An uncountable countable set
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From: Tony Orlow on 12 Oct 2006 22:11 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, which is supposed to be the limit of this sequence? Why even mention the gedanken at all then? I suppose every vase is empty at noon, or just whatever you feel like declaring. You're playing silly magic tricks. I'm ashamed for the planet.
From: Tony Orlow on 12 Oct 2006 22:12 Virgil wrote: > In article <452e882a(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> 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. > > If a particular ball is not removed it remains in the vase and if it is > removed it does not remain in the vase. After iteration n, n+1 through 10n remain. > > The set of numbered balls is well ordered by their numbering. If any > numbered balls remain in the vase at noon, then there must be one with > the least number of any remaining. n+1, or aleph_0+1, if you prefer. > Which one would that be TO? > Or does TO go around with an eraser erasing those numbers as the balls > are put into the vase? > Yes, that's what I do, to prevent them from disappearing, because I so hate magic. Uh huh. >>> Specifically, that for each particular ball (whatever you >>> want to label it), there is a time when it comes out. >>> >>> - Randy >>> >> Specifically, that for every ball removed, 10 are inserted. > > And later, but before noon, also removed. > > So TO what is the number on the first ball NOT removed??? n+1
From: Tony Orlow on 12 Oct 2006 22:15 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. > > You must take the >> limit as the number of iterations approach oo. > > No such 'limit' is defined. > It's defined as being divergent. > > >>>> ... 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. >>> Certainly does. I mean that sum from x=1 as x increases 2, 3, 4, ... >>> without limit of (10-1) diverges. >> 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. >>> Let me ask you another question, Tony, as I don't think you answered >>> the last one. >> I don't see any previous question at this point, but I'm relatively sure >> I answered what was asked. > > TO never answers what he was asked, he always goes off on some sort of > tangent, usually an irrelevant one. You don't even know what the subject is half the time. >> Here is an argument, ending with a conclusion I don't >>> personally swallow. Can you tell me at what point it goes wrong? >>> (Or do you think it is valid?) >>> >>> Consider the function step0: R -> R mapping x to 0 if x<0 and mapping x >>> to 1 if x>=0. >> A discontinuous function at x=0. >> >>> FWIW, we can write this function in a C-like way (taking 'TRUE' and >>> 'FALSE' to have the numeric values 1 and 0 respectively), so it is just >>> a simple expression: >>> >>> step0(x) = (x>=0) >>> >>> OK, for n a positive integer, now consider the sequence of values of >>> step0(p) for p=-1, -1/2, -1/3, ... -1/n, ... without end >>> >>> For any n, -1/n < 0, therefore step0(-1/n) = 0. >>> >>> So the sequence of values is simply the constant sequence 0, 0, 0, 0, >>> .... without end >>> >>> The limit of a constant sequence of values is the single value itself. >>> >>> Therefore lim(n->oo) step0(-1/n) = 0 >>> >>> By the Orlovian limit-swapping axiom, therefore: >>> >>> step0(lim(n->oo) -1/n) = 0 >>> >>> But lim (n->oo) -1/n = 0. >>> >>> Thus step0(0) = 0. >>> >>> But by definition, step0(0) = 1 >>> >>> Therefore 0 = 1. >> A function with such a declared discontinuity has two limits at that >> point, depending on the direction of approach. So, what else is new? >> That proves nothing. >> >> What causes a discontinuity at noon? I'll tell you. The von Neumann >> limit ordinals. That's schlock. You're concluding that a linear increase >> results in nothing, when it's clearly a series of operations which diverges. > > 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.
From: Tony Orlow on 12 Oct 2006 22:16 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?
From: Tony Orlow on 12 Oct 2006 22:19
Alan Morgan wrote: > In article <452e8c2a(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> 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)? > > I think you actually mean, what is 10-1+10-1+10-1.... > > It was recognized long before Cantor that there isn't a simple answer to > that question. > > Alan There is if you prohibit rearranging the terms to change the relative frequencies of the two terms. Group all you like without rearranging. This series is (+10-1)+(10-1)+(10-1)+... |