An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Randy Poe on 13 Oct 2006 13:44 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
From: cbrown on 13 Oct 2006 13:47 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com 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. > >>>> > >>> 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. But what do either of those statements have to do with whether or not ball 15 is in the vase at t=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
From: MoeBlee on 13 Oct 2006 13:54 Han de Bruijn wrote: > No more glibberish than defining the ordinals by putting ever more curly > braces around the empty set: That's not how we define 'ordinal'. MoeBlee
From: MoeBlee on 13 Oct 2006 14:01 Han de Bruijn wrote: > Virgil wrote: > > Axiom of infinity: There exists a set x such that the empty set is a > > member of x and whenever y is in x, so is S(y). > > Which is actually the construction of the ordinals. Right? Wrong. > So the axiom of infinity says that you can get everything from nothing. > This is contradictory to all laws of physics, where it is said that you > pay a price for everything. E.g. mass and energy are conserved. Sets are abstract objects, not physical objects. So of course they are not expected to obey laws of mass and energy. MoeBlee
From: stephen on 13 Oct 2006 14:53
Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Randy Poe <poespam-trap(a)yahoo.com> wrote: >> >>> 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. >> >>> No, there's one of your leaps again. >> >>> That's a particularly weird one. >> >>> "If the value at noon doesn't have THIS to do with the >>> sequence, then it must not have ANYTHING to do with >>> the sequence". >> >>> There's no reason to make such a leap. >> >>> - Randy >> >> Actually I think Tony is right on this one. The >> sequence Tony is talking about is >> 1, 9, 18, 27, ... > Uh, starts with 0, but do go on... >> This sequence represents the number of balls at times before >> noon. The sequence has nothing to do with the number of >> balls at noon, as the value for noon does not appear in >> the sequence. This is why nobody who argues that the >> vase is empty at noon ever mentions such a sequence, and >> instead point out the simple fact that each ball added >> before noon is removed before noon. >> >> Stephen >> > So, the infinite sequence of finite iterations where we can actually > tell exactly how many balls are in the vase has nothing to do with the > vase's state at noon, which is supposed to be the limit of this > sequence? 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. The whole argument is simply that if -(1/2)^floor(n/10) is less than zero (the minutes before noon that the ball is added), then -(1/2)^n is less than zero (the minutes before noon the ball is removed). This really does not rely on set theory. > 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. 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 |