An uncountable countable set
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From: David Marcus on 23 Oct 2006 20:31 cbrown(a)cbrownsystems.com wrote: > stephen(a)nomail.com wrote: > > With the added surreal twist that the limit of the number > > of unnumbered balls in the vase as we approach noon is 0, > > but the number of unnumbered balls in the vase at noon is > > infinite. :) > > I think his response, when I pointed this out to him, was either "Oh, > shut up!" or "Whatever." That is consistent with my suggestion that Tony is reasoning by imagining a vase filling up. If you visualize the vase filling up in your mind, you don't see the unnumbered balls in the picture. -- David Marcus
From: Virgil on 23 Oct 2006 20:58 In article <453d52d1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Randy Poe wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>> > >>>>>>>> At every time before noon there are a growing number of balls in the > >>>>>>>> vase. The only way to actually remove all naturally numbered balls > >>>>>>>> from > >>>>>>>> the vase is to actually reach noon, in which case you have extended > >>>>>>>> the > >>>>>>>> experiment and added infinitely-numbered balls to the vase. All > >>>>>>>> naturally numbered balls will be gone at that point, but the vase > >>>>>>>> will > >>>>>>>> be far from empty. > >>>>>>> By "infinitely-numbered", do you mean the ball will have something > >>>>>>> other > >>>>>>> than a natural number written on it? E.g., it will have "infinity" > >>>>>>> written on it? > >>>>>>> > >>>>>> Yes, that is precisely what I mean. If the experiment is continued > >>>>>> until > >>>>>> noon, > >>>>> The clock ticks till noon and beyond. However, the explicitly- > >>>>> stated insertion times are all before noon. > >>>>> > >>>>>> so that all naturally numbered balls are actually removed (for at > >>>>>> no finite time before noon is this the case), > >>>>> There is no finite time before noon when all balls have > >>>>> been removed. > >>>>> > >>>>> However, any particular ball is removed at a finite > >>>>> time before noon. > >>>>> > >>>>>> then any ball inserted at > >>>>>> noon must have a number n such that 1/n=0, > >>>>> However, there is no ball inserted at noon. > >>>>> > >>>>>> which is only the case for > >>>>>> infinite n. If the experiment does not go until noon, not all naturaly > >>>>>> numbered balls are removed. > >>>>> The experiment goes past noon. No ball is inserted at noon, > >>>>> or past noon. > >>>>> > >>>>> - Randy > >>>>> > >>>> Randy, does it not bother you that no ball is removed at noon, > >>> ... I agree with that... > >>> > >>>> and yet, when every ball is removed before noon, > >> I should have said "each"... > > > > Each ball is removed before noon. Every ball is removed > > before noon. All the balls are removed before noon. Given > > any ball, that ball is removed before noon. > > > >>> ... I agree with that... > >>> > >>>> balls remain in the vase? > >>> .. I don't agree with that. > >>> > >>> When did I ever say balls remain in the vase? Every ball > >>> is removed before noon. No balls remain in the vase at > >>> noon. > >> Do you disagree with the statement that, at every time -1/n, when ball n > >> is removed, for every n e N, there remain balls n+1 through 10n, or 9n > >> balls, in the vase? > > > > At every time before noon, there are not only finitely many balls > > in the vase, there are still infinitely many balls yet to be put in > > the vase. Of course, every one of those balls will be removed > > before noon. Without exception. > > > >>>> How do you explain that? > >>> I would certainly have difficulty understanding how the vase > >>> could be non-empty at noon, given that every ball in the vase > >>> is removed before noon. > >> There is no ball, in all of N, for which the vase is empty at its > >> departure from the vase. > > > > Yes. There is no last ball. > > > > But there is no ball which fails to be removed. > > > >>> But YOU are the one who says the vase is non-empty at > >>> noon. I never said such a thing. I'm certainly not going to > >>> defend YOUR illogical position. > >> I was saying it is non-empty at every one of the finite times before > >> noon where any ball is inserted or removed. Do you argue against THAT > >> statement? > > > > I agree with that statement. For t<0, the vase is non-empty. > > > > At t=0, the vase is empty. > > > >>> So you now agree that it makes no sense that the vase > >>> could be non-empty at noon? That the vase must, in other > >>> words, be empty? > >> No, you misread. > > > > You didn't ask me "does it make sense that the vase is > > non-empty at noon"? Ah well, I'll answer that question anyway. > > No, it doesn't make sense to me that the vase would be > > non-empty at noon. Of course it's empty. > > > > - Randy > > > > Even though it didn't become empty at noon, nor before... For it to "become" anything implies a gradual and continuous change spread over some time interval of positive length, which is not the case here. The vase is non-empty at every time before noon and empty at noon in much the same discontinuous way that Sign(x) is non-zero at ever real x except x = 0.
From: Virgil on 23 Oct 2006 21:12 In article <453d5565(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <453d0c4e(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> David Marcus wrote: > > > >>> If I read this correctly, you agree that at all times every ball that is > >>> in the vase has a natural number on it, but at noon you say that there > >>> is a ball in the vase that does not have a natural number on it. Is that > >>> correct? > >> No. I am saying that if only finite iterations of the ball process > >> occur, then noon never occurs in the experiment to begin with. If noon > >> DOES exist in the experiment, then that can only mean that some ball n > >> exists such that 1/n=0, which would have to be greater than any finite n. > > > > What part of the gedanken experiment statement says anything like that? > > The part that says that ball n is removed at t=-1/n, combined with t=0, > or t=-0. Then 1/n=0, true only for infinite n. Then is TO claiming that 1/n = 0 for some number, n, reachable from 1 by repeated successorship, as these are the only n's allowed. > > > > >>> Now, please > >>> explain what "emptying" means. > >>> > >> "Empty" means not having balls. To become empty means there is a change > >> of state in the vase ("something happens" to the vase), from having > >> balls to not having balls. > > > > Does "emptying" (going from a state with specific balls in the vase to > > a state with no balls in the vase) occupy a duration of time greater > > than zero? > > It doesn't even appear to have that single moment to occur, in this > experiment, since it can't occur before noon, nor at noon, nor > thereafter. Certainly, if the vase starts with some uncountably infinite > number of balls which are removed according to the Zeno schedule, it > will empty, the vast, infinite majority being removed AT noon. But, if > this experiment is to empty, and is an experiment in time, then you > should be able to say when that occurs. > > >> Now, when does this moment, or interval, occur? > > > > If it is an instantaneous process, it would have to "happen" at noon. > > So, you are saying that something does occur at noon. But, what causes > that? Surely there are no naturally-numbered balls being added or > removed at noon? The process of inserting and removing balls has noon as lub. So the process is over and completed at noon but not before. > > > > > But as every ball is removed strictly before noon, it does not have to > > happen at all. > > You mean the vase doesn't have to empty? TO may mean that , but no sensible person will try to claim that for any n in N, ball n is in the vase after noon - (1/n minuts) Then what makes you think it's > empty? I know, I know. You have your logic. But, it amounts to > artificially creating an upper bound to a boundless set, The only "upper bound" is on the times at which balls are to be moved, and that has least upper bound of noon. > The fact remains that it doesn't become empty before noon, and > nothing happens at noon, so it doesn't empty. It is emptying, in a sense, at each t = -1/n, in such a way that it is empty at noon. Let those who disagree name the natural number on any ball remaining in the vase at noon.
From: Ross A. Finlayson on 23 Oct 2006 21:12 MoeBlee wrote: > Lester Zick wrote: > > >I don't insist that conversations must be about set theory. However, > > >when various incorrect things are said about set theory, then I may see > > >fit to comment, as well as I may see fit to mention set theoretic > > >approaches to certain mathematical subjects. In the present case, a > > >poster mentioned certain things about set theory, then I responded, > > >then he asked me questions, then I gave answers, then you commented on > > >my answers. That hardly presents me as insisting that the scope of this > > >newsgroup be confined to set theory. > > > > Technically set "theory" is only a method not a theory since it can't > > be proven true or false. > > By a 'theory' I mean a set of sentences closed under entailment. That's > what *I* mean when I use the word in such contexts as this one. I deny > that there are other senses of the word; I'm just telling you exactly > what I mean in this context. > > > Moe, you get all bent > > out of shape and claim they're spouting nonsense on the internet for > > no better reason than they don't follow the party line on the subject. > > No, I have no problem with someone rejecting the axioms of any > particular theory, thus to reject the theory. What I post against is > people saying things that are incorrect about certain theories. Saying > that one rejects the axioms or even that one rejects classical first > order logic is fine with me (though, I'm interested in what > alternatives one offers). But that is vastly different from saying > untrue things about existing theories. > > MoeBlee Eh, ZF is consistent with itself. That's because anything can be proven in an inconsistent theory, except where it would conflict with A theory of course, because there are only and all true statements in A theory. ZF is inconsistent. A set is not a set in ZF set theory. So, then, that gets me back to thinking about the, how you say, countable and uncountable naturals. Basically it gets to that there are so many lists that one of them would be the list of reals, because there are more lists than reals. There is only so much information in the reals. Hey, did you find that counterexample in the book? So, if one if the "previous" sets, of a process over the naturals that completes, was disjoint its previous, then, each of those existed. Ross
From: Virgil on 23 Oct 2006 21:13
In article <453d5620(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > We are talking about at noon itself, not at any finite time before noon. By noon, it is all over, and every ball inserted has been removed. At least if one follows the rules as set by the original problem. |