[HN Gopher] Hilbert's Paradox of the Grand Hotel
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Hilbert's Paradox of the Grand Hotel
Author : belter
Score : 56 points
Date : 2021-06-12 09:00 UTC (14 hours ago)
(HTM) web link (en.wikipedia.org)
(TXT) w3m dump (en.wikipedia.org)
| shannifin wrote:
| The "paradox" stems from thinking of "infinity" as an amount, and
| treating it as somehow logically finite. Once you toss away the
| "logic" of the finite, there is no paradox at all.
|
| ETA: For instance: One infinite set does not have "as many" or
| "more" or "less" members than any other infinite set. You cannot
| compare amounts, there are not "amounts" to compare.
| chimeracoder wrote:
| > ETA: For instance: One infinite set does not have "as many"
| or "more" or "less" members than any other infinite set. You
| cannot compare amounts, there are not "amounts" to compare.
|
| Not exactly true - there are different infinities. For example,
| the number of integers is infinite, and the number of real
| numbers is infinite, but there are more real numbers than
| integers.
|
| The number of integers is "countably infinite" and the real
| numbers are "uncountably infinite".
| magnio wrote:
| Interesting that this is classified as self-refential paradox by
| Wikipedians.
| amelius wrote:
| Infinity is a bad abstraction. Mathematicians make them too.
| lixtra wrote:
| Have some mercy with the ultrafinitist[1].
|
| [1] https://en.m.wikipedia.org/wiki/Ultrafinitism
| username91 wrote:
| Had never encountered this. It's very refreshing. Thank you!
| rightbyte wrote:
| Ye. It is quite bogus to claim to be able to accommodate new
| guests even though the hotel is full, no matter how many rooms
| there are. In that case the hotel is not full ...
| bmm6o wrote:
| That's the paradox! There are no empty rooms, and yet the
| hotel can accommodate arriving guests. Whether the hotel is
| full depends on your definition of "full". This is quite
| different from the finite hotel case.
| [deleted]
| hibbelig wrote:
| Well, if the hotel isn't full then there must be a vacant
| room. But the first room is occupied, the second room is
| occupied, ... If you can't find a vacant room then the hotel
| must be full!
| benjohnson wrote:
| I think this is a fair point - if there's a one person to one
| room correlation, then moving people around isn't exposing
| any additional empty rooms.
|
| It seems we're using the 'hey-everybody-move' as a buffer to
| accommodate new guests.
|
| Two questions:
|
| Is it ok to think because there's an infinite amount of
| moving, this buffer is infinitely large?
|
| Why can't we tell guests to just keep moving to random rooms?
| Would that also solve the problem? If you say no - random
| moves don't work - then I would propose that some of those
| random moves would fall into the outcomes of Hilbert's
| formula. Is that not sufficient?
| lupire wrote:
| "Moving" has nothing to do with the paradox. Being
| reassigned to a new room could take 0 time, with no buffer.
| Mapping n to n+1 creates a hole at 1.
|
| It's an unphysical model, that only exists in abstract
| mathematics.
| rightbyte wrote:
| An interpretation could be that remapping infinite sets
| like that creating holes is an invalid or nonsense
| operation.
| benjohnson wrote:
| Thanks for answering! I think I get it.
|
| Any idea on if directing guests to move randomly would
| work as well as directing them to move in patterns?
| bmm6o wrote:
| If collisions can happen they will happen with
| probability 1, since there is infinite opportunity.
| There's also the issue of there not existing a uniform
| distribution on the integers, so it's hard to get the
| chances of collision low in the first place.
| [deleted]
| tantalor wrote:
| None of the rooms are vacant = hotel is full
| jacobwilliamroy wrote:
| Infinity actually pops up in the context of stream processing,
| since often times real computers need to process arbitrary
| amounts of data with finite memory resources. Sometimes the
| data is too big to fit into memory all at once, or maybe the
| data moves too slowly to all be available immediately. Because
| there is no upper bound to how long a stream may be, streaming
| algorithms tend to be designed to be able to handle infinite
| amounts of data which is a property that comes in handy
| surprisingly often.
| morelisp wrote:
| Infinity (both the "cardinality of sets" kinds and the "limit
| of an increasing/decreasing function" kind) are
| generalizations, not abstractions; and both have proven
| extremely fruitful generalizations both theoretically and
| practically.
| lupire wrote:
| What's the practical application of infinity?
| t0mek wrote:
| The infinity is an important concept which appears in most
| areas of the maths and physics. The maths is full of ideas
| that may seem counterintuitive (eg. "imaginary unit" - the
| square root value of -1), but as long as they are precisely
| defined, they can be used in reasoning that gives practical
| results.
|
| And even if some math concepts can't be applied to anything
| practical yet, we never know what will be used to formulate
| another theory about the stock market behaviour or
| processes inside the black hole.
|
| Coming back to your question, here's an example: many math
| constants (pi, e) and functions (sin, cos) can be defined
| as infinite sums (series). They converge, getting closer
| and closer to the limit, but they never achieve it.
|
| Using these series it's possible to calculate (in a
| calculator, Excel, Quake or CAD) values of these functions
| with any required precision.
| jlokier wrote:
| Calculus, differential equations, integrals, all rely on
| iterating towards infinite sums of infinitesimal values. It
| seems a bit dodgy at first but can be made rigourous by
| giving precise, consistent definitions of different
| infinite values and the operations between them. These are
| used extensively in many fields. This can be done by using
| limits "towards infinity / towards zero", but it can also
| be done more directly with precise defintions of infinities
| and infinitesimals:
| https://en.wikipedia.org/wiki/Nonstandard_calculus
|
| Infinite series, where an infinite sequence is summed (or
| multiplied, or ...). Many useful things fell out of
| reasoning about entire infinite sequences as if they are a
| single unit. I like this video about a method Newton found
| how to calculate pi efficiently, from playing with infinite
| series: https://www.youtube.com/watch?v=gMlf1ELvRzc
|
| Again it's possible to define everything about infinite
| series in terms of limits, but sometimes it's just a more
| useful thinking aid to think of infinite series as
| themselves. In a way, it's like dropping some unnecessary
| syntax sugar (the limits) as you realise you can still do
| useful things without it. And in fact it's possible to
| reason consistently about some kinds of divergent infinite
| series, where limits don't work but they are still
| logically consistent.
|
| Complex exponentials. The discovery and proof of Euler's
| formula, ex=cos(x)+i[?]sin(x) comes out of reasoning about
| infinite series, showing that it's a consistent and useful
| definition of a complex exponential. Nowadays we use these
| exponentials to calculate all sorts of things in
| engineering and physics, we just take them for granted
| because they work and everything fits together
| consistently.
|
| Projective geometry, for example used in computer graphics.
| There is the concept of "point at infinity", which is
| actually represented by real values in a vector but in some
| respects you can think of it as representing "x/0" in the
| context of that geometry, and finding that further
| calculations using it still work out consistently to non-
| infinite results. Including the point at infinity as a
| value simplifies the system. Unlike with natural numbers
| where "infinity" is a complication that obeys different
| rules to numbers, the point at infinity in projective
| geometry removes edge cases and simplifies the rules.
|
| Elliptic curve cryptography also uses the "point at
| infinity" concept which arises from a geometrical or
| algebraic interpretation of the elliptic curve operations,
| and like in projective geometry it's a useful and
| consistent value to include, which is done because it
| simplifies the system to include it.
| kolinko wrote:
| Anything that deals with calculus has infinities
| underneath. Much of the modern technology wouldn't exist if
| we wouldn't know calculus.
|
| Also, check out Fourier transform and try to rewrite it
| without infinities - https://bookdown.org/vshahrez/lecture-
| notes/fourier-transfor... . Without fourier transform we
| wouldn't have mp3, and the whole field of signal processing
| if I'm not mistaken.
|
| Without signal processing we would not have long distance
| digital communication.
| TchoBeer wrote:
| You can make calculus and Fourier transforms without
| infinite sets; all you need is limits.
| jlokier wrote:
| Indeed you can.
|
| But you can also make calculus without limits, using
| explicit definitions of infinitesimals (called non-
| standard analysis or non-standard calculus), and in some
| ways this is simpler to use.
|
| You can approach it either way. As limits, or by
| extending the number system with particular consistent
| definitions of infinitesimals and infinities.
|
| In practice using infinitesimals is often what people
| actually do when using calculus anyway, even when
| mentioning the limit in a sloppy hand-wave in the
| background, so it's good that non-standard analysis shows
| that the simpler method people actually use is usually
| logically coherent and gives the same answers as limits.
| syops wrote:
| There is a precise, consistent definition of what it means for
| a set to be infinite. How is this a bad abstraction more than
| other mathematical definition?
| amelius wrote:
| Precision doesn't matter. Software abstractions, once coded,
| are precise. The problem is with human thinking. If the mind
| can't grasp at all times all the relevant technicalities that
| come with an abstraction, then the abstraction is bad.
| CrazyStat wrote:
| This is a very anthropocentric view of mathematics. Why
| should we expect mathematics to be amenable to a puny human
| mind grasping all relevant technicalities of a concept?
| amelius wrote:
| Most of mathematics is abstractions invented by humans to
| make math more comprehensible by humans. E.g. you can do
| differential equations without complex numbers, but they
| make life so much easier. Infinity is a mathematical
| concept invented by humans to make thinking about large
| numbers easier in some cases, but it's deceptive in other
| cases.
| simonh wrote:
| Which cases? Is there mathematics of infinity that you're
| privy to that are unknown to mathematicians?
|
| Also I disagree that thinking about infinity is useful
| when thinking about large numbers. That seems to me to be
| a fallacy. Any finite large number is inconsequential,
| and might as well be a 'small' number for any practical
| purposes, compared to any infinity.
| morelisp wrote:
| The set-theoretic definition of infinity came about to
| make thinking about _small_ numbers easier.
|
| As trivial examples, if we reject this definition of
| cardinality, we can no longer speak of the natural
| numbers as a set (nor therefore even finite subsets of
| them), we would lose key set-theoretic definitions of
| both natural and real numbers, we could not talk set-
| theoretically about the domain/range of most interesting
| functions, etc. Honestly, that would be a much more
| confusing mathematical world than one with Hilbert's
| hotel or Banach-Tarski spheres.
| syops wrote:
| The set theoretic definition of infinity did not come
| about in order to make thinking about large numbers
| easier. You are wrong in your perception on this.
| [deleted]
| syops wrote:
| What technicalities with the definition of an infinite set
| can't be grasped by the mind? The definition is quite
| simple and easy to understand (for a mathematician).
| lini wrote:
| A video, explaining this paradox, was published by Veritasium
| last month - https://www.youtube.com/watch?v=OxGsU8oIWjY
| Exuma wrote:
| Very cool. I'm still trying to wrap my head around what makes
| ABAB uncountable though. I understand the diagonal bit (its
| excluding them one by one), but it almost feels like that
| uncountable infinity is defined by a rule ("every combination")
| while the doors are defined by a different rule ("every item
| going upwards in count")
| beervirus wrote:
| Huge fan of Veritasium. Even on the videos where I already know
| a decent amount about the subject, I invariably still learn
| something new.
| codetrotter wrote:
| I saw this video a while ago. It's got interesting 3D
| animations and explains everything in a very approachable
| manner. I recommend watching it.
| desperate wrote:
| Another great video on it, this one from TED-Ed
|
| https://youtu.be/Uj3_KqkI9Zo
| shagie wrote:
| This was a pair of videos in Veritasium recently - How An
| Infinite Hotel Ran Out Of Room - https://youtu.be/OxGsU8oIWjY and
| Math Has a Fatal Flaw - https://youtu.be/HeQX2HjkcNo .
| username91 wrote:
| I find this hard to follow. The "first" guest (who was in room 1,
| in the article) moves over one room. The "last" guest (showed as
| being in 'room n') has no room to go to; the hotel is full.
| beervirus wrote:
| There is no last guest. There are infinitely many guests.
| username91 wrote:
| Thanks for chiming in.
|
| I have trouble with that, though - if there's a "first
| guest", then I don't see there can't be a "last guest"..
| "room 1" and "room n" (as the article labels them) are
| equally arbitrary if there are infinitely many.
| beervirus wrote:
| That's the difference between all the integers (from
| negative infinity to positive infinity) vs just the natural
| numbers (from 0 to positive infinity). You can have
| infinitely many guests starting from a particular first
| guest.
| username91 wrote:
| Ah, cool! I can see that, thanks. :)
|
| In that case I'd probably rephrase my original objection
| to say the "next guest" has nowhere to go, except when
| the they depend on the "next next" (etc.) guest finding a
| room in a "full" hotel - but I can see how that situation
| "never" comes up since they can pass the burden along
| infinitely.
|
| I just feel like someday at the end of time, this poor
| unassuming new guest is gonna get cheated out of a room..
| beervirus wrote:
| Yep, this is what makes Hilbert's hotel (and infinity in
| general) counterintuitive. When we try to apply reasoning
| that's correct for finite sets to infinite sets, very
| often it becomes incorrect.
| snakeboy wrote:
| Well the hotel has [countably] infinite-many rooms, so the nth
| guest moves to room n+1.
| username91 wrote:
| Room n+1's guest, similarly, has nowhere to go; the hotel is
| full. It has countably infinitely many rooms, and all of them
| are occupied.
|
| I can only see a shift of guests being possible if you
| momentarily ignore the initial constraints.
| jejones3141 wrote:
| Sigh. I'd finish my filk of "Heartbreak Hotel", but it seems to
| be taking forever.
| z5h wrote:
| Stranger still is that one can find a rational number between any
| two irrational numbers, yet the irrationals vastly outnumber (are
| not countably infinite) the rationals (which are countably
| infinite).
|
| Cantor's diagonal argument is an easy foray into uncountable
| cardinalities
| https://en.m.wikipedia.org/wiki/Cantor's_diagonal_argument
| TchoBeer wrote:
| When I was much younger this was the source of a lot of wonder
| for me. Both the rationals and the irrationals are dense in the
| reals, and yet the irrationals greatly outnumber the rationals.
| t0mek wrote:
| A similar observation is that there's exactly as many natural
| numbers [0, 1, 2, 3, ...) as integer numbers (the same, but with
| negatives: ..., -2, -1, 0, 1, 2, ...).
|
| Normally we'd need to count things to say there's "as many" X as
| Y. But with infinities counting is a bit tricky. So, is the
| infinite count of natural numbers the same kind of infinity as
| count for integer numbers?
|
| To check this, we need to see if for every number in one set we
| can assign exactly one number in the other set (and the other way
| around). It's actually pretty simple: 0 -> 0
| 1 -> -1 2 -> 1 3 -> -2 4 -> 2 5
| -> -3 6 -> 3 ...
|
| Since this mapping ("bijective function") exists, we know that
| every number in one set has exactly one representative in the
| other set - so the set counts are identical.
|
| What's interesting, if we look into real numbers (think: double
| in C, but without problems with approximation), there's much,
| much more of them. The infinite count of real numbers is much
| larger than the infinite count of integers. But let's keep it for
| another comment.
| jodrellblank wrote:
| > " _The infinite count of real numbers is much larger than the
| infinite count of integers. But let 's keep it for another
| comment._"
|
| That is, beautifully, Cantor's diagonal argument, which goes:
| write some decimal numbers between two integers 0 and 1:
| 0.111 0.222 0.333
|
| Now work diagonally through the digits: 0.[1]
| 1 1 0. 2 [2] 2 0. 3 3 [3]
|
| and change those selected ones to other digits:
| 0.[2] 1 1 0. 2 [3] 2 0. 3 3 [4]
|
| And pull those diagonals out into their own new number
| 0.[2][3][4] or 0.234, and add that in to the list:
| 0.111 0.222 0.234 <- new number 0.333
|
| That number differs from the first entry in the first decimal
| position, from the second number in the second position, from
| the third number in the third position, ... and the Nth number
| in the Nth position, because those are the positions you
| changed in each one to make sure of it.
|
| If it's different from every existing decimal in at least the
| one place, it _cannot_ be a duplicate entry seen before in the
| list you wrote down, it _must_ be a new entry.
|
| You can always change the first number in the first position
| because there's only a single digit there and nine more to
| choose from. You can always change the Nth number in the Nth
| position because 0.2 is really 0.20000000... so changing 0.2 in
| the 5th decimal place makes 0.2 into 0.20007 or etc. This makes
| the decimals longer, sub-dividing into increasingly tiny
| pieces, without end - infinitely.
|
| Therefore with an infinity of integers, you can subdivide
| infinitely between any two of them.
|
| You can take your infinite list of decimals between 0 and 1 and
| map the integers to them, 1 for the first entry, 2 for the
| second, 3 for the third, and pair up both infinities 1:1. And
| then have no integers leftover to map onto the infinity of
| decimals between 2 and 3, and again none left over for the
| decimals between 3 and 4, etc.
|
| Conclusion: there are infinite integers, and infinite decimals,
| and there are more decimals than integers. The infinity of
| decimals is the larger infinity.
|
| (Which makes some intuitive sense looking at single digit
| integers 0-9 on the left of a decimal point, fan-out to single
| digit 0-9 _for each of those_ on the right of the decimal
| point. 10x more 2-digit decimals than 1-digit integers (of
| course). Infinite permutations of digits on the left of the
| decimal point, an infinite permutation of digits on the right
| _for each starting permutation on the left_ , means infinityx
| more decimals than integers).
|
| [ I wrote this more for the practise of pulling it out of
| memory and going over it, because doing that cements it more in
| my memory. It is one of the few bits of math I can more or less
| remember. It would surely be more beneficial and correct for
| you to read it elsewhere. This is the paradox of internet
| comments written for the author, not the reader. ]
| thethimble wrote:
| I've always been severely dissatisfied with this argument. It
| feels like a sleight of hand as opposed to something
| profound.
|
| Are there any other roads to "sizes of infinity" that are
| more palatable than the diagonalization argument?
| morelisp wrote:
| Do you also see the First Incompleteness Theorem, or
| Halting Problem, as sleights of hand?
|
| Informally, the answer to your question is no - the
| Schroder-Bernstein theorem, which lets us order the size of
| sets, is sufficient to derive the law of the excluded
| middle. Therefore if you don't like the "trick" i.e. proof
| by contradiction (even given the contradiction is "actually
| constructed" in this case), and instead demand constructive
| mathematics, you will not be able to say much about
| relative cardinality.
| morelisp wrote:
| > The infinite count of real numbers is much larger than the
| infinite count of integers.
|
| "Much larger," or just barely, the smallest possible amount,
| larger? :)
| catillac wrote:
| Maybe someone can help me understand, not a mathematician or
| engineer. It seems like the hotel is "full" but because there are
| infinite rooms, when new guests arrive they just move everyone to
| their room + 1 and make space at room 0.
|
| I don't think I understand the significance here, since it seems
| logical.
| jedimastert wrote:
| That point being made here is that infinity changes some pretty
| basic assumptions about how we think of things like "full" and
| "counting".
| lupire wrote:
| It's incompatible with physical reality, no such hotel could
| exist. The paradox is in applying infinite math to physically
| realistic problems.
| joppy wrote:
| There are two ways you can define the hotel as being full:
|
| 1) The hotel is "full" if all the rooms are occupied.
|
| 2) The hotel is "full" if there is no way to rearrange the
| existing guests to leave an empty room.
|
| These two definitions of "full" are equivalent for finite
| hotels. The 'paradox' is illustrating that these two
| definitions are not the same for infinite hotels. It serves as
| a useful reminder that there might be more "space" in infinite
| sets than we are used to in the finite case, and that if we are
| dealing with infinite sets we should keep in mind the fact that
| (1) and (2) (or whatever analogues are relevant for our
| problem) might be different, despite seeming the same based on
| our real-world experience.
| catillac wrote:
| This is really helpful, thank you for the thorough
| explanation
| simonh wrote:
| It is logical, and that's the point. It's simply explaining
| various properties of infinite sets and techniques for doing
| operations on them that are provably correct.
| xg15 wrote:
| I find it somehow interesting that the reordering procedure also
| never terminates (as is of course expected for an iteration over
| an infinite set)
|
| So, if you stay in the hotel analogy, each additionally arriving
| guest will create a unique "reordering wave" that ripples through
| all the infinite rooms. This wave would keep going for all
| eternity.
|
| So if the hotel had k additional guests check in at any time in
| the past, it will have k reorderings going on forever.
|
| (That also makes the difference between finitely many additional
| guests and infinitely many more remarkable: you can also
| accomodate infinitely many additional guests with a single wave
| of reorderings)
|
| Not sure if this is in any way a useful observation, but it made
| the whole hotel scenario somehow even weirder for me than it
| already was.
| morelisp wrote:
| This depends on if you count "information propagation" time or
| not. For example, if you inform every guest about the pairing
| function when they check in, and moving from one room to any
| other takes finite time, all you need is a single "please move
| now" bit and each guest can move simultaneously.
|
| (If you don't assume you have instantaneous transmission and
| finite travel time, our unintuition about this also collides
| with unintuitive, unrelated, and very physically-real concepts
| like light cones - so I think it's within the spirit of the
| problem to assume we have them.)
| k2enemy wrote:
| > So if the hotel had k additional guests check in...
|
| They still haven't finished checking in the initial round of
| guests!
| a4isms wrote:
| You're the manager, and you know an infinite number of guests
| are checking in tomorrow. how many receptionists do you
| schedule to be on duty when they arrive?
| rightbyte wrote:
| > moving every guest from their current room n to room n+1
|
| The whole setup of the paradox is bad. There a infinit amount
| of rooms, all are full, but somehow atleast one room is not.
| The guests and rooms operate under different rules.
| MereInterest wrote:
| That's sort of the entire point of the paradox. Every room is
| full at the start. Not "somehow at least one is not", but
| every single room is full. Then, by having guests move to
| different rooms, you can rearrange such that there is an
| empty room.
|
| The guests are markers to indicate a bijective function
| between room numbers. Each guest represents a (before, after)
| tuple.
| rightbyte wrote:
| A point of a paradox can also be to invalidate the
| assumptions made. I mean, mathematicians are getting away
| with way too much nonsense by using diffucult words.
|
| The hotel paradox is essentially equivalent to:
| I can add a natural number to the set of all natural
| numbers which is not in the set allready.
| morelisp wrote:
| No, the paradox is that there are the same number of even
| natural numbers and natural numbers, yet the former is a
| strict subset of the latter.
| rightbyte wrote:
| Ye sorry I was refering to the hotel paradox. Edited.
| MereInterest wrote:
| I think I would call that a contradiction rather than a
| paradox. A conclusion that is a contradiction means that
| one of your starting assumptions is incorrect. A
| conclusion that is a paradox means that your intuition
| about the problem was incorrect.
| danbruc wrote:
| No, you are trying to use your intuition for finite sets on
| infinite sets. Subsets of infinite sets can have the same
| size as the entire set. There are as many odd natural numbers
| as there are natural numbers even if your intuition tries to
| tell you that this can not be true because the natural
| numbers additionally contain all the even natural numbers.
| rightbyte wrote:
| I am buying that.
|
| In the hotel case, it would be like claiming I can add a
| natural number to the set of all natural numbers which is
| not in the set allready. All guest are allready at the
| hotel ...
| lupire wrote:
| Which number are you adding? In the problem, you are
| adding 1 to the set, but it's already there, so you
| remove it by replacing it by adding 2, and n by n+1 for
| all n
| rightbyte wrote:
| Ye well my point is there is no number to add that is not
| allrady in the set and that the hotel clerk can't change
| that by rearranging guests. I.e. there are no guests that
| are not allready checked in (each guest need a room with
| a natural number on the door).
|
| Edit: I.e. the error in the paradox is assuming that
| there can be any more guests to check in when the hotel
| is full.
| [deleted]
| danbruc wrote:
| Just imagine all the even numbers are currently in the
| hotel, zero is in room zero, two is in room one, four is
| in room two...2n is in room n. Now all the odd numbers
| arrive at the hotel, maybe one after another, maybe all
| at the same time, but you can apply one of the procedure
| to fit them in. You must not confuse the numbers
| occupying each room with the numbers of the rooms, the
| room numbers are all the natural numbers but they can all
| be occupied by the even numbers alone. Because, as I
| said, there are as many odd numbers as there are even
| numbers as there are natural numbers.
| rightbyte wrote:
| > the room numbers are all the natural numbers but they
| can all be occupied by the even numbers alone
|
| Hmm ... that is an interesting take on it. I.e. guests
| with even customer id:s can have all rooms.
|
| Maybe I just have to accept the paradox to be a property
| of infinite sets by definition.
| danbruc wrote:
| It's not really a paradox, just some things that work for
| finite sets no longer work for infinite sets. The sets {
| A, B, C, D } and { 1, 2, 3, 4 } have the same size
| because they both have four elements. Another way to see
| that they have the same size is that you can provide a
| one to one mapping between them, for example A - 4, B -
| 3, C - 2, D - 1.
|
| For infinite sets the first option no longer works, you
| can not write down a specific number for the number of
| elements in a set and then see that two sets have the
| same size because you wrote down the same number for both
| of them. But the second option still works, you can
| provide a one to one mapping between the elements of two
| sets, for example n - 2n to match all natural numbers
| with all even numbers. Each natural number n has an
| associated even number 2n and each even number 2n has an
| associated natural number n.
|
| And this is then just the definition of what it means for
| two sets to have the same size, there is a one to one
| mapping between their elements. And this works for finite
| sets as well as for infinite sets. Everything else are
| consequences of that. Take the natural numbers and take
| the natural numbers with the first k of them removed, the
| two sets still have the same size because you can pair n
| with n + k even if it is against your intuition that the
| size of a set does not change when you remove some of its
| elements.
| danbruc wrote:
| No, there is no reordering wave, it happens all in parallel.
| Everyone steps out of his room at the same time, everyone walks
| to his new room, whether it is n + 1, n + k, 2n, or whatever,
| and everyone steps into his new room at the same time. So it is
| not what you probably imagined, that the person steps out of
| room 1, knocks on the door of room two, tells the person in
| there to move to room three, and so on.
| dave333 wrote:
| Alternately, the rooms can simply be renumbered with no guest
| motion.
| beervirus wrote:
| There's no requirement for it to be instantaneous. It could
| be, buy it could also be sequential. The thought experiment
| works fine either way.
| [deleted]
| nightcracker wrote:
| And how is everyone supposed to know when to leave their room
| and where to go?
| danbruc wrote:
| There is a screen in each room that shows you a message
| when to move where. Or you get a call from the
| receptionist, but not an individual call but one
| broadcasted to all rooms, announcing the formula for your
| new room number. Make up whatever you want, its a thought
| experiment.
| [deleted]
| benjohnson wrote:
| It's a very unique universe that quite a bit different
| that ours - It would require an infinite amount of energy
| to communicate that call leading to infinite entropy.
| Given that the moves happen instantaneously, the first
| guest to arrive would instantly cause the heat-death of
| this universe.
| codeflo wrote:
| They all hear an announcement at the same time via the
| hotel's speaker system. (And we're already ignoring so many
| laws of physics that the speed of light is the least of our
| problems.)
| lisper wrote:
| The laws of physics get chucked out the window the moment
| you say "infinitely many rooms".
| Jyaif wrote:
| In a universe with an infinite speed of light, can there
| even be any kind of waves? Do physical waves not also
| propagate instantly?
| danbruc wrote:
| What kind of waves? I guess you could at least have a
| light source that periodically gets brighter and dimmer
| and a detector at the other end of the universe would
| detect the brightness oscillation and it would not really
| make any difference whether the light propagates with an
| finite or infinite speed besides of course the
| propagation time.
| state_less wrote:
| They always leave their room after 24hrs and go to the room
| next door?
| mason55 wrote:
| I know we're already assuming a hotel with infinite rooms but
| doesn't this break causality by assuming FTL information
| propagation?
| danbruc wrote:
| Just make each room half the size of the preceding one and
| you can fit all the rooms in twice the length of the first
| room. Also solves the problem that, when moving from room n
| to 2n, the guy in room one has only to walk to next room
| while the poor guy in room Graham's number has quite a
| journey ahead of him. Please address complaints about
| narrow rooms that can not even fit a single proton to the
| manager.
| simonh wrote:
| We consider different universes with different properties
| to our own in mathematics, and indeed in physics, all the
| time.
| mason55 wrote:
| Oh, I'm not arguing that the thought experiment is wrong.
| Just extending it in the direction of "reasons this
| doesn't work in the real world"
| morelisp wrote:
| Where would you even find the fuel for the infinitely-
| long bus! Mathematicians must be idiots.
| bmm6o wrote:
| I don't see an issue if you want the communication to not
| be instantaneous. The only requirements i see are that it
| has to be faster than the guests, so that everyone's room
| is empty when they get there.
| sandworm101 wrote:
| Why assume that the hotel's hallway is strait? Maybe it has
| an infinite number of rooms all around a circular lobby, a
| lobby that grows in diameter/circumference every time someone
| books a room. The message from the desk (in the middle of the
| lobby) then hits every room at the same time. No waves. Just
| instant simultaneous movement in every room.
| morelisp wrote:
| As soon as the hotel has taken in even a small infinity of
| patrons, this layout is not possible - a circle must have a
| finite circumference (and if we did try to generalize our
| definition of circle it would result also in an infinite
| radius).
|
| Maybe a hypersphere with finite bounds but infinite
| dimensions? My geometry is fairly poor but I think this is
| what is called, for reasons not directly related to the
| hotel paradox, a Hilbert sphere.
| ukj wrote:
| Distributed synchronisation is lost upon mathematicians.
| barnaclejive wrote:
| This was the topic of an episode of Omnibus
| https://www.omnibusproject.com/83
|
| "Twice a week, Ken Jennings and John Roderick add a new entry to
| the OMNIBUS, an encyclopedic reference work of strange-but-true
| stories that they are compiling as a time capsule for future
| generations."
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