it like that so I will take it out. I meant the part of "all rooms are taken" – Prudencio Aug 14 '23 at 03:45
  • 1
    "there is always a room," No one claims that. The claim is that we can always rearrange things to free up a room. This feels funny, because it's not something that can be done in the finite case. But a "rearrangement"is definitely required. – JonathanZ Aug 14 '23 at 04:13
  • 2
    "if we agree that both sets are equal," That is not claimed either. The claim is that they can be put in bijection with each other. Knowing the difference between those two concepts is key to understanding how infinity works with ordinality. I strongly encourage you to read more and gain an understanding of this. – JonathanZ Aug 14 '23 at 04:17
  • @JonathanZ I accept that "we can always rearrange things to free up a room." but have a problem with the idea that there can be a need for it. More precisely, If the set of guests is mapped to the set of rooms ("all rooms are occupied"), then where is this new guest coming from? As I see it is like you Tax ID. You are born, you get one. Same in this hotel, you are born, you get a room assigned.

    I mean that the two sets have the same size of course. I will clarify that.

     – Prudencio Aug 14 '23 at 07:04
  • 1
    This is not "the set of all guests" in a way that is fixed and unchanging, imagine it as just some infinite subset of an infinite set of people, and they happen to be guests. You can take one person that wasn't already in the set of guests, and that will be your new guest. You are left with a new, different set of guests. Of all things one can be confused by, I wouldn't think this would be it. – Uretki Aug 14 '23 at 08:05
  • @Uretki, once I invoke this 1-to-1 correspondence, then both sets are paired. The bijective function is the registrar-receptionist so to say, every time a new person "joins" the set of guests (gets a GID), the function assigns a space as room for them (with a corresponding RNr. eg: n to n). In that sense: "all rooms are taken" means the process is not finished (we are still in "finite" space). We are in the process. If we are always in the process, the Hotel is never full and there is never need to move anybody. We can argue one way, or the other, not both at the same time. – Prudencio Aug 14 '23 at 09:58
  • 1
    "every time a new person "joins" the set of guests (gets a GID), the function assigns a space as room for them ... We are in the process." You could come up with a mathematical formalism for this process, but it wouldn't correspond to the concept of function. In particular, like @Uretki says, the set of current guests is fixed for one particular set of assignments (function), but people come and go, so later there will be a different set of current guests, requiring a different assignment function. Maybe reread the answer to the third dup you found, where they discuss the "reassignment"? – JonathanZ Aug 14 '23 at 13:13
  • "current guests". Thank you @JonathanZ, and everyone else for their feedback. I have learned a lot. I rewrote the question now. I understand better and can express my issue better now. I hope. – Prudencio Aug 14 '23 at 16:24
  • "have no need to come up with fancy algorithms to reassign every new addition to the set guests to host tonight." There is no guarantee that a new reassignment exists, so we have to explicitly exhibit one. Specifically, if an uncountable number of new guests show up, the countable number of rooms will not be able to accommodate them all . – JonathanZ Aug 14 '23 at 17:43
  • I have been avoiding this topic of "higher" infinites. I prefer to keep talking about a countable infinite hotel hosting countable infinite set(s) of guests. Now, I just saw this similar question, but I don't see the need to "kick people out", just reassign them. https://math.stackexchange.com/questions/4410240/hilberts-hotel-kick-everyone-out?rq=1 – Prudencio Aug 14 '23 at 20:13

1 Answers1

6

To me, it sounds like you are considering two possible statments about a hotel

  1. The hotel has a guest in every room
  2. The hotel can fit in another guest

and feeling that the two statements can never be simultaneously true. Indeed, for real-world finite hotels, the two statements are opposites of each other. But for an infinite hotel, it is possible for both to be true at the same time (indeed the second statement is always true for an infinite hotel).

Put another way: for finite hotels, we use "full" to mean "there's a guest in every room", and we also use "full" to mean "they can't fit in another guest". But these two senses of the word "full" are not equivalent for infinite hotels. Part of the value of the Hilbert hotel thought experiment is to help us notice when our intuitions about finite sets are incorrect intuitions for infinite sets.

This clash with intuition is a perfectly reasonable cognitive dissonance to have while learning about infinite sets! It seems like that discomfort is making you lean in the direction of not accepting infinite sets as well-defined mathematical objects (and you wouldn't be the first to feel that way). But infinite sets are crucial to mathematics—without them we couldn't even talk about intervals of real numbers like $[0,1]$. I would encourage you instead to lean into the discomfort and use it to help refine your intuition in ways that will pay off for future mathematical studies.

Greg Martin
  • 78,820
  • Thank you Greg. I don't have problems with the idea of infinity in general. I think 2. is clear: you could always accept a new object in an infinite set. But more with 1. I don't think you can so easily start by saying that an infinite set is full. It sounds to me that they are seeing the infinite set as bounded. For me, if it's infinite, then it cannot be full. And could only be consider full if there is no objects left to let in (i.e: all potential guests are already in). Put another way: if it has a guest in every room, there are no future guests possible. – Prudencio Aug 13 '23 at 18:04
  • 1
    "if it has a guest in every room, there are no future guests possible". This is certainly not true if the hotel is finite. Why do you think it becomes true once it becomes infinite? I don't really understand your beliefs, but it almost sounds like you think there is only one infinite set? – JonathanZ Aug 13 '23 at 23:02
  • @JonathanZ I clarified that now. Let's stay in the infinite (countable) hotel. I think the issue comes down to this idea of a somehow unmapped new customer. For me is not possible. If it is the case that all rooms are taken, then all possible customers are already checked in. – Prudencio Aug 14 '23 at 04:06
  • @Prudencio "If it is the case that all rooms are taken, then all possible customers are already checked in." No! Let's say the rooms are numbered $1, 2, 3, 4, 5, \ldots$, and the possible guests are also numbered $1, 2, 3, 4, 5, \ldots$. Now you could have guest $n$ in room $n$, but you could just as easily have guest $2n$ in room $n$ for every $n$. All rooms are filled, and you still have infinitely many odd-numbered guests who are not checked in. – Robert Israel Aug 14 '23 at 05:02
  • @RobertIsrael, please clarify: if I "could just as easily have guest 2 in room for every ", then half the guests won't get a room. (and if I have guest n in room 2n, then half the rooms will be empty), no? I still think I get what you mean, but the problem remains!: how can an unmapped guest arrive? – Prudencio Aug 14 '23 at 06:43
  • 1
    "Unmapped guest" point out one source of your confusion. There is more than one mapping, and the set of guests changes from day to day. – JonathanZ Aug 14 '23 at 14:03