An uncountable countable set
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From: Randy Poe on 12 Oct 2006 17:11 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. - Randy
From: Randy Poe on 12 Oct 2006 17:14 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
From: stephen on 12 Oct 2006 17:42 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, ... 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
From: Lester Zick on 12 Oct 2006 18:01 On 12 Oct 2006 12:17:21 -0700, imaginatorium(a)despammed.com wrote: > >Virgil wrote: >> In article <1160669820.603144.288450(a)e3g2000cwe.googlegroups.com>, >> mueckenh(a)rz.fh-augsburg.de wrote: > ><bibble-babble-bobble> > >> > 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". > >No, it will defeat him, game, set, and math. Not bad, Brian. A little less of the former and a little more of the latter and I may yet have to resurrect "technically" in your case. ~v~~
From: Virgil on 12 Oct 2006 18:04
In article <452e882a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> 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. If a particular ball is not removed it remains in the vase and if it is removed it does not remain in the vase. The set of numbered balls is well ordered by their numbering. If any numbered balls remain in the vase at noon, then there must be one with the least number of any remaining. Which one would that be TO? Or does TO go around with an eraser erasing those numbers as the balls are put into the vase? > > > > 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. And later, but before noon, also removed. So TO what is the number on the first ball NOT removed??? |