An uncountable countable set
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From: Tony Orlow on 13 Oct 2006 12:23 Virgil wrote: > In article <452ef2bc(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> 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. > > 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? > The iterations occurring in the immediate vicinity of noon are indistinguishable, though infinite in number. That's why your theory loses focus on the problem. > What a weird world TO lives in. It is a weird world, when transfinitology can be considered the standard of anything.
From: Tony Orlow on 13 Oct 2006 12:26 Virgil wrote: > In article <452ef411(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> The whole point of the Zeno machine is to conceive of completing this >> infinite series of events, and yet, it compresses the vast majority of >> events into a single moment at noon, making it impossible to distinguish >> them. > Actually, in either version of the original problem, NONE of the > transactions take place AT noon. Each of them precedes noon. And, after each of those transactions, before noon, there is an increased finite number of balls in the vase. So, it's nothing but finite and growing before noon. Then, at noon.....what? The linear growth implodes? It's true hogwash at its worst, Virgil, and you know it.
From: Tony Orlow on 13 Oct 2006 12:30 Virgil wrote: > In article <452ef4a0(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> Randy Poe wrote: >>>>> Tony Orlow 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. >>>>> That's a red herring. It's not the name of the ball that's relevant, >>>>> but whether for any particular ball it is or isn't removed. >>>> The "name" is the identity. It doesn't matter which ball you remove, >>>> only how many at a time. >>>> >>>>>> Likewise, adding labels to the balls in this infinite case >>>>>> does not add any information as far as the quantity of balls. >>>>> No, but what the labels do is let us talk about a particular >>>>> ball, to answer the question "is this ball removed"? >>>> We care about the size of the collection. If replacing the elements with >>>> other elements changes the size of the set, then you are doing more than >>>> exchanging elements. >>>> >>>>> If there is a ball which is not removed, whatever label >>>>> is applied to it, then it is still in the vase. >>>> How convenient that you don't have labels for the balls that transpire >>>> arbitrarily close to noon. You don't have the labels necessary to >>>> complete this experiment. >>>> >>>>> If there is a ball which is removed, whatever label is >>>>> applied to it, then it is not in the vase. >>>> If a ball, any ball, is removed, then there is one fewer balls in the >>>> vase. >>>> >>>>>> That is >>>>>> entirely covered by the sequence of insertions and removals, >>>>>> quantitatively. >>>>> Specifically, that for each particular ball (whatever you >>>>> want to label it), there is a time when it comes out. >>>>> >>>> Specifically, that for every ball removed, 10 are inserted. >>> All of which are eventually removed. Every single one. >>> >>> - Randy >>> >> Every single one, each after another ten are inserted, of course. Come on! > > So lets put them all in one minute earlier so they are all in before any > have to be removed and each ball will be in for a longer time, and then > remove them one at a time according to the original schedule. > > According to TO, putting them in earlier and taking them out as before > leaves FEWER in the vase at noon, even though there is no change in > removals. If you decouple the series of insertions with the series of removals, each series having its own point of condensation (say, fill up to noon and empty up to 12:01), then you have a different problem. If the series, which is a sequence, specifies that only one is removed for every ten added, in alternation, then that creates a relation between the insertions and removals that's so obvious, it really is weird that it even merits discussion. Then again, people haggle over creationism and religion, too. Is not all one? According to Boole it is. :)
From: Tony Orlow on 13 Oct 2006 12:31 Virgil wrote: > In article <452ef5a6(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> stephen(a)nomail.com wrote: >>> Randy Poe <poespam-trap(a)yahoo.com> 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. >>>> No, there's one of your leaps again. >>>> That's a particularly weird one. >>>> "If the value at noon doesn't have THIS to do with the >>>> sequence, then it must not have ANYTHING to do with >>>> the sequence". >>>> There's no reason to make such a leap. >>>> - Randy >>> Actually I think Tony is right on this one. The >>> sequence Tony is talking about is >>> 1, 9, 18, 27, ... >> Uh, starts with 0, but do go on... >> >>> This sequence represents the number of balls at times before >>> noon. The sequence has nothing to do with the number of >>> balls at noon, as the value for noon does not appear in >>> the sequence. This is why nobody who argues that the >>> vase is empty at noon ever mentions such a sequence, and >>> instead point out the simple fact that each ball added >>> before noon is removed before noon. >>> >>> Stephen >>> >> So, the infinite sequence of finite iterations where we can actually >> tell exactly how many balls are in the vase has nothing to do with the >> vase's state at noon > > Right. > > > >> which is supposed to be the limit of this >> sequence? > > Why is it the limit of any sequence? > And since the set of balls removed by noon includes every ball, how > does TO come up with any balls still waiting to be removed at noon? You tell me how many were removed, and I'll tell you how many remain.
From: Tony Orlow on 13 Oct 2006 12:37
Virgil wrote: > 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? By being coupled with the insertion of ten additional balls. >>> 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. > The sum diverges. You know what I'm saying. There is no limit, much less of 0. >>>> 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. When does the last ball go in? Is it in by noon? > >>> 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. If at most one ball is removed at any time, and the vase becomes empty, then there is no other possibility but that there is a final ball removed. However, that's impossible. > > 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. Your point of condensation makes any distinction between the finite iterations in N and infinite iterations. Who can distinguish any of the infinite events happening within infinitesimal time of noon? |