An uncountable countable set
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From: Virgil on 12 Oct 2006 23:21 In article <452ef471(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus 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. 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. > > > > How about this problem: Start with an empty vase. Add a ball to a vase > > at time 5. Remove it at time 6. How many balls are in the vase at time > > 10? > > > > Is this a nonsensical question? > > > > Not if that's all that happens. However, that doesn't relate to the ruse > in the vase problem under discussion. So, what's your point? The point is that after a ball is removed from the vase it is no longer in the vase, and every ball is removed before noon.
From: Virgil on 12 Oct 2006 23:26 In article <452ef4a0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> 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. > > > > - Randy > > > > Every single one, each after another ten are inserted, of course. Come on! So lets put them all in one minute earlier so they are all in before any have to be removed and each ball will be in for a longer time, and then remove them one at a time according to the original schedule. According to TO, putting them in earlier and taking them out as before leaves FEWER in the vase at noon, even though there is no change in removals.
From: Virgil on 12 Oct 2006 23:27 In article <452ef4f0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe 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 > > > > "noon is not a part of the sequence of iterations" > > Uh, then what does the sequence have to do with noon? Everything takes place before noon, so that by noon, it is all over and done with.
From: Virgil on 12 Oct 2006 23:32 In article <452ef5a6(a)news2.lightlink.com>, 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 Right. > which is supposed to be the limit of this > sequence? Why is it the limit of any sequence? And since the set of balls removed by noon includes every ball, how does TO come up with any balls still waiting to be removed at noon?
From: Virgil on 12 Oct 2006 23:39
In article <452ef610(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > 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. > > After iteration n, n+1 through 10n remain. > > > > > 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. > > n+1, or aleph_0+1, if you prefer. As there was no original ball with that number, when did any such non-existent ball get put into the vase? Or did it, like Topst, just grow there? > > > 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? > > > Yes, that's what I do, to prevent them from disappearing, because I so > hate magic. Uh huh. That logicians' and mathematicians hands are quicker than TO's eyes has long been apparent. > > > >>> 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??? > > n+1 Any ball not removed could never have been inserted. Since every ball inserted before noon has a time of removal before noon, it must be that any balls not removed before noon are also not inserted at all. |