An uncountable countable set
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From: Virgil on 11 Oct 2006 22:50 In article <452d8ef0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. The labeling is as relevant in those cases as in the case of two balls, numbered one and two, resepectively, with ball two being put in and left there and ball one not being put in at all.
From: David Marcus on 12 Oct 2006 03:19 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? -- David Marcus
From: David Marcus on 12 Oct 2006 03:21 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. -- David Marcus
From: Tony Orlow on 12 Oct 2006 10:48 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
From: Tony Orlow on 12 Oct 2006 10:49
Virgil wrote: > In article <452d8ef0(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> 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. > > The labeling is as relevant in those cases as in the case of two balls, > numbered one and two, resepectively, with ball two being put in and left > there and ball one not being put in at all. If you put ball 1 instead of 2, does that change the number of balls left in the vase? It shouldn't. |