An uncountable countable set
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From: David Marcus on 24 Oct 2006 22:02 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> Either something happens an noon, or it doesn't. Where do you stand on > >>>>>>>> the matter? > >>>>>>> What does "something happens" mean, please? I really don't know what you > >>>>>>> mean. > >>>>>> ??? Do you live in the universe, or in a static picture? When "something > >>>>>> happens" o an object, some property or condition of it "changes". That > >>>>>> occurs within some time period, which includes at least one moment. > >>>>>> There is no moment in this problem where the vase is emptying, > >>>>>> therefore, that never "occurs". If you are going to insist that time is > >>>>>> a crucial element of this problem, then you should at least be familiar > >>>>>> with the fact that it's a continuum, and that events occurs within > >>>>>> intervals of that continuum. > >>>>> Thanks. That explains what "something happens" means. Now, please > >>>>> explain what "emptying" means. > >>>> "Empty" means not having balls. To become empty means there is a change > >>>> of state in the vase ("something happens" to the vase), from having > >>>> balls to not having balls. > >>>> > >>>> Now, when does this moment, or interval, occur? > >>> A reasonable question. Before I answer it, let me ask you a question. > >>> Suppose I make the following definitions: > >>> > >>> For n = 1,2,..., define > >>> > >>> A_n = -1/floor((n+9)/10), > >>> R_n = -1/n. > >>> > >>> For n = 1,2,..., define a function B_n by > >>> > >>> B_n(t) = 1 if A_n < t < R_n, > >>> 0 if t < A_n or t > R_n, > >>> undefined if t = A_n or t = R_n. > >>> > >>> Let V(t) = sum{n=1}^infty B_n(t). > >>> > >>> Then V(-1) = 1 and V(0) = 0. If we consider V to be a function of time, > >>> at what time does it become zero? > >> Just answer the question, and stop beating around the bush. > > > > To recap, you wrote, "When 'something happens' to an object, some > > property or condition of it 'changes'. That occurs within some time > > period, which includes at least one moment." You also wrote, "To become > > empty means there is a change of state in the vase ('something happens' > > to the vase), from having balls to not having balls." > > > > From the definition of V, if t equals A_n or R_n, then V(t) is not > > defined. > > From your definition, but I would say that B_n(t) is 1 at A_n and 0 at > R_n, and let V(t) be defined at every point. There are always a specific > number of balls, if additions and removals occur instantaneously. OK, let's do that. Then all the functions are defined for all real numbers. > > For other 1 <= t < 0, V(t) is positive. > > Yes, there are balls in the vase (a growing number) at every time before > noon. > > And, V(0) = 0. > > I notice you summed to oo: "Let V(t) = sum{n=1}^infty B_n(t)." > > Can you sum to oo when the set is limited to N? Sure. It is just notation. sum_{n=1}^infty B_n(t) is defined to mean lim_{N=infty} sum_{n=1}^N B_n(t) And, this limit can also be defined in a way that only mentions natural numbers. Have you ever taken Calculus? > When n e N and t=-1/n, does t=0 exist? This is the kind of fallacy you > folks try to accuse me of. t=0 ^ t=-1/n ^ n e N = FALSE. > > > You asked when does the vase change from having balls to having no > > balls? Since, V(t) is positive (or undefined) for 1 <= t < 0 and V is > > zero at time zero, it would seem that according to *your* definition of > > "become empty", the vase becomes empty at noon. > > Correct, as much as I could expect. If the vase is empty at noon, but > not before, then it must have become empty at that point in time. > > > > > The only reason I say "seem" is that I don't know whether your > > definition allows V to be undefined at the times of addition and > > removal. We could either agree that at the addition and removal > > instants, the ball is not in the vase (thus changing the definition of > > V) or we could agree that "becomes empty" requires V to be defined at > > all times (in which case the vase never becomes empty). > > Just like we are saying that the vase becomes empty at t=0 since it is > not empty at t<0, at the moment of insertion or removal we should > consider the event completed from that moment forward. That's > consistency, and it removes any undefined states of the vase at any time > t<0. > > Regardless, V(0) > > = 0, so there are no balls in the vase at noon. > > > > No, t=0 is proscribed by n e N and t=-1/n. Contradiction. Sorry. You > can't have it both ways. Either noon is in the experiment and something > occurs, which involves infinitely numbered balls, or you stick with the > original constraint that n e N and so t=0 is not allowed. Your choice. Before we discuss the experiment, let's look at my definitions of A_n, R_n, B_n, and V. Did you just agree above that they are fine mathematical definitions? Is there anything wrong with them mathematically (ignoring for the moment whether they give a correct interpretation of the problem)? Are the functions {B_n} and V defined for all real numbers? -- David Marcus
From: David Marcus on 25 Oct 2006 00:50 Virgil wrote: > In article <453e824b(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > Virgil wrote: > > > In article <453e4a85(a)news2.lightlink.com>, > > > Tony Orlow <tony(a)lightlink.com> wrote: > > >> If the vase exists at noon, then it has an uncountable number of balls > > >> labeled with infinite values. But, no infinite values are allowed i the > > >> experiment, so this cannot happen, and noon is excluded. > > > > > > So did the North Koreans nuke the vase before noon? > > > > > > The only relevant issue is whether according to the rules set up in the > > > problem, is each ball inserted before noon also removed before noon?" > > > > > > An affirmative confirms that the vase is empty at noon. > > > A negative directly violates the conditions of the problem. > > > > > > How does TO answer? > > > > You can repeat the same inane nonsense 25 more times, if you want. I > > already answered the question. It's not my problem that you can't > > understand it. > > It is a good deal less inane and less nonsensical than trying to > maintain, as TO and his ilk do, that a vase from which every ball has > been removed before noon contains any balls at noon that have not been > removed. Ah, you are forgetting the balls labeled with "infinite values". Those balls haven't been removed before noon. Although, I must say I'm not too clear on when they were added. -- David Marcus
From: Virgil on 25 Oct 2006 01:16 In article <MPG.1fa8b708fa0b1ac0989760(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: > > In article <453e824b(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > Virgil wrote: > > > > In article <453e4a85(a)news2.lightlink.com>, > > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> If the vase exists at noon, then it has an uncountable number of balls > > > >> labeled with infinite values. But, no infinite values are allowed i > > > >> the > > > >> experiment, so this cannot happen, and noon is excluded. > > > > > > > > So did the North Koreans nuke the vase before noon? > > > > > > > > The only relevant issue is whether according to the rules set up in > > > > the > > > > problem, is each ball inserted before noon also removed before noon?" > > > > > > > > An affirmative confirms that the vase is empty at noon. > > > > A negative directly violates the conditions of the problem. > > > > > > > > How does TO answer? > > > > > > You can repeat the same inane nonsense 25 more times, if you want. I > > > already answered the question. It's not my problem that you can't > > > understand it. > > > > It is a good deal less inane and less nonsensical than trying to > > maintain, as TO and his ilk do, that a vase from which every ball has > > been removed before noon contains any balls at noon that have not been > > removed. > > Ah, you are forgetting the balls labeled with "infinite values". Those > balls haven't been removed before noon. Although, I must say I'm not too > clear on when they were added. Well, they don't seem to have been added at any time before noon, so I suppose they must have been added right at noon, maybe in order to avoid a having a discontinuity at noon, and probably by one of TO's Zeno machines,
From: imaginatorium on 25 Oct 2006 01:30 David Marcus wrote: > Tony Orlow wrote: > > David Marcus wrote: > > > Tony Orlow wrote: > > >> As each ball n is removed, how many remain? > > > > > > 9n. > > > > > >> Can any be removed and leave an empty vase? > > > > > > Not sure what you are asking. > > > > If, for all n e N, n>0, the number of balls remaining after n's removal > > is 9n, does there exist any n e N which, after its removal, leaves 0? > > I don't know what you mean by "after its removal"? Oh, I think this is clear, actually. Tony means: is there a ball (call it ball P) such that after the removal of ball P, zero balls remain. The answer is "No", obviously. If there were, it would be a contradiction (following the stated rules of the experiment for the moment) with the fact that ball P must have a pofnat p written on it, and the pofnat 10p (or similar) must be inserted at the moment ball P is removed. Now to you and me, this is all obvious, and no "problem" whatsoever, because if ball P existed it would have to be the "last natural number", and there is no last natural number. Tony has a strange problem with this, causing him to write mangled versions of Om mani padme hum, and protest that this is a "Greatest natural objection". For some reason he seems to accept that there is no greatest natural number, yet feels that appealing to this fact in an argument is somehow unfair. > > Sure. But it's easily explainable and resolvable once a proper measure > > is applied to the situation. Omega doesn't lend itself to proper > > measure. Infinite series do. Bijection loses measure for infinite sets. > > N=S^L and IFR preserve measure. Oh, right, well Tony has a number of "explanations" for things, most of them equally mysterious. Brian Chandler http://imaginatorium.org
From: Virgil on 25 Oct 2006 01:46
In article <1161754218.785144.91070(a)e3g2000cwe.googlegroups.com>, imaginatorium(a)despammed.com wrote: > David Marcus wrote: > > Tony Orlow wrote: > > > David Marcus wrote: > > > > Tony Orlow wrote: > > > >> As each ball n is removed, how many remain? > > > > > > > > 9n. > > > > > > > >> Can any be removed and leave an empty vase? > > > > > > > > Not sure what you are asking. > > > > > > If, for all n e N, n>0, the number of balls remaining after n's removal > > > is 9n, does there exist any n e N which, after its removal, leaves 0? > > > > I don't know what you mean by "after its removal"? > > Oh, I think this is clear, actually. Tony means: is there a ball (call > it ball P) such that after the removal of ball P, zero balls remain. > > The answer is "No", obviously. If there were, it would be a > contradiction (following the stated rules of the experiment for the > moment) with the fact that ball P must have a pofnat p written on it, > and the pofnat 10p (or similar) must be inserted at the moment ball P > is removed. > > Now to you and me, this is all obvious, and no "problem" whatsoever, > because if ball P existed it would have to be the "last natural > number", and there is no last natural number. > > Tony has a strange problem with this, causing him to write mangled > versions of Om mani padme hum, and protest that this is a "Greatest > natural objection". For some reason he seems to accept that there is no > greatest natural number, yet feels that appealing to this fact in an > argument is somehow unfair. > > > > > Sure. But it's easily explainable and resolvable once a proper measure > > > is applied to the situation. Omega doesn't lend itself to proper > > > measure. Infinite series do. Bijection loses measure for infinite sets. > > > N=S^L and IFR preserve measure. > > Oh, right, well Tony has a number of "explanations" for things, most of > them equally mysterious. > > Brian Chandler > http://imaginatorium.org And many of them downright wrong! |