Exploring Hilbert's Paradox: Vacancies and Infinite Summations

[AKG]

page 1 post 1.

moving the first guest makes the first room available and no one fills it, why doesn't that make a vacany? because he has to move into room 2 and the person there has to then move and so on in an infinite number of moves that can't occur "really"? The issue isn't about the physical meaning of moving rooms, it's just about bijections from an infinite set to itself, not about doing algorithms one step at a time: all the guests move simultaneously, what's wrong with that?

Why are you still on about page 1, post 1? We've already gone over why that's wrong. Honestly, this is really pathetic.

 

[matt grime]

Ok, so we can leave the algebra and focus on the post 18 argument about the each guest vacating one room and taking one room and there being a net change in vacancies of zero, yet there is room 1 free.

We can at least agree that the model is wrong because it does not reflect the 'reality' so there must be soemthing wrong in the belief that *counting* the rooms that are empty can be done by *counting* the rooms that are full. The net effect you are calculating is exactly showing that there are an infinite number of rooms (that is assuming the axiom of choice) as it just shows that the set of guests/rooms is in bijection with a proper subset of itself. That is all that shows, why must it show how many vacancies are created? we can see that it doesn't, therefore the argument doesn't apply there. So, by what argument do you claim that your counting model is correct?

 

[master_coda]

I gave you the bucket analogy. What's more, your attitude seems very closed-minded. What about another number characteristic to a set (like the cardinal number) but gave us some other information. Saying that it simply can't be done because no one has before is a poor attitude. I was reading some history of set theory, and remember reading that Kronecker was seriously discouraging Cantor from publishing his set theory because of the way it dealt with infinites, however, it's a good thing he went ahead. Later, paradoxes in the theory were discovered, e.g. Russel's and Cantor's paradoxes, but these were later solved using a reformulation of set theory (ZFC I believe). The assumption that we can't find a better way to deal with infinites (as Cantor and then Zermelo and Frankel and whoever else did) is a useless and unhelpful one.

Your bucket analogy doesn't tell us anything useful, either, since we have no way of knowing how an infinite number of buckets would do anything. Your analogy amounts to nothing more than "finite things in reality work this way, so shouldn't math always give the same results, even for infinite things?"

You aren't exactly providing helpful suggestions. You've really done nothing but point out results from set theory that seem counter-intuitive to you and then complain that we should be doing more to make set theory work the way you think it should.

There are lots of things that can be done to find different formulations of set theory, but the particular suggestions that you are giving are not useful. The only way to implement them would be to forbid large classes of operations to avoid having those operations contradict the properties you want. So we should just throw away all of these operations so that you have have a more philosophically pleasing version of set theory?

And that "poor attitude" remark is really pathetic. That's the same kind of thing I hear from people who insist they have a perpetual motion machine, or a solution to the halting problem, or have a bijection from that natural numbers to the real numbers. Apparently, being unwilling to waste time on something that's provably impossible is just a "poor attitude", or being "closed minded".

 

[AKG]

Your bucket analogy doesn't tell us anything useful, either, since we have no way of knowing how an infinite number of buckets would do anything. Your analogy amounts to nothing more than "finite things in reality work this way, so shouldn't math always give the same results, even for infinite things?"

No way of knowing? If you're stumped as to how much water an infinite number of empty buckets will contribute to a tub when poured into the tub, I don't see any reason to discuss this with you. Of course you can easily suggest that without a rigourous model for an infinite number of buckets, any common sense argument is inadmissible. I suppose that's fair enough, but not very constructive.

You aren't exactly providing helpful suggestions. You've really done nothing but point out results from set theory that seem counter-intuitive to you and then complain that we should be doing more to make set theory work the way you think it should.

I'm complaining about nothing.

There are lots of things that can be done to find different formulations of set theory, but the particular suggestions that you are giving are not useful.

Which suggestions?

The only way to implement them would be to forbid large classes of operations to avoid having those operations contradict the properties you want. So we should just throw away all of these operations so that you have have a more philosophically pleasing version of set theory?

Do you have any proofs for what is the only way to do this?

And that "poor attitude" remark is really pathetic. That's the same kind of thing I hear from people who insist they have a perpetual motion machine, or a solution to the halting problem, or have a bijection from that natural numbers to the real numbers. Apparently, being unwilling to waste time on something that's provably impossible is just a "poor attitude", or being "closed minded".

It's great you can be so quick to rule out any new ideas. The fact is, I'm not even providing any new ideas, I suggested they may be helpful. You do have a poor attitude, and if it makes you feel better to class my objections as the same as those suggesting they can build a perpetual motion device, that's fine. What's wrong with suggesting another way to characterize sets that allows for what we have about bijections, but gives some quantifiable meaning to what happens when you add one element to an infinite set?

 

[master_coda]

No way of knowing? If you're stumped as to how much water an infinite number of empty buckets will contribute to a tub when poured into the tub, I don't see any reason to discuss this with you. Of course you can easily suggest that without a rigourous model for an infinite number of buckets, any common sense argument is inadmissible. I suppose that's fair enough, but not very constructive.

How is it more constructive to pretend that there is a common sense argument to be made? If there isn't any way test something physically, then pretending that one particular interpretation has more physical validity than another is even less constructive. At least the "common sense arguments are worthless" point of view is honest.

It's great you can be so quick to rule out any new ideas. The fact is, I'm not even providing any new ideas, I suggested they may be helpful. You do have a poor attitude, and if it makes you feel better to class my objections as the same as those suggesting they can build a perpetual motion device, that's fine. What's wrong with suggesting another way to characterize sets that allows for what we have about bijections, but gives some quantifiable meaning to what happens when you add one element to an infinite set?

Hmm, I see. So despite the fact that you've made no effort to learn anything, or even listen to anything that anyone has said to you, you are going to insist that it is I who has the poor attitude. That's actually the most interesting paradox you've mentioned so far.

 

[AKG]

Here's an argument along your lines of reasoning. The fact that there exists a bijection between N and N\{1} proves nothing. It proves that for each guest, n, there exists a unique room n+1, but this doesn't prove that guest n can enter that room. If we have a finite set of rooms and guests, then we can say that if for some guest, n, there exists a room r(n), then guest n can occupy r(n), but there's no reason to believe the same logic that works with finite sets can be applied to infinite sets. The same "reason" that suggests that we cannot treat the effect of moving an infinite number of individual guests as the net effect of an infinite number of individual moves is the reason that suggests that treating the relationship between the existence of a room and the ability to enter a room in a finite case as the same relationship in the infinite case. Of course, no such sensible "reason" exists, as far as I can tell.

Hmm, I see. So despite the fact that you've made no effort to learn anything, or even listen to anything that anyone has said to you, you are going to insist that it is I who has the poor attitude. That's actually the most interesting paradox you've mentioned so far.

Of course I've listened and learned. I've found a satisfactory reason as to why the 0 = 1 argument doesn't hold (i.e. 0 = 0 + 0 + ... = 1 - 1 + 1 - 1 + ..., etc). However, all you've managed to say is that any other formulation of set theory is provably wrong, and there is no point in trying.

 

[master_coda]

Here's an argument along your lines of reasoning. The fact that there exists a bijection between N and N\{1} proves nothing. It proves that for each guest, n, there exists a unique room n+1, but this doesn't prove that guest n can enter that room. If we have a finite set of rooms and guests, then we can say that if for some guest, n, there exists a room r(n), then guest n can occupy r(n), but there's no reason to believe the same logic that works with finite sets can be applied to infinite sets. The same "reason" that suggests that we cannot treat the effect of moving an infinite number of individual guests as the net effect of an infinite number of individual moves is the reason that suggests that treating the relationship between the existence of a room and the ability to enter a room in a finite case as the same relationship in the infinite case. Of course, no such sensible "reason" exists, as far as I can tell.

Except that all we are trying to prove is that every guest has a room. We don't have to refer to an infinite number of guests to do that. Your proof required you to not only show that every move did not create a vacancy, but to assume that the aggregation of all the moves takes on certain properties just because its individual moves do.

Besides, you've entirely missed the point I was making earlier. If you dig deep enough, you'll probably find out that there is some definition that just basically says your interpretation is wrong and the standard one is right. This is not based on any "fundamental truth", but simply the fact that the standard interpretation allows us to do a lot of useful work with infinite sets, while your interpretation does not (because if forces us to discard so many powerful concepts as inconsistent).

I've found a satisfactory reason as to why the 0 = 1 argument doesn't hold (i.e. 0 = 0 + 0 + ... = 1 - 1 + 1 - 1 + ..., etc). However, all you've managed to say is that any other formulation of set theory is provably wrong, and there is no point in trying.

You haven't understood the point I'm making at all. The problem with your interpretation not that established theories are "right" or that your theory is "wrong". It's just that all your interpretation logically contradicts a lot of existing theory and constructions, and doesn't really give us any useful new constructions in return.

There's certainly nothing sacred about ZF set theory - there are a lot of other approaches to set theory (such as von Neumann-Bernays-Goedel set theory, New Foundations, etc.) and none of them are more "right". But most of the more recent formulations were attempts to strengthen set theory, not to weaken it by forcing us to discard large classes of constructions.

And we really would have no choice by to weaken set theory (any version of it) to accept your interpretation. By the very definition of consistency, a consistent theory cannot have both a statement and a separate, contradictory statement be true. If anything in the existing theory contradicted your interpretation, we would have no other choice but to discard it (and any other equivalent construction). So incorporating your interpretation really would require us to throw stuff away and not be able to replace it - no amount of trying will allow us to make a contradiction go away, all we can do is throw out one of the statements (and of course anything equivalent to that statement). Even then, there's no guarantee we can still produce a consistent theory, but that's the least we have to do.

 

[AKG]

Except that all we are trying to prove is that every guest has a room. We don't have to refer to an infinite number of guests to do that. Your proof required you to not only show that every move did not create a vacancy, but to assume that the aggregation of all the moves takes on certain properties just because its individual moves do.

Yours does the same. You show that each individual has a room, but don't show that the aggregation of all the moves follows normal finite rules, where if there exists a room for a guest, he can occupy it. You've given me no reason to accept your assumption.

As an additional thought, consider the set $\omega + 1$. It is both an infinite set, with cardinality $\aleph _0$ I believe, and it has a last element $\omega$, and in this set the successor of the last element does not exist. The same may be true for Hilbert's Hotel, if this interpretation of ordinals and cardinals is correct.

 

[matt grime]

By assumption, Hilberts hotel's room has order type w, there is no last room and we assume every guest may occupy a room; it's a thought experiment, not a real situation. Moreover, it appears you don't understand ordinals, for even if we relabelled the rooms to have order type w+1, then there is still an initial segment of order type w, and we don't need to make the person in room w+1 do anything, and we can still move people up one room in this initial segment where there is no last element.

Instead of thinking of set theory as explaining Hilbert's Hotel, try and understand the hotel as an explanation of the set theory, which is perhaps nearer the mark.

 

[master_coda]

Yours does the same. You show that each individual has a room, but don't show that the aggregation of all the moves follows normal finite rules, where if there exists a room for a guest, he can occupy it. You've given me no reason to accept your assumption.

Why would I care about showing that the aggregation of all moves follows finite rules?

Besides, proving that every guest can have a room and still leave room #1 empty contradicts the idea that the aggregation of all moves follows finite rules. So why would I even want to make that assumption?

 

[AKG]

Moreover, it appears you don't understand ordinals, for even if we relabelled the rooms to have order type w+1, then there is still an initial segment of order type w, and we don't need to make the person in room w+1 do anything, and we can still move people up one room in this initial segment where there is no last element.

Sorry, my mistake.

 

[AKG]

Why would I care about showing that the aggregation of all moves follows finite rules?

If you don't, you don't justify the assumption that just because for each guests, n, there exists a room n+1, that a guest can occupy that room.

 

[master_coda]

If you don't, you don't justify the assumption that just because for each guests, n, there exists a room n+1, that a guest can occupy that room.

No. I can prove that every guest can be put into a unique room, and still leave an empty room. I don't need to prove (or assume) anything about process of the guests moving to do that.

Your proof was based on proving something about every individual move, and using that plus an assumption to prove something about the aggregation of all moves.

My proof is only trying to prove something about individual people. I don't take any extra steps after that.


This isn't really going anywhere. The approach is entirely backwards. Taking a physical situation and trying to draw conclusions about math from it is a waste of time, since the argument always tends to break down into an argument about what mathematical theory is the most appropriate for modelling this particular situation. The actual math never gets discussed.

 

[AKG]

No. I can prove that every guest can be put into a unique room, and still leave an empty room. I don't need to prove (or assume) anything about process of the guests moving to do that.

Your proof was based on proving something about every individual move, and using that plus an assumption to prove something about the aggregation of all moves.

My proof is only trying to prove something about individual people. I don't take any extra steps after that.

This isn't really going anywhere. The approach is entirely backwards. Taking a physical situation and trying to draw conclusions about math from it is a waste of time, since the argument always tends to break down into an argument about what mathematical theory is the most appropriate for modelling this particular situation. The actual math never gets discussed.

No, I accept the actual mathematical part that for each n there exists an n+1. That's trivial, that is how the naturals are defined, and inductive set where each member has a successor in the set. But when you make assumptions as to how the actual guests and rooms will behave, you run into problems because, well, you make faulty assumptions. But I agree, this won't go any further.

 

[master_coda]

No, I accept the actual mathematical part that for each n there exists an n+1. That's trivial, that is how the naturals are defined, and inductive set where each member has a successor in the set. But when you make assumptions as to how the actual guests and rooms will behave, you run into problems because, well, you make faulty assumptions. But I agree, this won't go any further.

Yes, that's the problem. You've made different assumptions, and so you're going to complain that my assumptions are wrong. We never get to discuss set theory because there's worrying about what the proper way to model this situation is. And worrying about the proper modelling of a physically absurd situation is clearly not a productive use of time.

 

[ChristopherL]

I just don't understand.

The main problem I have with this paradox is it makes no sense. If there is an infinite number of rooms, then an infinite number of guests should be no problem. At a hotel you don't move guests room to room as they check in, you put them in the next available room, so the whole concept behind this paradox is farfetched to me.

 

[LukeD]

Sweet zombie paradox!

You're right, there is no paradox here. It is essentially just that 1+countable infinity is still countable infinity (as far as sets are concerned at least).

Read some basic set theory if you want to understand where this comes from and why there is no problem here.

 

[SW VandeCarr]

I haven't read all the posts, but it seems that with aleph null infinity there is a one to one correspondance between guests and rooms if and only if you specify one guest to each room and there are no empty rooms. In this case the hotel is always full regardless of how guests move by definition. However you may also allow many to one guests to some or all rooms and also allow for empty rooms. It's still all aleph null infinity.

 

[CRGreathouse]

My argument says that no single guest creates a vacancy, so it doesn't make sense to say that the whole movement somehow does.

This corresponds almost precisely to the fact that none of {1}, {1, 2}, {1, 2, 3}, ... are infinite, but {1, 2, 3, ...} is infinite.

 

[D H]

This is a twice-necromanced thread, CR. You are arguing with posts AKG made five years ago. He has progressed a lot since then.