An uncountable countable set
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From: Virgil on 27 Oct 2006 19:32 In article <454287d9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > As we approach noon, the limit is 0. We don't reach noon. If TO never reaches noon, he must still be less than one day old. I must say he often acts like it. The rest of us manage to reach noon on the close order of once a day, with some variations for those circumnavigating the Earth at high speeds.
From: Virgil on 27 Oct 2006 19:37 In article <4542888c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > > (The vertical lines denote "cardinality".) > > > > Um, before I answer that question, I think you need to define what you > mean by "|IN - OUT|" =0. How are you measuring IN and OUT, and how do > you define '-' on these "numbers"? I suspect that David means by IN - OUT the set difference defined as {x in IN: not x in OUT} In this NG, this set difference is sometimes indicated with the backslash, as "IN \ OUT".
From: Tony Orlow on 27 Oct 2006 19:43 David R Tribble wrote: > Tony Orlow wrote: >>> You have agreed with everything so far. At every point before noon balls >>> remain. You claim nothing changes at noon. Is there something between >>> noon and "before noon", when those balls disappeared? If not, then they >>> must still be in there. > > David R Tribble wrote: >>> Of course there is a "something" between "before noon" and "noon" where >>> each ball disappears. At step n, time 2^-n min before noon, ball n is >>> removed. This happens for every ball, since there is a step n for >>> every ball. The balls are removed, one by one, one at each step, >>> before noon. > > Tony Orlow wrote: >> As each ball n is removed, how many remain? Can any be removed and leave >> an empty vase? > > Each ball n is placed into the vase at time 2^int(n/10), and then later > removed at time n. This happens for every ball before noon. So every > ball is inserted and then later removed from the vase before noon. > > At any given time n before noon, ten balls are added to the vase and > then ball n (which was added to the vase in a previous step) is > removed. Your entire confusion results from assuming a "last" time > prior to noon, but there is no such time. > At no time prior to noon are all balls removed. Nor are any removed at noon. It cannot be empty, then.
From: David Marcus on 27 Oct 2006 19:44 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> Mike Kelly wrote: > >>>>>>> Now correct me if I'm wrong, but I think you agreed that every > >>>>>>> "specific" ball has been removed before noon. And indeed the problem > >>>>>>> statement doesn't mention any "non-specific" balls, so it seems that > >>>>>>> the vase must be empty. However, you believe that in order to "reach > >>>>>>> noon" one must have iterations where "non specific" balls without > >>>>>>> natural numbers are inserted into the vase and thus, if the problem > >>>>>>> makes sense and "noon" is meaningful, the vase is non-empty at noon. Is > >>>>>>> this a fair summary of your position? > >>>>>>> > >>>>>>> If so, I'd like to make clear that I have no idea in the world why you > >>>>>>> hold such a notion. It seems utterly illogical to me and it baffles me > >>>>>>> why you hold to it so doggedly. So, I'd like to try and understand why > >>>>>>> you think that it is the case. If you can explain it cogently, maybe > >>>>>>> I'll be convinced that you make sense. And maybe if you can't explain, > >>>>>>> you'll admit that you might be wrong? > >>>>>>> > >>>>>>> Let's start simply so there is less room for mutual incomprehension. > >>>>>>> Let's imagine a new experiment. In this experiment, we have the same > >>>>>>> infinite vase and the same infinite set of balls with natural numbers > >>>>>>> on them. Let's call the time one minute to noon -1 and noon 0. Note > >>>>>>> that time is a real-valued variable that can have any real value. At > >>>>>>> time -1/n we insert ball n into the vase. > >>>>>>> > >>>>>>> My question : what do you think is in the vase at noon? > >>>>>> A countable infinity of balls. > >>>>> So, "noon exists" in this case, even though nothing happens at noon. > >>>> Not really, but there is a big difference between this and the original > >>>> experiment. If noon did exist here as the time of any event (insertion), > >>>> then you would have an UNcountably infinite set of balls. Presumably, > >>>> given only naturals, such that nothing is inserted at noon, by noon all > >>>> naturals have been inserted, for the countable infinity. Then insertions > >>>> stop, and the vase has what it has. The issue with the original problem > >>>> is that, if it empties, it has to have done it before noon, because > >>>> nothing happens at noon. You conclude there is a change of state when > >>>> nothing happens. I conclude there is not. > >>> So, noon doesn't exist in this case either? > >> Nothing happens at noon, and as long as there is no claim that anything > >> happens at noon, then there is no problem. Before noon there was an > >> unboundedly large but finite number of balls. At noon, it is the same. > > > > So, noon does exist in this case? > > Since the existence of noon does not require any further events, it's a > moot point. As I think about it, no, noon does not exist in this problem > either, as the time of any event, since nothing is removed at noon. It > is also not required for any conclusion, except perhaps that there are > uncountably many balls, rather than only countably many. But, there are > only countably many balls, so, no, noon is not part of the problem here. > As we approach noon, the limit is 0. We don't reach noon. To recap, we add ball n at time -1/n. We don't remove any balls. With this setup, you conclude that noon does not exist. Is this correct? -- David Marcus
From: Virgil on 27 Oct 2006 19:44
In article <45428aeb(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> You are mentioning balls and time and a vase. But, what I'm asking is > >>>>> completely separate from that. I'm just asking about a math problem. > >>>>> Please just consider the following mathematical definitions and > >>>>> completely ignore that they may or may not be relevant/related/similar > >>>>> to the vase and balls problem: > >>>>> > >>>>> -------------------------- > >>>>> For n = 1,2,..., let > >>>>> > >>>>> A_n = -1/floor((n+9)/10), > >>>>> R_n = -1/n. > >>>>> > >>>>> For n = 1,2,..., define a function B_n: R -> R by > >>>>> > >>>>> B_n(t) = 1 if A_n <= t < R_n, > >>>>> 0 if t < A_n or t >= R_n. > >>>>> > >>>>> Let V(t) = sum_n B_n(t). > >>>>> -------------------------- > >>>>> > >>>>> Just looking at these definitions of sequences and functions from R > >>>>> (the > >>>>> real numbers) to R, and assuming that the sum is defined as it would be > >>>>> > >>>>> in a Freshman Calculus class, are you saying that V(0) is not equal to > >>>>> 0? > >>>> On the surface, you math appears correct, but that doesn't mend the > >>>> obvious contradiction in having an event occur in a time continuum > >>>> without occupying at least one moment. It doesn't explain how a > >>>> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is > >>>> that all finite naturals are removed by noon. I never disagreed with > >>>> that. However, to actually reach noon requires infinite naturals. Sure, > >>>> if V is defined as the sum of all finite balls, V(0)=0. But, I've > >>>> already said that, several times, haven't I? Isn't that an answer to > >>>> your question? > >>> I think it is an answer. Just to be sure, please confirm that you agree > >>> that, with the definitions above, V(0) = 0. Is that correct? > >> Sure, all finite balls are gone at noon. > > > > Please note that there are no balls or time in the above mathematics > > problem. However, I'll take your "Sure" as agreement that V(0) = 0. > > > > Okay. > > > Let me ask you a question about this mathematics problem. Please answer > > without using the words "balls", "vase", "time", or "noon" (since these > > words do not occur in the problem). > > I'll try. > > > > > First some discussion: For each n, B_n(0) = 0 and B_n is continuous at > > zero. > > What??? How do you conclude that anything besides time is continuous at > 0, where yo have an ordinal discontinuity???? Please explain. For anyone except TO, it would be obvious that each B_n has only two points of discontinuity, A_n and R_n, neither of which occurs at t=0. > > > In fact, for a given n, there is an e < 0 such that B_n(t) = 0 for > > e < t <= 0. > > There is no e<0 such that e<t and B_n(t)=0. That's simply false. For any n, let e = R_n/2 < t then B_n(t) = 0. > > In other words, B_n is not changing near zero. > > Infinitely more quickly but not. That's logical. And wrong. Why is it that, according to TO, everything logical is wrong? > I'll consider answering that when you correct the errors above. As the errors above are all TO's, TO should correct them. |