An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Mike Kelly on 12 Oct 2006 20:54 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, ... ? -- mike.
From: Tony Orlow on 12 Oct 2006 21:58 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. > > For example, do you agree that, according to the definitions explicitly > given in the problem, we can conclude that if at time t, ball 15 is in > the vase, that therefore t <= -1/15? Uh, not exactly, but something like that. > > Cheers - Chas >
From: Tony Orlow on 12 Oct 2006 22:04 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 Marcus wrote: >>>>>>>>>>> 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). >>> Sorry, but I don't quite understand. When you stated the problem in >>> English, it ended with a question mark. But, your statement in >>> Mathematics does not end with a question mark. If it is a >>> problem/question, I think it should end with a question mark. Please >>> give the statement of the problem in Mathematics. >>> >> What is sum(n=1->oo: 9)? > > So, you are saying that the translation of "Given an unfillable vase and > an infinite set of balls, insert 10 balls in the vase, remove 1, and > repeat indefinitely. This part is the core statement of the problem. In the actual problem, time-wise, it does not repeat indefinitely, but only for a minute, however that is a diversion, and the sequence of events continues indefinitely, iteration-wise. 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 minus 1/2^n minutes. What will we find in > the vase at noon?" is "What is sum_{n=1}^{infty} 9?". 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. So, that part's irrelevant. You have an infinite series of events which diverges without bound. > > How would you translate this problem: "Add nine balls to a vase. Repeat > an infinite number of times. How many balls are in the vase?" > sum(n=1->oo: 9). It's no different from sum(n=1->oo: 10-1).
From: Tony Orlow on 12 Oct 2006 22:05 David Marcus 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. > > How about this problem: Start with an empty vase. Add a ball to a vase > at time 5. Remove it at time 6. How many balls are in the vase at time > 10? > > Is this a nonsensical question? > Not if that's all that happens. However, that doesn't relate to the ruse in the vase problem under discussion. So, what's your point?
From: Tony Orlow on 12 Oct 2006 22:06
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! |