Exploring Hilbert's Paradox: Vacancies and Infinite Summations

[AKG]

In Hilbert's famous paradox of the Grand Hotel, we have a hotel with an infinite number of rooms and an infinite number of guests, and we can create a vacancy by having each guest move over to the next room. However, I don't see how this works. For one, each individual guest moves, and each move by a guest creates a vacancy (when he leaves his room) and then eliminates a vacancy (when he occupies the next room). Each individual move changes the number of vacancies by zero. Why should an infinite number of such moves be any different? The sum of a countably infinite number of zeroes is zero, so how is the vacancy created?

Also, why is it permissible to say that all of those guests who move over actually do find a room (leaving one vacancy) and, there isn't always going to be one guest with no room (even if we can't say he's the "last" guest) but it is not permissible to do the following:

0 = 0 + 0 + 0 + 0 + ...
0 = (1 - 1) + (1 - 1) + (1 - 1) + ...
0 = 1 - 1 + 1 - 1 + 1 - 1 + ...
0 = 1 + (-1) + 1 + (-1) + 1 + (-1) + ...
0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ...
0 = 1 + 0 + 0 + 0 + ...
0 = 1 ?

 

[AKG]

Well I just read this:

The error here is that the associative law cannot be applied freely to infinite sums unless they are absolutely convergent. In fact, it is possible to show that in any field, 0 is not equal to 1.

(Source)

Why is it that we cannot do this with the associative law, but we can essentially do something similar with the guests and rooms. Actually, what does it mean to apply the associative law to an infinite sum? We are applying it to an infinite number of finite sums, are we not? Anyways, assuming someone clears that up, why is that something we cannot do, but we can essentially shift the association of guests and rooms, i.e. if you think of each room as +1, and each guest as -1, then originally we have (+1-1) + (+1-1) + ... and have zero vacancies. Then we shift the brackets over, and get 1 vacancy, as shifting the brackets over is similar to associating a guest with the next room.

 

[mathman]

Hilbert: As you know, every real hotel has only a finite number of rooms. Once you get to infinite, funny things happen. It is no different from saying the number of integers is the same as the number of even integers.

Series: Your second line (1-1+1...) is not absolutely convergent, so it is not surprising that rearranging terms gives a different answer.

 

[jcsd]

It's an interting problem, but the very fact your moving from sum whose value is a defined, to one whose value is undefined back to one which is defined, eems simlair to divison by zero to me.

 

[robert Ihnot]

If it is difficult to see the vacancy, look at it like this: The hotel manager tells every guest to move from room n to room 2n. Thus 1 goes to 2, 2 goes to 4, 3 goes to 6, and what do you know: We have empty rooms 1, 3, 5,...2n+1, +++, why we have half the rooms vacant! This way of working is a lot faster than just creating only 1 new vacancy per move!

 

[AKG]

Series: Your second line (1-1+1...) is not absolutely convergent, so it is not surprising that rearranging terms gives a different answer.

I haven't rearranged the terms at all. Please explain why the associative law cannot be applied to infinite sums that aren't absolutely convergent. Actually that sounds strange to me (it's what was quoted from Wikipedia). Could you explain exactly what's meant by that? And please also explain why we can essentially do an analogous thing with the people and rooms, but not with numbers.

 

[AKG]

Or better yet, prove to me that all guests can find a room after shifting over one. You can say that for all guests originally in room n, there must be a room n+1, but I can say that for every term t_n in the series such that t_n = -1, there is a term t_(n+1) = 1, so they can always cancel out.

 

[robert Ihnot]

As for the sum, so as 0=1; the problem with that is that an alternating series converges to diffent sums depending on how the terms are grouped, as you have shown. If the absolute value of the terms was convergent we have a different matter, but here, of course, the absolute value term by term is infinite. The error here is taking an infinite series, calling its sum 0, the making it an alternating series 1 -1, 1, -1 and then rearranging the terms, well, now the sum is not 0.

 

[AKG]

As for the sum, so as 0=1; the problem with that is that an alternating series converges to diffent sums depending on how the terms are grouped, as you have shown. If the absolute value of the terms was convergent we have a different matter, but here, of course, the absolute value term by term is infinite. The error here is taking an infinite series, calling its sum 0, the making it an alternating series 1 -1, 1, -1 and then rearranging the terms, well, now the sum is not 0.

This is the same "error" Hilbert seems to be making. And you're not saying why it's wrong, simply that it's wrong. Personally, I can't see how it makes any sense that the order in which terms are added should make a difference just because the terms are infinite and the series is not absolutely convergent. As long as you still ultimately add the same terms, why would order matter? But anyways, I'll take it that it does matter without an explanation for now. But then tell me why a reordering does matter, and thus is not allowed with 1 -1 + 1 + ... but is allowed with the guests and rooms.

In the attached image, consider the black dots to be guests, and the squares to be rooms. The red lines show the original associations of rooms to guests, and the blue lines show the associations after the move. Shifting association like this seems similar to shifting the brackets around with the series I presented. And, of course, if anyone can give a reason as to why the associative law cannot be used, or that a non-absolutely-convergent series cannot have the associative law applied to some of the terms, that would be nice.

EDIT: Maybe another way of putting it: we know that if we have a finite number of terms, we can freely associate the terms in the series and perform additions. The same is true with finite rooms/guests, i.e. if we have a 6 room hotel (more like a motel), we can puts guests 1 through 6 in rooms 1 through 6, or put guest 1 in room 2, and mix it up in general. In both cases, we can freely associate terms or room/guests. Now we can't freely associate them with an infinite series that's not absolutely convergent. Along the same lines, what makes Hilbert or you or anyone think they can freely associate guests like that with an infinite number of rooms?

ANOTHER EDIT: Please, also keep in mind the question : if each move made by a guest from one room to another can neither create nor eliminate a vacancy, how can an infinite number of them?

The most sensible thing to me is this: the ordering or association of a series does not matter. 1 + 1 - 1 + 1 - 1 + ..., strange as it sounds, should depend on whether there can be a pairing between the +1's and -1's or not. Essentially, the infinity of terms would have to be either odd or even (yes, it's weird, but no one said we're dealing with non-weird stuff). The arrangment should not matter. If we do not know whether it's even or odd, it's like asking what sin(x) is as x approaches infinity, and let's say x is in the set of all multiples of pi/2. Now if we say that a hotel is occupied, then moving the guests around shouldn't affect the occupied state. If we can actually pair the guests to the room, then it's occupied.

Of course, this isn't perfect, I'd have to think about it some more. But I think it's a little better. If anything, it at least provides us a criteria to determine if there are vacancies or not, whereas the normal approach does not. Whether the hotel is full or "half"-empty, we can always draw a one-to-one correspondence between rooms and guests, and in fact we can always draw a "five-to-one" correspondence too. So what criteria is there?

 

[master_coda]

This is the same "error" Hilbert seems to be making. And you're not saying why it's wrong, simply that it's wrong. Personally, I can't see how it makes any sense that the order in which terms are added should make a difference just because the terms are infinite and the series is not absolutely convergent. As long as you still ultimately add the same terms, why would order matter? But anyways, I'll take it that it does matter without an explanation for now. But then tell me why a reordering does matter, and thus is not allowed with 1 -1 + 1 + ... but is allowed with the guests and rooms.

The reason it's wrong to rearrange the terms is because if you assume that an infinite number of additions is associative, then you can show that 1=0 which is a contradiction. So your assumption that addition is associative in the infinite case must have been wrong.

Anyway, a lot of things break down when you just take the finite case and naively extend it to the infinite case.

 

[AKG]

The reason it's wrong to rearrange the terms is because if you assume that an infinite number of additions is associative, then you can show that 1=0 which is a contradiction. So your assumption that addition is associative in the infinite case must have been wrong.

That's like saying if you rearrange the guests and create one vacancy out of none, you have a contradiction, so you can't rearrange the guests. Again, you're not showing why it's wrong, only that it's wrong and leads to a contradiction, not explaining what's fundamentally wrong with applying the associative law like that. And what is really being done that's considered "association in the infinite case?" I'm applying the associative law to only a finite nubmer of terms (2 terms), simply doing it an infinite number of times. It seems to me that a series that is absolute convergent allows this sort of association, so it's not that we can't use the association law an infinite number of times. It simply leads to a contradiction in certain cases. Again, I think it's contradictory to state that 1=0 is a contradiction, but creating 1 vacancy in a hotel with zero vacancies is not a contradiction.

 

[Hurkyl]

In Hilbert's famous paradox of the Grand Hotel, we have a hotel with an infinite number of rooms and an infinite number of guests, and we can create a vacancy by having each guest move over to the next room. However, I don't see how this works.

If everybody moves to the next room, does anybody move into the first room?



As for your questions about infinite sums, allow me to stress that there is more involved in infinite sums than addition. If you start with the first number, then add the second number, then the third number, and so on, you will never have added an infinite number of terms; such an approach is simply inadequate.

 

[master_coda]

That's like saying if you rearrange the guests and create one vacancy out of none, you have a contradiction, so you can't rearrange the guests. Again, you're not showing why it's wrong, only that it's wrong and leads to a contradiction, not explaining what's fundamentally wrong with applying the associative law like that. And what is really being done that's considered "association in the infinite case?" I'm applying the associative law to only a finite nubmer of terms (2 terms), simply doing it an infinite number of times. It seems to me that a series that is absolute convergent allows this sort of association, so it's not that we can't use the association law an infinite number of times. It simply leads to a contradiction in certain cases. Again, I think it's contradictory to state that 1=0 is a contradiction, but creating 1 vacancy in a hotel with zero vacancies is not a contradiction.

Creating a vacany out of none isn't a contradiction, unless you assume that a hotel with an infinite number of rooms is supposed to behave exactly like a hotel with a finite number of rooms - there's no reason to make that assumption, so we don't. On the other hand, 1=0 is a contradiction according to any useful definition of numbers.

The reason addition is not associative in the infinite case is because defining it to be associative in an infinite case is because it cannot be done without rendering numbers useless.

 

[hello3719]

Suppose we arrange the sum 1-1+1-1+1...

by grouping them in packs of two we do get (1-1) +(1-1)+...
but this sum yields zero if and only if the number of terms is EVEN, Since this is an infinite sum then there is an infinite of terms. But infinite isn't a number so we can't judge if it is even or not, meaning that the grouping of these terms is inconclusive.

 

[wisky40]

starting with this identity 0=0, then 0=0+0+0+0+...

it's true that when every 0 is broken into (1-1), the alternating ones (positives & negatives) are even. I think AKG forgot one (-1) of the last pair.

I can work it out in this way also:
0= n0
0= n(1-1)
0= (n-1+1)(1-1)
0= (n-1)(1-1)+(1-1)
0=1+(n-1)(1-1)-1
0=1+0+0+0+......-1=1-1=0 AKG is not showing (-1) from the last pair.

 

[matt grime]

I have yet to see any compelling reason from AKG why the 1=0 sum paradox is equivalent to the Hilbert hotel. The best so far is that they 'seem' the same.

Let us prove that there is not problem in the Hilbert hotel:

let S be the set of people who do not find a new room after rearrangement. If S is non;empty it has a least element, s, say. However by COnstruction s was asked to move to room s+1 # so S is empty.

 

[jcsd]

starting with this identity 0=0, then 0=0+0+0+0+...

it's true that when every 0 is broken into (1-1), the alternating ones (positives & negatives) are even. I think AKG forgot one (-1) of the last pair.

I can work it out in this way also:
0= n0
0= n(1-1)
0= (n-1+1)(1-1)
0= (n-1)(1-1)+(1-1)
0=1+(n-1)(1-1)-1
0=1+0+0+0+......-1=1-1=0 AKG is not showing (-1) from the last pair.

But by definition there is no 'last pair'.

 

[AKG]

I have yet to see any compelling reason from AKG why the 1=0 sum paradox is equivalent to the Hilbert hotel. The best so far is that they 'seem' the same.

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:

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

 

[matt grime]

That sum is not allowed or rather it won't be if only finitely many of the entries are zer, as you must want in order to do the paradoxical thing., so your model does not hold. Algebric sums must be finite, or they are formal series that do not represent a natural number. Try again.

 

[AKG]

What's not allowed? Formal series do not represent a natural number? Why is that? What do they represent? Surely they can represent a real number. Now would you suggest then that the 1 of the reals is not the zero of the 1 of the naturals?

 

[matt grime]

They do not necessarily have to represent anything. The alternating sums of plus and minus ones that you want is not a valid operation in the integers (nor in the reals, but we don't need to even think about the reals for this question). You've still not offered me any reason to suppose that your attempt to model the hilbert hotel leads you to the 0=1 paradox.

 

[matt grime]

Or let, me clairfy that, I don't see why the 0=1 paradox not being 'allowed' means you don't understand how the hilbert hotel works.

 

[AKG]

I offered a suggestion in post 18. Then you said a formal series cannot represent a natural number. I'd like you to elaborate on this. I'm no longer talking about the alternating sums of plus and minus ones.

What is wrong with the following

$0 \in \mathbb{W}$
$\sum_{n=1} ^{\infty} 0 = 0$
$\therefore \sum_{n=1} ^{\infty} 0 \in \mathbb{W}$

It actually sounds somewhat reasonable to me to suggest that infinite series cannot represent natural number, but I'd like you to elaborate. Thanks.

 

[matt grime]

Let us at least talk about the integers, which are a ring. The algebraic operations are defined only on pairs of numbers, and hence by induction on any finite collection of numbers. They are not defined for infinite sums. Full stop. It may be reasonable to say that an infinite sum of elements of Z will be in Z if and only if finitely many of the terms are non;zero, and indeed that is acceptable, but it does not make sense to sum infinitely many of them and expect to stay in the integers. nor is it supposed to nor is it necessary for us to attempt to do so.

THere are such things as formal infinite sums. These are algebraic, and you've probably met them in terms of generating functions.

In short, there is no need to sum an infinite set of integers, and there is no framework for us to be even allowed to do this. Please note this is an algebraic not an analytic statement.

 

[AKG]

Let us at least talk about the integers, which are a ring. The algebraic operations are defined only on pairs of numbers, and hence by induction on any finite collection of numbers. They are not defined for infinite sums.

For one, associativity is only being applied to pairs of numbers, it just so happens that it is being applied to an infinite number of distinct pairs. What's technically wrong in that?

Full stop. It may be reasonable to say that an infinite sum of elements of Z will be in Z if and only if finitely many of the terms are non;zero, and indeed that is acceptable, but it does not make sense to sum infinitely many of them and expect to stay in the integers.

Yes, but I'm summing an infinite number of zeroes. Please review post # 18 and show me where I summed anything but zeroes together (this would constitute proving that $\exists n \in \mathbb{W}\mbox{ such that } v_n \neq 0$).

 

[matt grime]

USing the associativity is a red herring. That you are only doing things to finite parts of an infinite sum is immaterial, it's the fact that you're using an infinite sum inside Z that is wrong. I would also point out that you may add 1 to 1, that involves two elements of Z and gives an answer, yet adding the result of that up an infinite number of times doesn't mena you get an element of Z, does it?

Where in the axioms of a Ring do you see something that tells you how to add together an infinite number of elements in the ring? We are not dealing in analytic results here, remember, and it is a finitely generated ring, before some one starts to cite the infinite product of copies Z, as opposed to the infinite direct sum of copies of Z...

Post 18 is completely irrelevent, not to say wrong. v(1) is not zero, that is the whole point. the first person vacates room one and no one takes it from the existing set of guests thus freeing it for the new guest.

 

[AKG]

Post 18 is completely irrelevent, not to say wrong. v(1) is not zero, that is the whole point. the first person vacates room one and no one takes it from the existing set of guests thus freeing it for the new guest.

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.

 

[master_coda]

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.

This argument is based on the erroneous assumption that $\infty-\infty=0$.

You can't simply conclude that because the number of guests leaving the room is the same as the number of guests entering a room then number of vacancies created must be zero, because that makes the assumption that "number of guests leaving" and "number of guests entering" are in fact numbers. In the infinite case they are not.

 

[AKG]

master_coda

This argument is based on the erroneous assumption that $\infty - \infty = 0$.

You can't simply conclude that because the number of guests leaving the room is the same as the number of guests entering a room then number of vacancies created must be zero, because that makes the assumption that "number of guests leaving" and "number of guests entering" are in fact numbers. In the infinite case they are not.

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:

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

 

[hello3719]

still 0*infinity isn\t equal to 0.

 

[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.