An uncountable countable set
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From: Virgil on 12 Oct 2006 14:14 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?
From: cbrown on 12 Oct 2006 14:21 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 example, do you agree that, according to the definitions explicitly given in the problem, we can conclude that if at time t, ball 15 is in the vase, that therefore t <= -1/15? Cheers - Chas
From: Tony Orlow on 12 Oct 2006 14:23 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. > > - Randy > Specifically, that for every ball removed, 10 are inserted.
From: Tony Orlow on 12 Oct 2006 14:26 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. I think there are three, you know, the number of licks it takes to get to the tootsie roll center of a tootsie pop. That makes about as much sense as saying an infinite number of them vanish. If noon is not part of your sequence, then it's a nonsensical question, and if it is, then the limit applies.
From: Virgil on 12 Oct 2006 14:26
In article <1160669646.780939.326190(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > I wrote: > > > The axiom of infinity does only state n+1 exists if n is given. It is > > > realized by he numbers of my list. > You replied: > > That is *not* what the axiom of infinity states. The axiom states that > > there exists a set that contains all the successors of its elements. > And I reply: > That is wrong. The axiom says: For all possible n in all possible > cases: If n is a natural number, then n+1 is a natural number too. It > does *not* say that it is meaningful to speak of *all* natural numbers. > It does not say that the set N does actually or completely exists. http://en.wikipedia.org/wiki/ZFC 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). > It > does not say that this infinity is finished, i.e., that all natural > numbers can exist together. http://en.wikipedia.org/wiki/ZFC 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). > Well: "there is a set". But the meaning of > these three words depends heavily on the interpretation. "There exists a set..." makes it hard to say that no such set exists in ZFC. But "Mueckenh" says it anyway. |