An uncountable countable set
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From: Virgil on 14 Oct 2006 15:10 In article <4531023e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Where is there defined in the problem any mention of a discontinuity in > the process? There isn't. The discontinuities follow from the statement of the problem. The "number of balls in the urn" has a discontinuity each time it changes. > > > > >> What causes a discontinuity at noon? I'll tell you. The von Neumann > >> limit ordinals. That's schlock. > > > > Um, that's foaming at the mouth. Von Neumann limit ordinals have > > nothing to do with it - the very simplest notion of the natural numbers > > being an unending sequence - Wolf Kirchmeir's six? eight?-year old > > grandson's understanding - is absolutely all that is needed. You can't > > grasp the notion of an unending sequence, which is why you are in such > > a total tangle. > > I am in no tangle. That you cannot see that your conclusion is based on > the notion of omega as the first limit ordinal and first discontinuity > in the ordinals is disappointing. If the sequence never ends, then noon > never comes. The notion that an infinite sequence cannot take occur within a finite time interval has been dead since Zeno. > > The process at noon is not well defined, since the distinction between > iterations disappears. How do you know there are countably many > iterations, and not some uncountably number? You don't. You base your > argument on all iterations being finite, but there is no least upper > bound to the finites, because there is no least infinite. In TO's world, only TO can say what is or is not there, but in ZF and NBG, there is a least infinite ordinal.
From: Virgil on 14 Oct 2006 15:14 In article <453102b5(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: > >>>>>> 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. > >>>>>> > >>>>>>>> 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. > >>>>>>> > >>>>>> Specifically, that for every ball removed, 10 are inserted. > >>>>> All of which are eventually removed. Every single one. > >>>>> > >>>> Every single one, > >>> Yes. > >>> > >>>> each after another ten are inserted, of course. > >>> And I can tell you the time that each of those is removed. > >>> > >>>> Come on! > >>> Come on yourself. You *know* there is a removal time > >>> associated with every ball. > >>> > >> I know that at no time > > > > Crucial phrase missing here: "at no time BEFORE noon" > > > >> have all the balls previous inserted been > >> removed, but only 1/9th of them, since 1 is removed for every 10 > >> inserted. > > > > You have correctly described the situation at every one > > of the infinite values of 1 < t < 0. > > > >> What is the flaw in that logic? > > > > That you somehow think f(x), x<0 forces a value of f(0). > > > > - Randy > > > > lim(t->0: balls(t))<>0 lim(t->0: balls(t)) does not exist, but balls(0) does exist and balls(0)=0 Functions can exist at points at which their limits do not. There are even functions with domain R which are discontinuous at every rational argument but continuous at every irrational one.
From: Virgil on 14 Oct 2006 15:17 In article <45310346(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > But what do either of those statements have to do with whether or not > > ball 15 is in the vase at t=0? > > Nothing to do with ball 15. That has a specific time of removal. Every > specific ball does. The balls at noon are not distinguishable nor specific. Every ball is as distinguishable and specific as the number on it, and they all have numbers on them. > > > > > Do you believe that we cannot state whether ball 15 is in the vase at > > 1/pi seconds before midnight, because there is no step associated with > > 1/pi? > > > > Cheers - Chas > > > > That's a dumb question, made to make me look dumb. It backfired. On the contrary, it makes TO look almost as dumb as he actually is.
From: cbrown on 14 Oct 2006 15:26 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> cbrown(a)cbrownsystems.com wrote: <snip> > >>>>> Putting aside the question of /how/ (limit? sum of binary functions?) > >>>>> one determines the /number/ of balls in the vase at time t for a > >>>>> moment... > >>>>> > >>>>> Do you then agree that there is some explicit relationship described in > >>>>> the problem between what time it is, and whether any particular > >>>>> labelled ball, for example the ball labelled 15, is in the vase at that > >>>>> time? > >>>> For any finite time before noon, when iterations of the problem are > >>>> temporally distinguishable, yes, but at noon, no. > >>>> > >>> I don't understand why you think this would be the case. > >>> > >>> Why do you think the relationship holds for t < 0? > >>> > >>> Why you do think it does not hold for t >= 0? > >>> > >>> Cheers - Chas > >>> > >> Because for t>=0, n>=oo. > > > > Actually, for t>=0, there is /no/ natural number n such that t = -1/n. > > Similarly, for t = -1/pi, there is no natural number n such that t = > > -1/n. > > Yeah, no idding. Who said oo was a natural number? > You just implied it; when you claimed that at t>=0, n>=oo; where I presume that by "n", you refer to the statement in the problem: "At time t = -1/n, where n is a natural number, we add balls labelled (10*(n-1)+1) through 10*n inclusive, and remove the ball labelled n". That statement obviously does not refer to removals or additions of balls at time t = 0, becuase there is no natural number n such that -1/n = 0. Do you agree with this conclusion? > > > > But what do either of those statements have to do with whether or not > > ball 15 is in the vase at t=0? > > Nothing to do with ball 15. That has a specific time of removal. Every > specific ball does. This contradicts what you said above regarding whether we can determine if a particular ball is in the vase at some time t; you wrote: > >>>> For any finite time before noon, when iterations of the problem are > >>>> temporally distinguishable, yes, but at noon, no. Do you now agree that we can conclude from the problem statement that, for each ball labelled with a natural number, that ball is not in the vase at noon? > The balls at noon are not distinguishable nor specific. Since the only statement in the problem regarding putting balls in the vase is a statement about putting balls which are labelled with a specific and unique natural number in the vase, I don't see how you justify this conclusion /purely in terms of the given problem/. At any rate, for each natural number n, the number of balls in the vase which are "indistinguishable and not specific" at time t = -1/n is 0. By your own logic, why can't we conclude that therefore the number of balls in the vase which are "indistinguishable and not specific" at t=0 is also 0 (i.e., lim n->oo 0 = 0)? > > > > > Do you believe that we cannot state whether ball 15 is in the vase at > > 1/pi seconds before midnight, because there is no step associated with > > 1/pi? > > > > Cheers - Chas > > > > That's a dumb question, made to make me look dumb. It backfired. You're a bit feisty today. There are no dumb questions; only dumb answers. Cheers - Chas
From: Virgil on 14 Oct 2006 15:30
In article <45310552(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > Who ever said it was the limit of this sequence? > > > >> Why even mention the gedanken at all then? > > > > I am not the one who brought it up. I am not even sure > > why people think it has anything to do with set theory. > > It doesn't. It's a distinct SEQUENCE of events, not a set without order. > Set theory doesn't apply. It's just another example of set theorists > trying to claim that everything falls under set theory. This experimant > obviously does not. Set theory is incapable of handling the concept of > sequence in a well-defined way over such a set. Set theory does a much better job of it than TO does, since set theory does not require balls which have been removed from a vase to be in it at a later time. > No, set theory confuses the issue with its concentration on omega. There > is no such distinct size of the finite naturals. There is in ZF and NBG. The ordinal of the set of all finite naturals is precisely that omega which TO cannot comprehend. >The infinite iterations > are all compressed to a point in this experiment, and since those > operations are a combination of additions and subtractions, set > theorists feel entitled to rearrange the events any way that gets them > their magical results. It's pitiful. it is the statement of the gedankenexperiment itself that does all the arrangement, we who read what it requires only follow what it says. TO, on the other hand, tries to read into it things not said. > > > > >> 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. > > > > The only argument I am making is that each ball that is added > > before noon is removed before noon. > > I don't disagree with that. After the removal of every such ball before > noon, nine times as many balls remain as have been removed. That is true > for every moment before noon. The conclusion as to what happens AT noon > either does, or doesn't, have to do with this fact. It does not. What it does have to do with is that every ball has a time before noon at which it is removed, so that no ball which has been removed will still be in the vase at noon. Unless TO, or others, can refute this statement, they must accede to having no balls in the vase at noon. > > Of course by supposing that > > an infinite number of actions can be performed we are playing > > silly magic tricks. This is not a physical problem. Insisting > > on a physical answer to an unphysical problem is pointless. > > > > Stephen > > > > If we start with a vase full of any number of balls, and remove one ball > at each of these -1/2^n times, then it becomes empty at noon, or before > if the number of balls is finite. There is no argument about that. > However, in this case, no balls is removed without ten more being > inserted, so the vase cannot become empty, despite set theoretical > shenanigans. So that TO again makes the argument that, without changing the removals at all, delaying the insertion of some balls until others have been removed leaves more balls in the vase than if the balls are all put in at the start. |