An uncountable countable set
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From: Tony Orlow on 12 Oct 2006 14:51 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> David Marcus wrote: > > <snip-snop: the valiant shall see> > >> Time is actually irrelevant. > > If you are trying to determine the limit of the sequence of operations, > time does appear to be irrelevant, yes. Individual operations are indistinguishable at noon. You must take the limit as the number of iterations approach oo. Then what do you get? Why do you have a conflict between looking at it in terms of iterations vs. time? Because of the clever little Zeno machine. Nice obfuscation. > >> ... 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. > > 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. 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. > > Corollary: set theory is inconsistent. That's not my conclusion, especially since that's not my logic. > > > Brian Chandler > http://imaginatorium.org >
From: Tony Orlow on 12 Oct 2006 14:54 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.
From: imaginatorium on 12 Oct 2006 15:17 Virgil wrote: > In article <1160669820.603144.288450(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: <bibble-babble-bobble> > > No proof possible because every proof must be dismissed unless the game > > of set theory should perish. > > The "game of set" theory, as defined by ZF or NBG or something similar, > will survive "Mueckenh". No, it will defeat him, game, set, and math. Thank you. Brian Chandler http://imaginatorium.org
From: David Marcus on 12 Oct 2006 16:57 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. 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?". 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?" -- David Marcus
From: David Marcus on 12 Oct 2006 17:02
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? -- David Marcus |