An uncountable countable set
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From: Virgil on 12 Oct 2006 23:48 In article <452ef6c6(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452e8ea4(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> Individual operations are indistinguishable at noon. > > > > Individual operations all occur prior to noon. > > > > All operations prior to noon result in a growing but finite number of > balls in the vase. How does the removal of a ball from the vase (one or the many operations before noon) increase the number in the vase. TO? > > > > > You must take the > >> limit as the number of iterations approach oo. > > > > No such 'limit' is defined. > > > > It's defined as being divergent. Wrong! Only convergent sequences have limits. For divergent sequences, there is no limit. That is what diverging means, no limit. See any standard calculus text for a definition of limits of sequences. > > > > >> Right, and that characterizes the salient features of the gedanken. > > > > It ignores one monumentally salient feature: > > that there is a time before noon at which each ball is removed. > > Yes, directly after a set of ten has been inserted. But I suppose that's > irrelevant. Does it prevent every ball from having a precise time before noon at which it is removed? Unless it does, it IS irrelevant. > > > > The number of balls as a function of the number of insertion-removal > > operations completed certainly diverges, but how this prevents any > > specific ball from being removed before noon is not apparent, > > particularly when there is a specific rule determining when each ball is > > removed. > > Yes, directly after ten others are inserted. The only way this could > result in an empty vase is if it had, at some point, -9 balls in the vase. There goes TO's phoney "last ball" argument again. Let S be the set of balls for which there is a specific time prior to noon for removal. Clearly S is a subset of N, the set of balls inserted. So TO is claiming N\S is not empty, but cannot find any members of it, even using both hands.
From: Virgil on 12 Oct 2006 23:52 In article <452ef6e8(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452e8f61(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <452e5862$1(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> David Marcus wrote: > >>>>> Virgil wrote: > >>>>>> In article <452d11ca(a)news2.lightlink.com>, > >>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>> > >>>>>>>> I'm sorry, but I can't separate your statement of the problem from > >>>>>>>> your > >>>>>>>> conclusions. Please give just the statement. > >>>>>>> The sequence of events consists of adding 10 and removing 1, an > >>>>>>> infinite > >>>>>>> number of times. In other words, it's an infinite series of (+10-1). > >>>>>> That deliberately and specifically omits the requirement of > >>>>>> identifying > >>>>>> and tracking each ball individually as required in the originally > >>>>>> stated > >>>>>> problem, in which each ball is uniquely identified and tracked. > >>>>> It would seem best to include the ball ID numbers in the model. > >>>>> > >>>> Changing the label on a ball does not make it any less of a ball, and > >>>> won't make it disappear. If I put 8 balls in an empty vase, and remove > >>>> 4, you know there are 4 remaining, and it would be insane to claim that > >>>> you could not solve that problem without knowing the names of the balls > >>>> individually. Likewise, adding labels to the balls in this infinite case > >>>> does not add any information as far as the quantity of balls. That is > >>>> entirely covered by the sequence of insertions and removals, > >>>> quantitatively. > >>> > >>> If, as in the original problem, each ball is distinguishable from any > >>> other ball, TO should be able to tell us which ball or balls, if any, > >>> remain in the vase at noon. > >>> > >>> Suppose, for example, the balls are all of different sizes, with each in > >>> sequence being only 9/10 as large as its predecessors. Then each > >>> iteration consists of putting that largest 10 balls that have not yet > >>> been in the vase into it and then taking the largest ball in the vase > >>> out. > >>> > >>> Since the balls are well ordered by decreasing size, any non-empty set > >>> of them must have a largest ball in it. So what is the largest ball in > >>> the vase at noon, TO? > >> It's obviously going to be infinitesimal. > > > > But since none of the actual balls are infinitesimal (for every n in N, > > (9/10)^(n-1) is finitesimal, not infinitesimal) that means there are no > > balls in the vase at noon. > > What is the smallest ball inserted? That is a "Have you stopped beating your wife?" question, which assumes a condition contrary to fact. As there is no last ball inserted, there is no smallest ball inserted, nor any last ball removed, nevertheless, all balls are inserted, and all balls are removed, and all before noon.
From: Virgil on 12 Oct 2006 23:55 In article <452ef7e7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>>> cbrown(a)cbrownsystems.com wrote: > >>>>> Tony Orlow wrote: > >>>>>> Virgil wrote: > >>>>>>> In article <452d11ca(a)news2.lightlink.com>, > >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>> > >>>>>>>>> I'm sorry, but I can't separate your statement of the problem from > >>>>>>>>> your > >>>>>>>>> conclusions. Please give just the statement. > >>>>>>>>> > >>>>>>>> The sequence of events consists of adding 10 and removing 1, an > >>>>>>>> infinite > >>>>>>>> number of times. In other words, it's an infinite series of (+10-1). > >>>>>>> That deliberately and specifically omits the requirement of > >>>>>>> identifying > >>>>>>> and tracking each ball individually as required in the originally > >>>>>>> stated > >>>>>>> problem, in which each ball is uniquely identified and tracked. > >>>>>> The original statement contrasted two situations which both matched > >>>>>> this > >>>>>> scenario. The difference between them was the label on the ball > >>>>>> removed > >>>>>> at each iteration, and yet, that's not relevant to how many balls are > >>>>>> in > >>>>>> the vase at, or before, noon. > >>>>> Do you think that the numbering of the balls is not relevant to > >>>>> determining the answer to the question "Is there a ball labelled 15 in > >>>>> the vase at 1/20 second before midnight?" > >>>>> > >>>>> Cheers - Chas > >>>>> > >>>> If it's a question specifically about the labels, as that is, then it's > >>>> relevant. It's not relevant to the number of balls in the vase at any > >>>> time, as long as the sequence of inserting 10 and removing 1 is the > >>>> same. > >>>> > >>>> Tony > >>> Ah, but noon is not a part of the sequence of iterations. No more than > >>> 0 is an element of the sequence 1, 1/2, 1/4, 1/8, .... > >>> > >>> The question asks how many balls are in the vase at noon. Not at some > >>> iteration. > >>> > >> Ah, but if noon is not part of the sequence, then nothing from the > >> sequence has anything whatsoever to do with how many balls are in the > >> vase at noon. I think there are three, you know, the number of licks it > >> takes to get to the tootsie roll center of a tootsie pop. That makes > >> about as much sense as saying an infinite number of them vanish. If noon > >> is not part of your sequence, then it's a nonsensical question, and if > >> it is, then the limit applies. > > > > So, do you think 0 is an element of the sequence 1, 1/2, 1/4, 1/8, ... > > ? > > > > No, Iv'v come around. I agree with you now. :) 0 has nothing whatsoever > to do with that sequence at all. Isn't that right? So TO concedes that the situation at noon has nothing to do with the sequence of numbers of balls at times before noon.
From: Ross A. Finlayson on 13 Oct 2006 00:02 Virgil wrote: > > So that TO says that at noon, after a particular ball, like number 15, > has been removed, it may not be distinguishable whether ball 15 is in > the vase of not? > > What a weird world TO lives in. And you don't? Ross
From: cbrown on 13 Oct 2006 01:00
Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> Virgil wrote: > >>>>> In article <452d11ca(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>>> I'm sorry, but I can't separate your statement of the problem from your > >>>>>>> conclusions. Please give just the statement. > >>>>>>> > >>>>>> The sequence of events consists of adding 10 and removing 1, an infinite > >>>>>> number of times. In other words, it's an infinite series of (+10-1). > >>>>> That deliberately and specifically omits the requirement of identifying > >>>>> and tracking each ball individually as required in the originally stated > >>>>> problem, in which each ball is uniquely identified and tracked. > >>>> The original statement contrasted two situations which both matched this > >>>> scenario. The difference between them was the label on the ball removed > >>>> at each iteration, and yet, that's not relevant to how many balls are in > >>>> the vase at, or before, noon. > >>> Do you think that the numbering of the balls is not relevant to > >>> determining the answer to the question "Is there a ball labelled 15 in > >>> the vase at 1/20 second before midnight?" > >>> > >>> Cheers - Chas > >>> > >> If it's a question specifically about the labels, as that is, then it's > >> relevant. It's not relevant to the number of balls in the vase at any > >> time, as long as the sequence of inserting 10 and removing 1 is the same. > >> > > > > Putting aside the question of /how/ (limit? sum of binary functions?) > > one determines the /number/ of balls in the vase at time t for a > > moment... > > > > Do you then agree that there is some explicit relationship described in > > the problem between what time it is, and whether any particular > > labelled ball, for example the ball labelled 15, is in the vase at that > > time? > > For any finite time before noon, when iterations of the problem are > temporally distinguishable, yes, but at noon, no. > I don't understand why you think this would be the case. Why do you think the relationship holds for t < 0? Why you do think it does not hold for t >= 0? Cheers - Chas |