An uncountable countable set
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From: Ross A. Finlayson on 2 Nov 2006 23:24 Tony Orlow wrote: > stephen(a)nomail.com wrote: > > Randy Poe <poespam-trap(a)yahoo.com> wrote: > > > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>>> Mike Kelly wrote: > >>>> Nothing is allowed to happen at noon in either experiment. > >>> Nothing "happens" at noon? I take this to mean that there is no > >>> insertion or removal of balls at noon, yes? Well, I agree with that. > >>> But what relevence does this have to the statement "noon does not > >>> exist"? What does that even *mean*? > >>> > >>> When you've been saying "noon doesn't exist", you actually mean to say > >>> "no insertion or removal of balls occurs at noon"? > >>> > >>> How about this experiment, does noon "exist" in this experiment : > >>> > >>> Insert a ball labelled "1" into the vase at one minute to noon. > >>> > >>> ? > > > >> I think that when Tony and Han say "noon doesn't exist" they > >> really mean "there is no noon on the clock in that experiment", > >> as a way of saying "I have no idea how to answer questions about > >> noon in that experiment, so I'll say that there is no noon and that > >> way I don't have to answer any such questions." > > Or, we say that introducing noon into the situation as the time of an > event creates a contradiction, since 1/n cannot be zero for any natural > n. Since the vase can only become empty (correct me if I'm wrong) if > balls are removed, and no balls are removed at noon, it cannot become > empty at noon. On the other hand, at every finite time -t before noon=0, > there are a finite but exponentially growing number of balls in the > vase, with respect to t. That is of course because n, the count of > iterations of adding 10 and removing 1, has an exponential relationship > with respect to t (well, using 1/2^n anyway), and a linear relationship > to the "size of the set". With each new iteration n, 9 balls are added. > Now, I understand you may not agree with the idea that the size of a set > might "change", but if you are talking about sequences, either over t or > over n, then you are not talking about static sets, but about > progressions. You are getting into measure, but cardinality affords no > measure except in the finite case, where it's unavoidable. Basically, > the n and the t are buried in the noon. > > > That sounds about right. It is also interesting that for some > > reason "noon" is not a necessary part of the problem, but "rates" > > and "iterations" are. Any formulation that ignores rates is > > incorrect, but it is okay to ignore "noon", despite the fact > > the actual question is about "noon". > > > > Stephen > > > > It's about the pertinent variables not being lost. > > Tony Hi Tony, Stephen, y'all, At the time each is removed ten were just added since the last was removed. The supertask, infinitely many discrete events in a finite amount of time, here this ball and vase scenario aka the Ross-Littlewood paradox, is interesting to some extent when compared against the general predilection here against the notion of dense discrete events in their normal ordering. There isn't a point at this case where the events are dense, instead the standard would not have the time actually reach noon/zero/one but merely approach it with the inductive guarantee that for however close you get you can get closer, but not actually reach: look but don't touch as it were. The points are dense to the left of noon, in that between any select and noon there are infinitely many others, else the process doesn't complete. Time is presupposed to proceed at a constant rate, where various excursions into physical theories ascribe to time various conditional rates. Since Zeno is known the dichotomy of the supertask and constant time. Some people assuage their mathematical guilt that that limit is not the sum by considering these notions of for example the hyperintegers, with some expectation of completion in those infinite terms. While that is so, there is only successor on these finite naturals, so the hyperintegers could only be reached by somehow becoming infinite from finite with the only supplied operation: unit increment. So, that's denial. I wonder what the reaction is to a statement along the lines of that the points are dense in their normal ordering in the reals, in their normal ordering of the reals. I believe it would be similar here the reaction of the consideration that there is basically infinity that is the compactification of the naturals: the covering of eyes, ears, and mouth. Those monkeys are blind, deaf, and dumb. Now I'm certainly not calling anyone here except Virgil a monkey. Yet, in the example of the counterexamples book which MoeBlee, who is reasonable, polite, and effusive, has had access to read and who claimed to be examining said source, nothing has been heard back of the matter. You can get that for free from the library. Nature is basically assumed continuous. Now, I'm aware with some layman concepts of gauge invariance and the Planck length and so on, quantized charge and etc., yet the atom used to be atomic, indivisible, and it has been said the superconductor is one big atom. Some aspects of the particle/wave duality in nature, physics, are seen with the reals, comprised as they are of points on a, or in the, line, as has been discussed here with regards to points on a line. The more closely (sub-) atomic particles are examined, the smaller they appear to be, similarly the universe' size always increases upon knowledge of it. Does that not in simile match examining a lesser positive real? Is there a universe ( of sets, or in reality)? The platonist has that they exist. Are functions between physical objects, eg fields, physical objects? Then the universe is infinite and as example its own powerset. Ross
From: David Marcus on 2 Nov 2006 23:38 Tony Orlow wrote: > Or, we say that introducing noon into the situation as the time of > an event creates a contradiction, since 1/n cannot be zero for any > natural n. As far as I know, you are the only one trying to introduce a ball insertion or removal at noon. The rest of us just want to know how many balls are in the vase at noon, since that is the question the problem asked. > Since the vase can only become empty (correct me if I'm wrong) if > balls are removed, and no balls are removed at noon, it cannot > become empty at noon. On the other hand, at every finite time -t > before noon=0, there are a finite but exponentially growing number > of balls in the vase, with respect to t. Since you said, "correct me if I'm wrong", let's examine your logic. You say the following are true. 1) The vase is non-empty at every time between 11:59 and noon. 2) No ball removals occur at noon. 3) The vase can only become empty, after having contained balls, if balls are removed. You say these imply the following. 4) The vase is non-empty at noon. Let's try the same logic on Stephen's example. You say the following are true. 1a) The vase contains a finite number of balls at every time between 11:59 and noon. 2a) No ball insertions occur at noon. 3a) The number of balls in the vase can only become infinite, after having been finite, through the insertion of balls. Therefore (using your logic), these should imply the following. 4a) The number of balls in the vase at noon is finite. But, you say the number of balls in the vase at noon is infinite. Or, consider the following translation of what you said into mathematics. First some definitions: For j = 1,2,..., let a_j = -1/floor((j+9)/10), b_j = -1/j. For j = 1,2,..., define a function f_j: R -> R by f_j(x) = 1 if a_j <= x < b_j, 0 if x < a_j or x >= b_j. Let g(x) = sum_j f_j(x). Then the following are true. 1b) g(x) > 0 for -1 <= x < 0. 2b) For all j, b_j <> 0. 3b) If t1 < t2, g(t1) > 0, and g(t2) = 0, then there is a j such that t1 < b_j <= t2. Therefore (using your logic), these should imply the following. 4b) g(0) > 0. However, this is false. In fact, g(0) = 0. Or, let's try the same logic on the modified problem where all the balls are inserted at one minute before noon. You say the following are true. 1c) The vase is non-empty at every time between 11:59 and noon. 2c) No ball removals occur at noon. 3c) The vase can only become empty, after having contained balls, though the removal of balls. Therefore (using your logic), these should imply the following. 4c) The vase is non-empty at noon. However, you've said that this vase is empty at noon. These examples show that the logic that you are using is not logical. -- David Marcus
From: David Marcus on 2 Nov 2006 23:42 Ross A. Finlayson wrote: > Hi Tony, Stephen, y'all, > > At the time each is removed ten were just added since the last was > removed. > > The supertask, infinitely many discrete events in a finite amount of > time, here this ball and vase scenario aka the Ross-Littlewood paradox, > is interesting to some extent when compared against the general > predilection here against the notion of dense discrete events in their > normal ordering. > > There isn't a point at this case where the events are dense, instead > the standard would not have the time actually reach noon/zero/one but > merely approach it with the inductive guarantee that for however close > you get you can get closer, but not actually reach: look but don't > touch as it were. The points are dense to the left of noon, in that > between any select and noon there are infinitely many others, else the > process doesn't complete. > > Time is presupposed to proceed at a constant rate, where various > excursions into physical theories ascribe to time various conditional > rates. Since Zeno is known the dichotomy of the supertask and constant > time. > > Some people assuage their mathematical guilt that that limit is not the > sum by considering these notions of for example the hyperintegers, with > some expectation of completion in those infinite terms. While that is > so, there is only successor on these finite naturals, so the > hyperintegers could only be reached by somehow becoming infinite from > finite with the only supplied operation: unit increment. So, that's > denial. > > I wonder what the reaction is to a statement along the lines of that > the points are dense in their normal ordering in the reals, in their > normal ordering of the reals. I believe it would be similar here the > reaction of the consideration that there is basically infinity that is > the compactification of the naturals: the covering of eyes, ears, and > mouth. > > Those monkeys are blind, deaf, and dumb. > > Now I'm certainly not calling anyone here except Virgil a monkey. > > Yet, in the example of the counterexamples book which MoeBlee, who is > reasonable, polite, and effusive, has had access to read and who > claimed to be examining said source, nothing has been heard back of the > matter. You can get that for free from the library. > > Nature is basically assumed continuous. Now, I'm aware with some > layman concepts of gauge invariance and the Planck length and so on, > quantized charge and etc., yet the atom used to be atomic, indivisible, > and it has been said the superconductor is one big atom. Some aspects > of the particle/wave duality in nature, physics, are seen with the > reals, comprised as they are of points on a, or in the, line, as has > been discussed here with regards to points on a line. The more closely > (sub-) atomic particles are examined, the smaller they appear to be, > similarly the universe' size always increases upon knowledge of it. > Does that not in simile match examining a lesser positive real? > > Is there a universe ( of sets, or in reality)? The platonist has that > they exist. Are functions between physical objects, eg fields, > physical objects? Then the universe is infinite and as example its own > powerset. Not your best poetry. Doesn't even rhyme. Want to try again? -- David Marcus
From: Virgil on 3 Nov 2006 00:08 In article <454ab927(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > Randy Poe <poespam-trap(a)yahoo.com> wrote: > > > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>>> Mike Kelly wrote: > >>>> Nothing is allowed to happen at noon in either experiment. > >>> Nothing "happens" at noon? I take this to mean that there is no > >>> insertion or removal of balls at noon, yes? Well, I agree with that. > >>> But what relevence does this have to the statement "noon does not > >>> exist"? What does that even *mean*? > >>> > >>> When you've been saying "noon doesn't exist", you actually mean to say > >>> "no insertion or removal of balls occurs at noon"? > >>> > >>> How about this experiment, does noon "exist" in this experiment : > >>> > >>> Insert a ball labelled "1" into the vase at one minute to noon. > >>> > >>> ? > > > >> I think that when Tony and Han say "noon doesn't exist" they > >> really mean "there is no noon on the clock in that experiment", > >> as a way of saying "I have no idea how to answer questions about > >> noon in that experiment, so I'll say that there is no noon and that > >> way I don't have to answer any such questions." > > Or, we say that introducing noon into the situation as the time of an > event creates a contradiction, since 1/n cannot be zero for any natural > n. Since the vase can only become empty (correct me if I'm wrong) if > balls are removed, and no balls are removed at noon, it cannot become > empty at noon. How can the vase be non-empty when every ball has been removed? Every ball is removed from the vase before noon, but TO claims that some of them must somehow sneak back into the vase when no one is looking to make the vase not empty after all balls have been removed.. > On the other hand, at every finite time -t before noon=0, > there are a finite but exponentially growing number of balls in the > vase, with respect to t. On the contrary, at most times nothing at all is happening. There are only countably many instants in an interval of uncountably many, at which any changes occur. TO cannot overcome the fact that every ball inserted before noon is removed before noon. > It's about the pertinent variables not being lost. It is pertinent that every ball inserted before noon is removed before noon. But TO tries to bury that fact.
From: David Marcus on 3 Nov 2006 00:32
Virgil wrote: > In article <454ab927(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > Or, we say that introducing noon into the situation as the time of an > > event creates a contradiction, since 1/n cannot be zero for any natural > > n. Since the vase can only become empty (correct me if I'm wrong) if > > balls are removed, and no balls are removed at noon, it cannot become > > empty at noon. > > How can the vase be non-empty when every ball has been removed? > Every ball is removed from the vase before noon, but TO claims that some > of them must somehow sneak back into the vase when no one is looking to > make the vase not empty after all balls have been removed.. Oh, I know that one: Tony shows the vase is not empty and you show it is. Therefore ZFC is inconsistent! -- David Marcus |