How Does Hilbert's Paradox Create Vacancies in an Infinite Hotel?

[master_coda]

No, you've misunderstood. I'm not assuming infinity - infinity is zero. It seems you're misinterpreting my argument to be something like this:

\sum _{k=1} ^{\infty} 1 - \sum _{k=1} ^{\infty}1 = 0

That's not what I'm saying at all. I'm saying:

\sum _{k=1} ^{\infty} (1-1) = 0

But you are making an "infinity - infinty = 0" argument. You're arguing that the number of people vacating a room is the same as the number of people taking a new room, so there must be zero new rooms available. You can't avoid that fact by trying to subtract the terms as you add them up instead of adding them all up first and subtracting them.


This "paradox" is just caused by the fact that when you have two infinite sets, you can compare their sizes in different ways to make it appear that one set or the other is smaller.

For example, if you have A = {0,1,2,...} and B = {1,2,...} then if you pair up every x in B with x in A, then A appears to be larger, since every element in B has been paired with an element in A, but there is still a 0 left over in A that has been paired up with nothing. On the only hand, if we pair up every x in B with x-1 in A, then the sets appear to be the same size, since there are no elements left over after we pair everything up.

So imagine that A = hotel rooms and B = people. Now according to one arrangement (x in B matched with x in A) room 0 is free, and according to the other arrangement there are no free rooms. You could also make an arrangement where every room is filled and there are still people left over with no room.

 

[hello3719]

and these 2 sums are the same, sum #1 is only a developped form of sum#2

 

[AKG]

But you are making an "infinity - infinty = 0" argument. You're arguing that the number of people vacating a room is the same as the number of people taking a new room, so there must be zero new rooms available. You can't avoid that fact by trying to subtract the terms as you add them up instead of adding them all up first and subtracting them.

No, if you're going to repeat that, I'm not going to bother repeating that you're misinterpreting. I'm not saying that the number of people leaving the room is the number of people entering the room, I'm saying that each move accounts for a zero change in vacancies, and a countably infinite number of moves which each result in a net change of zero in vacancies will create a net change of zero in vacancies. Find the sentence in post 18 that is flawed.

 

[AKG]

and these 2 sums are the same, sum #1 is only a developped form of sum#2

If you think that Infinite_Sum 1 - Infinite_Sum 1 is the same thing as Infinite_Sum (1-1), you're wrong. Infinite_Sum 1 is undefined, and there's no way undefined - undefined has any meaning, especially not the meaning Infinite_Sum (1-1).

 

[master_coda]

Let v_n represent the number of vacancies created by the room change made by the n^{th} guest. A guest cannot enter an occupied room. No more than one guest can be in a room at anyone time. If a guest is in a room, and moves to another room, then the room he/she was in becomes vacant by his or her leaving, and the vacant room he or she enters becomes occupied. \forall n \in \mathbb{N},\ v_n = 0. Now, we move all guests, so the number of vacancies created is:

\sum _{n=1} ^{\infty} v_n = \sum _{n=1} ^{\infty} 0 = 0

Well, one problem is that you assume that you can count the number of vacancies by adding up the vacancies created by each individual. All this really tells us is that a finite number of moves will not change the number of vacancies (which is correct).

Using an infinite sum does not automatically give you the result in the infinite case, just like you cannot always find the value of a function f(x) at x=a by taking the limit as x -> a.

 

[AKG]

Well, one problem is that you assume that you can count the number of vacancies by adding up the vacancies created by each individual.

What's wrong with this? So far you've simply said that I can't do it.

 

[master_coda]

What's wrong with this? So far you've simply said that I can't do it.

Because you are using limits. Just like limits as x -> a do not always give you the same answer as the result at x = a, limits as n -> infinity do not always give you the same answer as the result at infinity.

 

[AKG]

Where exactly am I using limits?

 

[master_coda]

Where exactly am I using limits?

\sum _{n=1} ^{\infty} v_n=\lim_{N\rightarrow\infty}\sum_{n=1}^N v_n

This is the definition of an infinite sum. This definition is used because, well, it works, and most of the time all we need is to take the limit as n -> infinity. Infinite sums aren't much use for working with the infinite case itself.

 

[AKG]

\sum _{n=1} ^{\infty} v_n=\lim_{N\rightarrow\infty}\sum_{n=1}^N v_n

This is the definition of an infinite sum. This definition is used because, well, it works, and most of the time all we need is to take the limit as n -> infinity. Infinite sums aren't much use for working with the infinite case itself.

I was going to note the fact that the limit is implied in the infinite series, but I didn't seriously think that you would object to that. And judging by what you said, I don't see a serious objection.

 

[master_coda]

I was going to note the fact that the limit is implied in the infinite series, but I didn't seriously think that you would object to that. And judging by what you said, I don't see a serious objection.

The fact that you're using a limit to describe the infinite case means you're assuming that the infinite case can be approximated by the finite case. If you want to do math, you have to prove that assumption. You can't just shrug off that problem with a "I don't see a serious objection".

 

[AKG]

Are you suggesting that evaluating an infinite series provides an "approximation" to the real answer? Please elaborate.

 

[master_coda]

Are you suggesting that evaluating an infinite series provides an "approximation" to the real answer? Please elaborate.

Usually the limit is the "real answer", in the sense that the limit has all the properties that we care about. A lot of the time, we even define the "real answer" as the one the limit gives us.

"Approximation" was really a poor word to use. I was trying to convey the idea that limits as N -> infinity are a way for trying to figure out what occurs in the infinite case by extrapolating from what happens when the finite case becomes arbitrarily large.

However, showing something it true for arbitrarily large finite cases isn't the same as showing it is true for the infinite case. Normally we don't even care, since the limit tells us everything we want to know anyway. But in cases where you can actually find an answer for the infinite case without using limits, you have to remember that the result of a limiting process may give a different answer since the limit does not actually prove anything about the infinite case itself.

 

[matt grime]

Rather than just make a redundant circular argument like this, point out the sentence that makes the false assumption. Upon leaving a room, a guest creates a vacancy. Upon entering a (vacant) room, a guest eliminates a vacancy. Each guest moves out of then into a room, thereby creating then eilminating a vacancy. How many new vacancies are created by this individual process? Zero. How do an infinite number of such moves create a vacancy? Or create an infinite number of vacancies? Or eliminate a finite or infinite number of vacancies? I suppose if you want to provide a useful answer, you would have to suggest a good reason as to why it is wrong to model the situation as an infinite number of guest movements. Or if you can pull it out some how, explain how creating a vacancy and the eliminating one results in something other than a zero change in the net vacancies.

Who moves into the first room after it's vacated? No one, that's where the vacancy comes from. Adding up and subtracting an infintie number of 1s does not model the situation because you can't add up and subtract an infinite number of ones in a well defined way just using the rules of the integers.

Got it? No one enters room 1, room 1 is then left empty...


Plus you're examing the guests. That is obviosuly the wrong thing to do: think about it, after the guests move there are still the same 'number' of guests in the same 'number' of rooms. What's important is the rooms they aren't in. Like the first room which is obviously left vacant for the new guest.


SOrry, to keep adding, but more things keep striking me about this. AKG, do you understand how to handle infinite sets? Particulary the ones ordered by N? You are almost going for the 'but the last guest has no where to go thing'. Do you at least see that room one becomes vacant? Now the only issue would be if some other guest had no room to go to. As i posted quite a while ago we can prove that there are no such guests by examining the first such, if there are some there must be a first one...

SO clearly a room becomes vacant, and your model doesn't allow this, hence your model is incorrect. Change the model.

 

[AKG]

Who moves into the first room after it's vacated? No one, that's where the vacancy comes from. Adding up and subtracting an infintie number of 1s does not model the situation because you can't add up and subtract an infinite number of ones in a well defined way just using the rules of the integers.

Got it? No one enters room 1, room 1 is then left empty...

I understand the paradox, this is unnecessary. I want to know what is wrong with my reasoning. And if you think I'm adding an infinite number of ones, then subtracting, please don't bother replying.

SOrry, to keep adding, but more things keep striking me about this. AKG, do you understand how to handle infinite sets? Particulary the ones ordered by N? You are almost going for the 'but the last guest has no where to go thing'. Do you at least see that room one becomes vacant? Now the only issue would be if some other guest had no room to go to. As i posted quite a while ago we can prove that there are no such guests by examining the first such, if there are some there must be a first one...

SO clearly a room becomes vacant, and your model doesn't allow this, hence your model is incorrect. Change the model.

Wrong. Your model assumes that you can freely associate rooms to guests, but that leads to a contradiction in that you've created 1 vacancy from none. If you don't see this as a contradiction, say what's wrong with my model. Don't say it's wrong because it doesn't fit your answer.

I understand that by the model using infinite sets it appears this is possible, and by my model which you haven't given any good reason to abandon, it is not. Another approach. Assume it takes zero seconds for a guest to move to the next room, but before guest 1 can move to room 2, room 2 must be vacated. For guest 2 to move to room 3, room 3 must be vacated. For guest n to move to room n+1, room n+1 must be vacated. Ultimately, this depends on the "last" room being vacated, but since no last room exists, this process is impossible. Set theory suggests that a bijection can be drawn between N and N\{1}, but I believe this leads to a contradiction, so either set theory is wrong, or I am and my argument is an inaccurate model of the situation. What is wrong with my post#18 argument. I'd like, if you can, pick our the line that is wrong. Is it wrong to assume that this can be modeled as an infinite number of moves? Is it wrong to assume that a guest moving into then out of a room can have no effect on the vacancies? I assume it's the latter, but if so, please give an answer other than "because set theory says so," I know that already. I'm not even assuming it's wrong, I just want to know how to defeat the possible objection that I've made.

 

[AKG]

master_coda

I think if assume (or define) that an infinite sequence of partial sums will converge on a real number, and you can show that for every arbitrarily small epsilon you can find an N such that the difference between the Nth partial sum and the proposed limit is less than epsilon, you can show that the sum cannot be any real number other than the limit. However, I don't know if it's right to assume that the sum of an infinite number of terms (even if they converge) must be a real number. Addition of infinite numbers is not part of the axioms, so I suppose it's just a definition that the sum will be real (if it converges).

 

[hello3719]

\sum _{n=1} ^{\infty} v_n = \sum _{n=1} ^{\infty} 0 = 0

The sum isn't equal to 0, it is indeterminate. since 0*infinity is indeterminate
unless you put limit

 

[master_coda]

master_coda

I think if assume (or define) that an infinite sequence of partial sums will converge on a real number, and you can show that for every arbitrarily small epsilon you can find an N such that the difference between the Nth partial sum and the proposed limit is less than epsilon, you can show that the sum cannot be any real number other than the limit. However, I don't know if it's right to assume that the sum of an infinite number of terms (even if they converge) must be a real number. Addition of infinite numbers is not part of the axioms, so I suppose it's just a definition that the sum will be real (if it converges).

The only problem is that there isn't really a good reason to assume that the infinite case can be modeled by taking the limit of the finite case. This isn't a problem when the only value you actually care about is what the series converges to, but if you actually care about the value produced by carrying out an infinite number of additions, then infinite series just don't tell you anything.

So basically, you could define the result of an infinite number of additions as the value that an arbitrarily large finite number of addition converges to, but since things tend to break we we assume that the infinite case must act like the finite case, this doesn't seem to be a good idea.

 

[pseudocarp]

AKG, here is a question for you. Have you played around much with that infinite sum you originally mentioned? Because there is an infinite group of 1 and -1's to play around with, you can group them and do things that seem misleading. What I mean is, you can say, "since we have an infinitude of +1 and an infinitude of -1, we can swap out any given finite number of signs."

So for instance, we could do this:

0 = (1-1) + etc
0 = 1 - 1 + 1 - 1 + etc
now you have the basic pieces and can rearrange them however you wish; in particular, you can go to infinity with ease:
0 = (1 + 1 - 1) + (1 + 1 - 1) + etc
which is
0 = 1 + 1 + 1 + ... = + infinity
and the same thing can be done to tend towards negative infinity. You can also group the 1's to make any number you wish - e.g.
0 = (1 + 1 + 1 + 1) - 1 + 1 - ... = (1 + 1 + 1 + 1) + (1 - 1) + (1 - 1) + ... = 4

I know seeing the last sum you want to say, but what happened to the two negative 1's that got switched? The answer is, with an infinite number of positive 1's, you can always 'make some more' to cancel out any finite group of negative 1's - in short, you can because there's no rule saying you can't.

Controversy over the use of such sums in proofs is the reason infinite arithmetic came to be separately studied. How convenient, to use a sum in your proof that can be made to equal any number.

You have to use the rules of infinite arithmetic in considering Hilbert's hotel - not looking at each finite step, but taking into account the fact that two infinities that differ by a finite number are commensurable.

 

[matt grime]

Wrong. Your model assumes that you can freely associate rooms to guests, but that leads to a contradiction in that you've created 1 vacancy from none. If you don't see this as a contradiction, say what's wrong with my model. Don't say it's wrong because it doesn't fit your answer.

i have no model, what model of mine are you referring to? as you are attempting to model a situation and your model says that something is impossible when it is possible (in the imaginary world where there are an infinite number of rooms) then your model of it is wrong, it doesn't fit the "observed" data.

I understand that by the model using infinite sets it appears this is possible, and by my model which you haven't given any good reason to abandon, it is not. Another approach. Assume it takes zero seconds for a guest to move to the next room, but before guest 1 can move to room 2, room 2 must be vacated. For guest 2 to move to room 3, room 3 must be vacated. For guest n to move to room n+1, room n+1 must be vacated. Ultimately, this depends on the "last" room being vacated, but since no last room exists, this process is impossible.

rubbish, this is an idealized situation, there is no need to consider these problems. it's an hotel with an infinite number of rooms, i don't think reality has any place in the discussion, that is those things that are constructible in a finite number of steps.


moreover, here's a way round it. send all the guests out for an hour, when they come back tell them to go to the next numbered room. their luggage may be moved at later convenient time for all parties

Set theory suggests that a bijection can be drawn between N and N\{1}, but I believe this leads to a contradiction, so either set theory is wrong, or I am and my argument is an inaccurate model of the situation. What is wrong with my post#18 argument. I'd like, if you can, pick our the line that is wrong. Is it wrong to assume that this can be modeled as an infinite number of moves? Is it wrong to assume that a guest moving into then out of a room can have no effect on the vacancies? I assume it's the latter, but if so, please give an answer other than "because set theory says so," I know that already. I'm not even assuming it's wrong, I just want to know how to defeat the possible objection that I've made.

you've not actually made any real objections, so why don't you repost here your alleged objection, rather than make people read back three pages. It simply appears that you don't know what the definition of cardinality is. Cardinality of infinite sets cannot be deduced from simply adding things up then subtracting them, otherwise there are no odd numbers are there?

the simple objection to your post 18 is that you are using the rules of finite arithemetic as if it applies to the infinite sums you cite. that is not true, and has no place in algebra. simply that.

I believe post 18 is error for the same reasons as I stated above: it deosn't matter waht rooms the people move into it only matters waht they vacate. room 1 is vacated, and no one enters it. your 'model' does not indicate this at any point. you're misusing infinite arithmetic. where do you have any evidence you're allowed to do this? you don't even need to look at the reals, and even there what you write is still wrong.

 

[pseudocarp]

Dear matt grime,

why not be civil? Are you writing in this forum in order to educate, instruct, clarify? It seems to me you are on some kind of petty power trip. If someone does not get your explanation, either go slower, or give up. It is unpleasant to read jerky posts, and surely is both obnoxious and unenlightening to repeat yourself. p-carp

 

[Hurkyl]

A correction:

<br /> \sum_{i=1}^{\infty} 0 = 0<br />

is actually correct. (and this has nothing to do with &infin; * 0)

 

[master_coda]

A correction:

<br /> \sum_{i=1}^{\infty} 0 = 0<br />

is actually correct. (and this has nothing to do with ∞ * 0)

Oh, that is the correct way to evaluate the infinite sum. It's just that the sum does not actually represent the "number of vacancies created" in the infinite case.

 

[hello3719]

yea sorry my mistake

 

[matt grime]

Dear matt grime,

why not be civil? Are you writing in this forum in order to educate, instruct, clarify? It seems to me you are on some kind of petty power trip. If someone does not get your explanation, either go slower, or give up. It is unpleasant to read jerky posts, and surely is both obnoxious and unenlightening to repeat yourself. p-carp

if people don't get the simple explanations, and apparently haven't taken time to consider them and think them through, what is wrong with repeating a simple argument? replies of mine depend upon the person i reply to. if they act lilke cranks they get treated like cranks.

want another interpretation? simple. hilbet's hotel is just offering an analogy of the fact that there is a bijection between N and N\{1}; why does the fact that you cannot do arithmetic with infinite sums have any bearing on this unless you're thinking about it in the wrong way; the moving of guests is not as important as the rooms that are freed.


so, shall we restate the objections one last time? you may not use the rules of arithmetic on infinite sums like this, in particular the alternating sum of plus and minus one. You do not evaluate infinite cardinalities in this fashion, you must use bijections, that is how cardinality is defined, not by adding things up and subtracting things.

 

[AKG]

matt grime

rubbish, this is an idealized situation, there is no need to consider these problems. it's an hotel with an infinite number of rooms, i don't think reality has any place in the discussion, that is those things that are constructible in a finite number of steps.

Of course you do. You argue that for any guest, n, there is a room, n+1. My objection is that he can only enter room n+1 if n+1 is empty.

moreover, here's a way round it. send all the guests out for an hour, when they come back tell them to go to the next numbered room. their luggage may be moved at later convenient time for all parties[/quote]That's the obvious answer, but then this suggests that when the uncountably infinite number of coaches with uncountably infinite number of guests arrive, everyone steps out into the courtyard, and then they all just go into the rooms, without having to do anything fancy like shift down all the guests in prime numbered rooms (or whatever the solution is for that case).

want another interpretation? simple. hilbet's hotel is just offering an analogy of the fact that there is a bijection between N and N\{1};

That's plainly obvious, my point is simply that current set theory provides this solution, and I suggest that it contradicts something that's more fundamentally true, or that it is an inappropriate model for this situation. So rather than saying "according to set theory, this is true, Q.E.D." show why it is a better model, or how it overcomes certain objections, etc. I highly doubt you'll be able to do this simply because of the fact that you aren't reading what I'm saying.

you may not use the rules of arithmetic on infinite sums like this, in particular the alternating sum of plus and minus one.

For the last time, this is irrelevant and I'm doing no such thing.

why does the fact that you cannot do arithmetic with infinite sums have any bearing on this unless you're thinking about it in the wrong way; the moving of guests is not as important as the rooms that are freed.

If you already presuppose that rooms are freed, I can see you'll be of no help. I'm not trying to be difficult. I understand the problem very well and I can understand we can easily map N to N\{1}, and thus it appears we can place each individual in one of those rooms. But at the same time, I suppose I'm saying that I don't think that addresses the whole issue. Perhaps, in another way, I believe that a better way to deal with infinites would be desirable. One that can make sense of the fact that we can map N to N\{1} but at the same time be able to distinguish the two sets; i.e. the size of a set after pulling out 4 of the elements can still be infinite but not the same infinity. Hopefully this isn't necessary, which is why I'm trying to see if set theory offers a more fundamental or better explanation as to why an infinite number of moves can change the number of vacancies no individual move can. At any rate, I'm willing to leave this at that; until I can think of a better way, I can live with the existing solution to Hilbert's paradox, I was looking for potentially better ways to deal with infinites, that's all.

 

[Hurkyl]

One that can make sense of the fact that we can map N to N\{1} but at the same time be able to distinguish the two sets;

Well, one contains the number one, and one does not. The latter is a subset of the former.

the size of a set after pulling out 4 of the elements can still be infinite but not the same infinity.

Well, the size of a set (more precisely, the cardinality of a set) is, by definition based on 1-1 correspondences, so there's not much hope here.

I was looking for potentially better ways to deal with infinites, that's all.

For the most part, the goal is to learn what you can't do with infinities. Then, the tools and techniques that still remain, are the good ways to deal with infinities.

 

[master_coda]

That's plainly obvious, my point is simply that current set theory provides this solution, and I suggest that it contradicts something that's more fundamentally true, or that it is an inappropriate model for this situation. So rather than saying "according to set theory, this is true, Q.E.D." show why it is a better model, or how it overcomes certain objections, etc.

Well, the "fundamentally true" argument is a complete waste of time argument. Anyone can assert that their interpretation, or definitions, or axioms, or whatever is more fundamentally true, so this kind of an assertion never goes anywhere useful.


Besides, most of your objections seem to just be demanding some sort of philosophical explanation as to why infinite sets don't behave the same as finite sets. Or why an infinite number of moving people doesn't behave like a finite number of moving people. Can you give us a reason why they should behave in the same way?

Math says infinite things don't work like finite things because when we try to force infinite things to behave like finite things, contradictions keep popping up. I don't understand why we need another reason to stop trying to model infinity with finite things.

 

[AKG]

Well, the "fundamentally true" argument is a complete waste of time argument. Anyone can assert that their interpretation, or definitions, or axioms, or whatever is more fundamentally true, so this kind of an assertion never goes anywhere useful.

If anyone can assert their axioms to be more fundamentally true, and two consistent logics with contradictory axioms exist, then we have problems, because both can't be true.

Besides, most of your objections seem to just be demanding some sort of philosophical explanation as to why infinite sets don't behave the same as finite sets. Or why an infinite number of moving people doesn't behave like a finite number of moving people. Can you give us a reason why they should behave in the same way?

I have only said thing along the lines of "a person leaving then going into a room doesn't effect the number of vacancies." Whatever room he's vacated, he's taken a room that was previously vacant, netting zero vacancies. I don't see a problem there, so why is there?

Math says infinite things don't work like finite things because when we try to force infinite things to behave like finite things, contradictions keep popping up. I don't understand why we need another reason to stop trying to model infinity with finite things.

I don't understand your last sentence. And does "math say" that infinite things simply cannot behave at all like finite things, or does "math" simply not know how to yet?

 

[master_coda]

If anyone can assert their axioms to be more fundamentally true, and two consistent logics with contradictory axioms exist, then we have problems, because both can't be true.

So what is more fundamentally true: Euclidean geometry, Hyperbolic geometry or Elliptic geometry? They are all consistent geometries, and have mutually contradictory axioms.

Or we could just recognize that arguing about whose axioms and definitions are more fundamentally true is a waste of time.

I have only said thing along the lines of "a person leaving then going into a room doesn't effect the number of vacancies." Whatever room he's vacated, he's taken a room that was previously vacant, netting zero vacancies. I don't see a problem there, so why is there?

Well, you're again assuming that a infinite number of moves should act like a finite number of moves.

I don't understand your last sentence. And does "math say" that infinite things simply cannot behave at all like finite things, or does "math" simply not know how to yet?

Well, we can prove that {1,2,3,...} people can be placed in {1,2,3,...} rooms and can be placed in {2,3,4,...} rooms. So any set theory that allows the use of infinite sets and functions is inconsistent with a theory that forces infinite sets to behave like finite sets.

You can produce a set theory where all sets act like finite sets. That would require you to throw away almost all of set theory, and probably require you to completely reformulate any theory that makes use of infinite sets (which probably isn't possible for most theories).

So what would this new forumation of set theory gain us?