An uncountable countable set
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From: Tony Orlow on 12 Oct 2006 10:56 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> David Marcus wrote: >>>>>>>>> Tony Orlow wrote: >>>>>>>>>> David R Tribble wrote: >>>>>>>>>>> Virgil wrote: >>>>>>>>>>>>> Except for the first 10 balls, each insertion follow a removal and with >>>>>>>>>>>>> no exceptions each removal follows an insertion. >>>>>>>>>>> Tony Orlow wrote: >>>>>>>>>>>>> Which is why you have to have -9 balls at some point, so you can add 10, >>>>>>>>>>>>> remove 1, and have an empty vase. >>>>>>>>>>> David R Tribble wrote: >>>>>>>>>>>>> "At some point". Is that at the last moment before noon, when the >>>>>>>>>>>>> last 10 balls are added to the vase? >>>>>>>>>>>>> >>>>>>>>>>> Tony Orlow wrote: >>>>>>>>>>>> Yes, at the end of the previous iteration. If the vase is to become >>>>>>>>>>>> empty, it must be according to the rules of the gedanken. >>>>>>>>>>> The rules don't mention a last moment. >>>>>>>>>>> >>>>>>>>>> The conclusion you come to is that the vase empties. As balls are >>>>>>>>>> removed one at a time, that implies there is a last ball removed, does >>>>>>>>>> it not? >>>>>>>>> Please state the problem in English ("vase", "balls", "time", "remove") >>>>>>>>> and also state your translation of the problem into Mathematics (sets, >>>>>>>>> functions, numbers). >>>>>>>> Given an unfillable vase and an infinite set of balls, we are to insert >>>>>>>> 10 balls in the vase, remove 1, and repeat indefinitely. In order to >>>>>>>> have a definite conclusion to this experiment in infinity, we will >>>>>>>> perform the first iteration at a minute before noon, the next at a half >>>>>>>> minute before noon, etc, so that iteration n (starting at 0) occurs at >>>>>>>> noon-1/2^n) minutes, and the infinite sequence is done at noon. The >>>>>>>> question is, what will we find in the vase at noon? >>>>>>> OK. That is the English version. Now, what is the translation into >>>>>>> Mathematics? >>>>>> Can you only eat a crumb at a time? I gave you the infinite series >>>>>> interpretation of the problem in that paragraph, right after you >>>>>> snipped. Perhaps you should comment after each entire paragraph, or >>>>>> after reading the entire post. I'm not much into answering the same >>>>>> question multiple times per person. >>>>> I snipped it because it wasn't a statement of the problem, as far as I >>>>> could see, but rather various conclusions that one might draw. >>>> I drew those conclusions from the statement of the problem, with and >>>> without the labels. >>> 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). > > So, you are saying that the problem translates into the following sum: > > sum_{i=1}^infty (10-1). > > But, when you translate a problem into mathematics, you need to say how > each part of the problem is represented in the mathematics. In > particular, how are the times represented in your model and what in your > model represents the number of balls in the vase? > Time is actually irrelevant. The sequence is measured in iterations as n->oo, and the number of balls in the vase at iteration n is represented by sum(x=1->n: 9). The limit of this sum as x diverges also diverges in linear fashion. When you bring time into the picture with this Zeno machine, all that really does is mush together all infinite iterations in a single moment, making it impossible to pinpoint any of those events, and allowing for this set-theoretic interpretation, which contradicts a clearly divergent infinite series.
From: Tony Orlow on 12 Oct 2006 11:00 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.
From: Randy Poe on 12 Oct 2006 11:09 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. > 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"? If there is a ball which is not removed, whatever label is applied to it, then it is still in the vase. If there is a ball which is removed, whatever label is applied to it, then it is not 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. - Randy
From: Mike Kelly on 12 Oct 2006 11:24 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. -- mike.
From: Ross A. Finlayson on 12 Oct 2006 11:28
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. > > > 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"? > > If there is a ball which is not removed, whatever label > is applied to it, then it is still in the vase. > > If there is a ball which is removed, whatever label is > applied to it, then it is not 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. > > - Randy I describe some conditions on the ball and vase problem that can help make it more realistic. The golem with the marker in the vase, where you can't reach into the vase, if you want one ball out for putting ten in, there would need to be infinitely many golems if each can only hold one ball. Recently in this discussion about infinite sets and so on one of the talking points about Cantor that has emerged is that he counts backwards from infinity. The empty-vasers construct the argument that for any ball labelled n, where each ball has some factory serial, they can denote some time 1/2^n where that number has been retrieved from the vase. By the same token, at time 1/2^n, ten balls were just placed in the vase. For each of those, the various times they are retrieved from the vase are exactly specified, and, at each of those ten more new ones are added to the vase. At each constructed time, for n many iterations, the count of balls in the vase is 9n. The count of balls in the vase is the difference of two divergent series. Ross |