An uncountable countable set
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From: Virgil on 12 Oct 2006 18:07 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)? What is the lowest natural number on a ball remaining in the vase at noon?
From: Virgil on 12 Oct 2006 18:16 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. You must take the > limit as the number of iterations approach oo. No such 'limit' is defined. > > > > >> ... 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. > > > > > 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. > > 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.
From: Virgil on 12 Oct 2006 18:19 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.
From: Lester Zick on 12 Oct 2006 18:19 On Thu, 12 Oct 2006 14:38:42 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: [. . .] >>> Circular arcs approach the straight line in the limit as radius->oo, but >>> other than than, no, pi's a quantity, a distance from the origin That's >>> what a real number is. >> >> Not possible, Tony, unless you want to pretend circular arcs are >> congruent with straight lines. Pi is an exact measure on a circle. > >Rolling the circle along the line gives a certain congruence and a >linear measure of the circumference. Pi is an exact congruence on a circle not a "certain" congruence. >> This is one reason I take issue with conventional classifications of >> transcendentals as irrationals. There is no "real" number line in >> formal terms and even Bob Kolker publicly admitted the point. No >> transcendental is perfectly congruent with any straight line segment. > >Well, I'm not sure what that means, and no offense to Bob, but his >assent doesn't mean my automatic agreement. I don't assume it would. But it is interesting that Bob would concur. > I really think you're >talking about a construction of the number rather than its raw quantity, >and in that sense you probably have a point. But, that doesn't affect >whether or not there isa point on the real number line corresponding to pi. Oh but you're wrong here, Tony. Every pair of points on a straight line is constructable with straight line segments and right angles. In terms of pairs of points on straight line segments pairs of points on curves aren't. >>>>> But specific infinite values and specific infinitesimals do. >>>> But those are things you never have and can never have. >>> I have them. >> >> Then show them. If you could have them they would be finite. > >They have two ends, if that's what you mean, but a potentially >uncountably long repeating string connecting the ends. This gives us >rational portions of declared infinities, with which we can actually do >some arithmetic. :) Well sure as long as the declarations hold up. Which isn't very long. >>> The idea of >>>> infinitesimals is only a conceptual device or limit used to denote the >>>> end of an ongoing process which never actually ends and can never end. >>>> >>> You and I are specks, Lester. >> >> You and I are not infinitesimals, Tony. You seem to be trying to make >> some kind of TOE out of arithmetic. It isn't and never will be. >> > >Never say never. I hardly ever say never, Tony. Only in the case of science. >>>> For comparison and classification of different infinite sets. >>>> >>> Please give an example, much as that may be distasteful for you. :) >> >> Nothing comes to mind offhand. L'Hospital's rule I just remember from >> college calculus. I've never done transfinite arithmetic but if sets >> are really infinite and comparable they have to have differentiable >> properties. I've heard people say there are non differentiable >> infinite sets but I've never seen one and they get mighty sketchy when >> it comes to details. >> > >Yeah, transfinitology is rather sketchy. Unfortunately there's always a resort to mathemagic to be had. >> You can eliminate self contradictory alternatives simply enough by >> staying with derivatives of contradiction. In any event if you just >> argue dialectically by examples all you wind up with is problematic >> philosophy and not science at all because examples are not and can >> never be exhaustive. As bad as axiomatic math may be at least it isn't >> reduced to arguing dialectically. Yet. >> >> ~v~~ > >Well, what we're doing, hopefully, is discussing which axioms lead to >acceptable results, which don't contradict basic logic or finite >mathematics in principle. That's not a deductive exercise, but an >inductive one. Well processes leading up to discovery are inductive. But the result has to be deductive. ~v~~
From: Alan Morgan on 12 Oct 2006 18:11
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 -- Defendit numerus |