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Hilbert's Paradox flaw

  1. Jun 7, 2009 #1
    Well first hello, I'm not really into physics, learning computer science at uni atm. However wanted to get something off my mind.

    Was browsing wikipedia a while ago and came across Hilbert's Paradox, the idea that a hotel with infinite rooms that are all full can still accept new guests by shifting guests over. It also said this (quoted from the wikipedia page):

    "This provides an important and non-intuitive result; the situations "every room is occupied" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms."

    Anyway it is my thought that this is not true, the hotel with infinite rooms that are all full cannot accept more guests. My reasoning is that there are infinite rooms and they all exist already. Any room that exists is occupied. Any guest you make leave a room for another guest is then means there is infinity +1 guests, for each new guest there is another left without a room. Or for each new guest there is a series of infinite room swaps with a guest always without a room. Or each new guest already has a room as they are a part of the infinitely many guests.

    Other thought is to view the hotel as a set, each guest as a dot, each dot is on a point (a room) in the set. The set has infinite points but you can't add points from outside the set, the set is already all points. So you can't add any more dots (aka guests) to this set as every point (aka room) already has a dot.

    Anyway thoughts on my ramblings here?
  2. jcsd
  3. Jun 7, 2009 #2

    the point is that you can make room in the hotel by changing the map occupant -> his_room_number.
    This does not work in a finite hotel.

    Of course, without resettling the occupants you will not find a room that is not occupied.
  4. Jun 7, 2009 #3


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    There's no such thing as "∞ + 1" because ∞ is not a number. If you want to use expressions like "∞ + 1", you must first define what that means.
  5. Jun 7, 2009 #4
    Keep in mind that "infinity" is not a number. It does not obey the rules of arithmetic for addition, division, and so forth. So be wary of a phrase like "infinity + 1", because it's misleading.

    [edit] dx, you beat me to it. :smile:
  6. Jun 7, 2009 #5
    Also keep in mind that there is no such thing as a hotel with infinite rooms, or infinitely many people, so the whole example makes no sense. It's supposed to illustrate a fact about bijections between N and N U {x}, and there's really no paradox.
  7. Jun 7, 2009 #6


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    But Hilbert gives a way for the hotel to accept more guests.

    Suppose there were only finitely many rooms -- maybe 100. If everyone moves to the next-numbered room, the former occupants of the last room (room 100) have nowhere to go. But there is no last room in Hilbert's Hotel: no one is in a room numbered "∞". Give me any room number and I'll tell you where that occupant goes to. Unlike the finite case, you can't find a roomless person.

    It's true that the 'paradox' doesn't hold for finitely many rooms. It's also true that the 'paradox' doesn't hold for finitely many moves (your "swap" reasoning). But for infinitely many it does work. Here you must abandon your finitistic intuition! Things work differently in Cantor's paradise.
  8. Jun 8, 2009 #7
    Yeah I know infinity +1 is not correct. Hrmm wouldn't that make it a paradox of the paradox?

    CRGreatHouse ->
    For each new guest that is accepted an infinite series of displacements are made, as each guest moves over to the next room, displaces that guest who moves to the next room, etc. So at any point the hotel cannot accept more guests as at any point for each guest that is accepted there is someone outside of a room swapping rooms. Infinitely swapping rooms does not make a guest suddenly have a room, someone will always be displaced.
    Also remember these aren't numbers but physical guests, they require time to move, so while you can say which room any guest will move to at any time there is still a guest without a room in the hotel.

    This is what I mean infinite +1, infinite guests with rooms +1 guest without one, and +1 for every new guest.

    Conversely about changing the room number, this is not correct, any room number is already occupied. Changing the number does not change the occupancy.

    or hrmmm using set theory again
    A is the set of infinite rooms
    B is the set of infinite guests occupying rooms
    H is the hotel

    again A, B and H are positive, whole, real numbers

    A + B = H
    (sum)A = (sum)B

    so adding a new guest is
    B + 1
    while (sum)A = (sum)B is still true?

    you can't do infinity + 1 right?

    A closed circuit superconductor describes the hotel good. You can keep adding infinitely many electrons to the circuit but the electrons never really have a place, they just keep moving.

    So now I've given some proof here (as bad as it may be :P) can you give some good proof that a new guest can actually get a room?

    Lastly sorry if I seem belligerent, I just love a good debate :D
  9. Jun 8, 2009 #8
    Give the new guest room number 1 and simultaneously tell all guests that if they are in room n, they must move to room n+1. Each guest after the move is completed has a well-defined room.
    The guests are not physical as we have an infinite amount of them. The time it takes for this to happen is also not relevant to the solution for the same reason, but if you want, install instantaneous room transporters in each room. This is already a ridiculous hotel with an infinite number of rooms anyway.
    The cardinality of a set, which is the subject in which this paradox occurs, has nothing to do with sums. It is only concerned with bijections. If, given sets A and B, a bijection exists between them, we say they have the same cardinality. It is common with finite numbers to associate sets bijective with the sets that define the natural (counting) numbers with the arabic script representing that natural number so that we have the shorthand notion that the cardinality of the set {0, 1} is 2 since it is bijective with 2 (the set).
  10. Jun 8, 2009 #9


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    ??No, they aren't! The only "paradox" I can see is that you don't seem to understand what is being said here. No, we are NOT talking about a physical hotel- there is no physical hotel with an infinite number of rooms- and so these are not physical guests. Saying that "they require time to move" is irrelevant.

    Your entire argument is one we have have seen many times: "I do not understand this, therefore it is wrong".

  11. Jun 8, 2009 #10
    To the OP:

    I wouldn't worry about things like this overmuch. Infinity isn't real, anyway. Well, no number is real, in a sense, but compared to things like 2, pi, etc...

    I always thought the hotel thing was a bad example. If you get that many people being made to switch rooms, you're going to have a lot of dissatisfied customers.

    Here's my paradox: the guy in the last occupied room must still be at the same distance from the first room, but you've added another person... so there are more people in the same space. Keep on doing this, and people are going to start becoming uncomfortable.

    Alright, we get the idea, Hilbert. Jeez. It's like he made this example to make it harder for people to believe that there exists a bijection between N and N (union) some other countable set.
  12. Jun 8, 2009 #11


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    If you're looking to set up a competing hotel, here's a thought. Have the guests in rooms 2, 4, 8, ..., 2^k, ... move to room 2^(2k), and the new guests can take rooms 2, 8, 32, ..., 2^(2k+1), .... If a second group of guests come, have guests in rooms 3^k move to 3^(2k), then 5^k to 5^(2k), and so on. Then even if an infinite number of infinite groups arrive, you can fit all of them in. But because of the rooms you've chosen, most guests never have to move and no guest has to move more than once.

    Of course the guests that do have to move tend to move pretty far, so I recommend licensing slider142's transportation technology.
  13. Jun 8, 2009 #12


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    Have them all move at the same time.

    Yes, it does. In fact, that's the whole point: infinite numbers work differently from finite ones.
  14. Jun 8, 2009 #13
    poor guys :(

    ok then here is another flaw with that theory, it must take time to swap rooms, it cannot be instant.

    First if its instant swapping- at any point there is a room with 2 guests, as the instant one moves in the other moves out, but at that instant there is 2 guests in the room. The sum of this is that there is always two guests in a room.

    The other view- At one instant the guest in the next room moves out, then the next instant the new guest moves in, this also cannot be, there are no adjacent points in time.

    hrmmm then, ignoring time

    you agree you cannot count the infinite set of whole, real, positive integers? You could count forever and not get them all right? Forgetting the time aspect, if you could write the numbers without time you still couldn't write them all.

    You can however write any number down. Or any number of numbers.

    Really the problem of the hotel clearly describes he is adding one to infinity (adding a guest to an infinite amount of guests).

    /sigh can see some people getting rather passionate about this :(

    I know the problem is about this and such but still

    show me you can add a guest to an infinite number of guests, write the mathematical proof :P

    Also please this is just philosophy debate. Meant to encourage and stimulate thought. Also my last post on the subject :P
  15. Jun 8, 2009 #14


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    It can take five minutes, if you like. As long as they all start moving at the same time that doesn't cause a problem. There's certainly no requirement that they move sequentially -- how would you enforce that, anyway?

    More like this:

    Time 0: Person 1 is in room 1, person 2 is in room 2, etc.
    Time 1: Person 1 leaves room 1, person 2 leaves room 2, etc.
    . . .
    Time 4: Person 1 arrives at room 2, person 2 arrives at room 4, etc.
    Time 5: Person 1 enters room 2, person 2 enters room 4, etc.

    What's wrong with that?
  16. Jun 8, 2009 #15
    I don't think anybody's banging their heads on their keyboards and swearing or anything :), but maybe you should have posted this in the philosophy section. You're taking relatively straightforward (albeit interesting and surprising) math and making it needlessly complicated. Infinite sets are fine. Infinite rooms and infinite people starts to get a little murky.

    It's like somebody putting two M.C Escher paintings side by side to illustrate 1+1=2 and you're asking them to show you a room with those dimensions.
  17. Jun 8, 2009 #16
    You're still somehow trying to monkey wrench time into set theory. The most you can do with your temporal argument is show that the physical existence of such a hotel is not consistent with the physical aspect of time, which is true anyway and has nothing to do with cardinal numbers.
    Here is a similar example which may illustrate why I am confused by your inclusion of time: Consider the positive x-axis, which is a ray in the usual Euclidean coordinate system. Now consider the function f(x) = x + 1 which shifts the ray one unit to the right. You seem to be objecting to the truth of this last statement by asking "How much time did it take all those points to move over and make room for the interval [0,1)? Since each point takes the same amount of time, the ray never moves over!" or "At each moment, two points are crowding each other!" In each case, you are making a new assumption about a currently feasible physical method of moving an infinite amount of people (a decidedly nonphysical object), where one successful method (that may not be currently feasible or physical) is all that is necessary. Ie., you're limiting a purely mathematical concept to particular physical terms where it does not directly apply.
    Your constant finding of a new physical fault is reminiscent of Zeno's paradoxes, though, which still stir the imagination every generation.
    Last edited: Jun 8, 2009
  18. Jun 9, 2009 #17


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    Because we are actually responding to a troll?

    No, every one moves, instantaneously, at the same time.

    Once again, irrelevant. This is not "physics", there is no time involved.

    All you are doing is showing, over and over again, that you do not understand the point of the argument. If adding one guest bothers you, think about this: have every guest in room n move to room 2n. Now only the even numbered rooms have guests. You have freed up every odd numbered room and now have room for an infinite number of new guests.

    Why? That has been done, in this thread, and you just ignored it.

    Refusing to pay attention to refutations of your point, as you have done here, does not "encourage and stimulate thought".
  19. Jun 9, 2009 #18
    I'll try my hand at a proof.

    Thm. There exists a bijection between the set N and NC = N union "Cat".

    Prf. Consider the following function from N to NC:

    if i > 0, f(i) = i+1
    if i = 0, f(i) = "Cat"

    Its inverse, f^(-1), from NC to N, is

    if i = "Cat", f^(-1)(i) = 0
    othwerise, f^(-1)(i) = i-1

    Both functions are injective (one element is mapped to one other element) and surjective (every element of the codomain is in the range). Ergo both bijections. QED

    Thm: If you have an infinite number of fake hotel rooms, inhabited by fake people who can communicate and move non-physically, and a new guest comes, you can fit the new guest (call him "Cat") in the same number of rooms (though they're all occupied already... just make room for the guy).

    Prf: Follows by the previous theorem. QED.
  20. Jun 10, 2009 #19
    EDIT: thanks for the replies

    I'm thinking about this waaaaaaay too much :(
    Last edited: Jun 10, 2009
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