An uncountable countable set
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From: Tony Orlow on 12 Oct 2006 14:27 Ross A. Finlayson 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. >> >>> 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"? >> >> If there is a ball which is not removed, whatever label >> is applied to it, then it is still in the vase. >> >> If there is a ball which is removed, whatever label is >> applied to it, then it is not 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. >> >> - Randy > > > I describe some conditions on the ball and vase problem that can help > make it more realistic. > > The golem with the marker in the vase, where you can't reach into the > vase, if you want one ball out for putting ten in, there would need to > be infinitely many golems if each can only hold one ball. > > Recently in this discussion about infinite sets and so on one of the > talking points about Cantor that has emerged is that he counts > backwards from infinity. > > The empty-vasers construct the argument that for any ball labelled n, > where each ball has some factory serial, they can denote some time > 1/2^n where that number has been retrieved from the vase. By the same > token, at time 1/2^n, ten balls were just placed in the vase. For each > of those, the various times they are retrieved from the vase are > exactly specified, and, at each of those ten more new ones are added to > the vase. At each constructed time, for n many iterations, the count > of balls in the vase is 9n. > > The count of balls in the vase is the difference of two divergent > series. > > > Ross > Exactly, though I dunno about the Golems.
From: Virgil on 12 Oct 2006 14:29 In article <1160669820.603144.288450(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > You question whether "all x in N" does exist, apparently. Based on > > > > what? > > > > > > Based on the impossibility to index the positions of our 0.111..., > > > > False. > > > > > based on the vase, based on many other contradictions arising from "all > > > x in N do exist". > > > > False. > > > > No proof given. > > 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".
From: Virgil on 12 Oct 2006 14:32 In article <1160675049.510897.169340(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > Tony Orlow wrote: > > > > 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"? > > By this nice example we see that this information is completely > irrelevant, as irrelevant as the possibility to find a bijection > between infinite sets and the mathematics built on this facility. > > > > If there is a ball which is not removed, whatever label > > is applied to it, then it is still in the vase. > > > > If there is a ball which is removed, whatever label is > > applied to it, then it is not in the vase. > > And if three balls without labels are in the vase, then they are inside > independent of someone's knowledge about their names or their different > properties. Don't forget: Cantor introduced set theory for use in > physics and chemistry. Don't forget: Originally mathematics was created > as a meaningful tool for science and economy. Mathematics is no more bound by that than modern chess is bound by its long forgotten beginnings. > > > > Specifically, that for each particular ball (whatever you > > want to label it), there is a time when it comes out. > > We are lucky that atoms do not carry labels, otherwise the universe > would probably be empty already. Whom does "Mueckenh" accuse of removing them?
From: Tony Orlow on 12 Oct 2006 14:38 Lester Zick wrote: > On Wed, 11 Oct 2006 20:38:53 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Wed, 11 Oct 2006 11:41:00 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Tue, 10 Oct 2006 13:56:43 -0400, Tony Orlow <tony(a)lightlink.com> >>>>> wrote: >>>>> >>>>>> Lester Zick wrote: >>>>>>> Tony, I'm going to strip as much as seems unessential. >>>>>>> >>>>>>> On Mon, 09 Oct 2006 15:56:32 -0400, Tony Orlow <tony(a)lightlink.com> >>>>>>> wrote: >>>>>>> >>>>>>>> Lester Zick wrote: >>>>>>>>> On Sun, 08 Oct 2006 22:07:23 -0400, Tony Orlow <tony(a)lightlink.com> >>>>>>>>> wrote: >>>>>>> [. . .] >>>>>>> >>>>>>>>> As far as transcendentals are concerned, Tony, the only thing that can >>>>>>>>> lie on a real number line in common with rationals/irrationals are >>>>>>>>> straight line segment approximations. That's the only linear order >>>>>>>>> possible. So either you give up transcendentals or a real number line. >>>>>>>> The trichotomy or real quantity itself defines a linear order. Each such >>>>>>>> value is greater than or less than every different value. Pi is >>>>>>>> transcendental - is it less than or greater than 3? Is there any doubt >>>>>>>> about that? >>>>>>> No but that only applies to linear approximations for pi, Tony. It has >>>>>>> nothing to do with the actual value of pi which lies off to the side >>>>>>> of any real number line. Imagine if you will the linear order you >>>>>>> speak of superimposed on a real number plane instead of a straight >>>>>>> line (which isn't even possible either). Then pi has to lie off to one >>>>>>> side of the straight line. So the metric for the straight line becomes >>>>>>> a non linear variable instead of linear. Thus the fact that the value >>>>>>> of pi lies between 3 and 4 doesn't show any value for pi and never >>>>>>> will. All you wind uyp with is a linear metric which approximates pi >>>>>>> which doesn't provide us with any real number line running along the >>>>>>> lines of 1, 2, e, 3, pi. So the real number line metric is variable. >>>>>>> >>>>>> Pinpointing particular points on the line may require varying degrees of >>>>>> computation/construction. A natural requires only a finite number of >>>>>> iterations of increment, a rational may require a division operation, >>>>>> and an irrational or a transcendental like pi may require an infinite >>>>>> computation to exactly pinpoint the value. That doesn't mean that value >>>>>> doesn't exist as a point on the line. It just mans specifying it >>>>>> requires an infinite process. >>>>> It requires an infinite process only because it isn't on the line. The >>>>> only thing an infinite process determines is how close the curve comes >>>>> to a straight line without ever getting there. Every other rational/ >>>>> irrational on the line can be located with finite processes because >>>>> they are on the line. >>>>> >>>> If the line is defined by trichotomy, and 3<pi<4, isn't pi on that line? >>> If by "trichotomy" you mean <=> on a straight line then no pi isn't on >>> that line, Tony. Pi lies on a circular curve not on a straight line. >>> Approximations for pi such as 3<pi<4 do lie on a straight line but >>> only indicate how close circular arcs come to straight line >>> approximations. >>> >> 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. > 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 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. > >>>>>>>>>> Perhaps not, but if, say, y=1/x is both oo and -oo at x=0, and oo=-oo, >>>>>>>>>> then the function is continuous in every respect, which is what we might >>>>>>>>>> desire in such a fundamental algebraic relation. >>>>>>>>> But for the division operation x never becomes zero. Which indicates >>>>>>>>> that there can be no plus or minus infinity and no continuity. >>>>>>>> Is 0 part of the continuum, or just another arbitrary "limit" >>>>>>>> discontinuity? When you ask yourself, "If I divide this finite space >>>>>>>> into individual points, how many will I have?", what answer do you get? >>>>>>>> How much of the space between 0 and 1 does each real in that interval >>>>>>>> occupy, and how many are there? >>>>>>> Well points don't occupy any space at all, Tony. >>>>>> Right, their measure is 0. >>>>>> >>>>>> Intersection ends in >>>>>>> points but subdivision never ends in points. >>>>>> In the limit, subdivision results in infinitesimal segments, where the >>>>>> values of the endpoints cannot be distinguished on any finite scale. >>>>> Oh I don't know about that, Tony. Certainly the first bisection in a >>>>> series is exactly half the length of the starting finite segment. >>>> Yes, and every finitely-indexed subdivision results in finite segments, >>>> but that cannot be truly when the segment is infinitely subdivided. >>> As far as that goes it never is. >> In essence, there is no end to subdivision. In the limit, the subsegment >> is a point, or a pair of identical points. > > No it isn't a point because the limit is never reached. Infinitesimal > subdivision is just a never ending process not a geometric thing or > arithmetic result. Nor is there any difference between one point and > two identical points. Either you have infinitesimals or nothing. > >>>>>> That's the difference >>>>>>> between intersection and differentiation. Although not a natural zero >>>>>>> certainly exists but the question is how it's used. Zero us
From: Tony Orlow on 12 Oct 2006 14:40
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)? |