An uncountable countable set
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From: Tony Orlow on 23 Oct 2006 15:32 Virgil wrote: > In article <453caf87(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: >>> Tony Orlow wrote: >>>> cbrown(a)cbrownsystems.com wrote: >>>>> Tony Orlow wrote: >>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> You have agreed with everything so far. At every point before noon >>>>>>>> balls >>>>>>>> remain. >>>>>>> To be precise, the assertions above all imply that at every time t = >>>>>>> -1/n, where n is a natural number, there are balls in the vase. >>>>>>> >>>>>>> But that *alone* does not even include every time t before noon; let >>>>>>> alone every time t. For example, notice that nowhere above do you or I >>>>>>> /explicitly/ assert: "at t=-2/3, the number of balls in the vase is a >>>>>>> positive finite number". >>>>>>> >>>>>>> We assert something specific about t = -2/4, and something specific >>>>>>> about t = -1/3, but nowhere do we directly state somthing about t = >>>>>>> -2/3. >>>>>>> >>>>>>> On the other hand, given the problem statement, I think we would both >>>>>>> /agree/ that there "should be" an obvious (perhaps even unique) >>>>>>> well-defined answer to the question : "what is the number of balls in >>>>>>> the vase at time t = -1/pi?" >>>>>>> >>>>>>> Assuming in the remaining statements that one agrees with the previous >>>>>>> statement, this leads us to the question: what are the unstated >>>>>>> assumptons that allow to agree that this must be the case? >>>>>>> >>>>>>> I attempted to describe those assumptions in my previoius post. Did you >>>>>>> read those assumptions? If so, do you agree with those assumptions? >>>>>> At this point I don't recall your previous post. I've been off a bit. >>>>> Well, allow me to repeat them here (with two minor changes): >>>>> >>>>> In order to interpret the problem >>>>> >>>>> "At each time t = -1/n where n is a (strictly positive) natural number, >>>>> we place the balls labelled 10*(n-1)+1 through 10*n inclusive in the >>>>> vase, and remove the ball labelled n from the vase. What is the number >>>>> of balls in the vase at time t=0?" >>>>> >>>>> I make the following simple (and I would claim, fairly uncontroversial >>>>> and natural) assumptions: >>>>> >>>>> --- (object permanence) >>>>> >>>>> (1) When we speak of a time t, we mean some real number t. >>>>> >>>>> (2) If a ball is in the vase at any time t0, there is a time t <= t0 >>>>> for which we can say "that ball was placed in the vase at time t". >>>>> >>>>> (3) If a ball is placed in the vase at time t1 and it is not removed >>>>> from the vase at some time t where t1 <= t <= t2, then that ball is in >>>>> the vase at time t2. >>>>> >>>>> (4) If a ball is removed from the vase at time t1, and there is no time >>>>> t such that t1 < t <= t2 when that ball is placed in the vase, then >>>>> that ball is not in the vase at time t2. >>>>> >>>>> ---- (obedience to the problem constraints) >>>>> >>>>> (5) If a ball is placed in the vase at some time t, it must be in >>>>> accordance with the description given in the problem: it must be a ball >>>>> with a natural number n on it, and the time t at which it is placed in >>>>> the vase must be -1/floor(n/10). >>>>> >>>>> (6) If a ball is removed from the vase at some time t, it must be in >>>>> accordance with the description given in the problem: it must be a ball >>>>> with a natural number n on it, and the time t at which it is removed >>>>> from the vase must be -1/n. >>>>> >>>>> (7) If n is a natural number with n > 0, then the ball labelled n is >>>>> placed in the vase at some time t1; and it is removed from the vase at >>>>> some time t2. >>>>> >>>>> --- (very general definition of "the vase is empty at noon") >>>>> >>>>> (8) the number of balls in the vase at time t=0 is 0 if, and only if, >>>>> the statement "there is a ball in the vase at time t=0" is false. >>>>> >>>>> --- >>>>> >>>>> Perhaps you would add other assumptions (9), (10), etc.; but my >>>>> question is: >>>>> >>>>> Given the problem statement, do you agree that /each/ of these >>>>> assumptions, /on its own/, is reasonable and not just some arbitrary >>>>> statement plucked out of thin air? >>>>> >>>>> If not, which assumption(s) is(are) not reasonable or is(are) >>>>> unneccessarily arbitrary? >>>>> >>>>> <snip> >>>> Those all look reasonable to me as I read them. I don't see any >>>> statement regarding the fact that ten balls are added for every one >>>> removed, though that can be surmised from the insertion and removal >>>> schedule. That's the salient fact here. You never remove as many as you >>>> add, so you can't end up empty. >>> What about #5? It says that every ball in the vase has a natural number >>> on it. Do you agree with that? >> That is in the problem statement. Therefore, nothing transpires at noon, >> since -1/n<0 for all n e N. > > That statement offers no problems to those who do not require anything > beyond the conditions of the original problem. The problem statement precludes the arrival of noon. When did you stop beating your wife? >>>> 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. > > In any physical world, something happening, or changing, requires an > interval of time of strictly positive duration to occur. Nothing can > "happen" instantaneously in that world. But even where infinitely fast processes occur, there has to be at least a moment where it occurs. > > So what does TO mean by "something happening" instantaneously in a > mathematical world, for which we have no physical world analog? I mean it can happen in a moment, a period of time less than any finite length. > >> There is no moment in this problem where the vase is emptying, >> therefore, that never "occurs". > > The process of "emptying" may not occur, in the sense of the number of > balls decreasing from one moment to another at any time before noon, but > the result does, in the sense of there being no ball which has not been > removed, at noon.
From: Tony Orlow on 23 Oct 2006 15:34 Virgil wrote: > In article <453cb03a(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: >>> Tony Orlow wrote: >>> >>>> At every time before noon there are a growing number of balls in the >>>> vase. The only way to actually remove all naturally numbered balls from >>>> the vase is to actually reach noon, in which case you have extended the >>>> experiment and added infinitely-numbered balls to the vase. All >>>> naturally numbered balls will be gone at that point, but the vase will >>>> be far from empty. >>> By "infinitely-numbered", do you mean the ball will have something other >>> than a natural number written on it? E.g., it will have "infinity" >>> written on it? >>> >> Yes, that is precisely what I mean. If the experiment is continued until >> noon, so that all naturally numbered balls are actually removed (for at >> no finite time before noon is this the case), then any ball inserted at >> noon must have a number n such that 1/n=0, which is only the case for >> infinite n. If the experiment does not go until noon, not all naturaly >> numbered balls are removed. If it does, infinitely-numbered balls are >> inserted. > > > And where do these allegedly infinitely numbered balls come from? > I do not recall any of them being mentioned in the original gedanken, so > that TO is creating his own separate gedanken. > > Note that whatever TO may require in his version of the gedanken, his > requirements do not alter that no such thing occurs in the original. > > TO takes the childish position that if he cannot have things his own > way playing by the rules, he will change the rules to get his own way. They come from the inclusion of noon in the experiment. If any ball is removed at noon, ten are inserted, and their numbers will be of a form that makes them infinite. Nothing occurs at noon in your experiment, and the vase is empty before noon.
From: Tony Orlow on 23 Oct 2006 15:35 Tony Orlow wrote: > Virgil wrote: >> In article <453cb03a(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: >> >>> David Marcus wrote: >>>> Tony Orlow wrote: >>>> >>>>> At every time before noon there are a growing number of balls in >>>>> the vase. The only way to actually remove all naturally numbered >>>>> balls from the vase is to actually reach noon, in which case you >>>>> have extended the experiment and added infinitely-numbered balls to >>>>> the vase. All naturally numbered balls will be gone at that point, >>>>> but the vase will be far from empty. >>>> By "infinitely-numbered", do you mean the ball will have something >>>> other than a natural number written on it? E.g., it will have >>>> "infinity" written on it? >>>> >>> Yes, that is precisely what I mean. If the experiment is continued >>> until noon, so that all naturally numbered balls are actually removed >>> (for at no finite time before noon is this the case), then any ball >>> inserted at noon must have a number n such that 1/n=0, which is only >>> the case for infinite n. If the experiment does not go until noon, >>> not all naturaly numbered balls are removed. If it does, >>> infinitely-numbered balls are inserted. >> >> >> And where do these allegedly infinitely numbered balls come from? >> I do not recall any of them being mentioned in the original gedanken, >> so that TO is creating his own separate gedanken. >> >> Note that whatever TO may require in his version of the gedanken, his >> requirements do not alter that no such thing occurs in the original. >> >> TO takes the childish position that if he cannot have things his own >> way playing by the rules, he will change the rules to get his own way. > > They come from the inclusion of noon in the experiment. If any ball is > removed at noon, ten are inserted, and their numbers will be of a form > that makes them infinite. Nothing occurs at noon in your experiment, and ACK!! I meant the vase is NOT empty before noon. > the vase is empty before noon.
From: Tony Orlow on 23 Oct 2006 15:40 Randy Poe wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>> >>>>>> At every time before noon there are a growing number of balls in the >>>>>> vase. The only way to actually remove all naturally numbered balls from >>>>>> the vase is to actually reach noon, in which case you have extended the >>>>>> experiment and added infinitely-numbered balls to the vase. All >>>>>> naturally numbered balls will be gone at that point, but the vase will >>>>>> be far from empty. >>>>> By "infinitely-numbered", do you mean the ball will have something other >>>>> than a natural number written on it? E.g., it will have "infinity" >>>>> written on it? >>>>> >>>> Yes, that is precisely what I mean. If the experiment is continued until >>>> noon, >>> The clock ticks till noon and beyond. However, the explicitly- >>> stated insertion times are all before noon. >>> >>>> so that all naturally numbered balls are actually removed (for at >>>> no finite time before noon is this the case), >>> There is no finite time before noon when all balls have >>> been removed. >>> >>> However, any particular ball is removed at a finite >>> time before noon. >>> >>>> then any ball inserted at >>>> noon must have a number n such that 1/n=0, >>> However, there is no ball inserted at noon. >>> >>>> which is only the case for >>>> infinite n. If the experiment does not go until noon, not all naturaly >>>> numbered balls are removed. >>> The experiment goes past noon. No ball is inserted at noon, >>> or past noon. >>> >>> - Randy >>> >> Randy, does it not bother you that no ball is removed at noon, > > ... I agree with that... > >> and yet, when every ball is removed before noon, I should have said "each"... > > ... I agree with that... > >> balls remain in the vase? > > .. I don't agree with that. > > When did I ever say balls remain in the vase? Every ball > is removed before noon. No balls remain in the vase at > noon. Do you disagree with the statement that, at every time -1/n, when ball n is removed, for every n e N, there remain balls n+1 through 10n, or 9n balls, in the vase? > >> How do you explain that? > > I would certainly have difficulty understanding how the vase > could be non-empty at noon, given that every ball in the vase > is removed before noon. There is no ball, in all of N, for which the vase is empty at its departure from the vase. > > But YOU are the one who says the vase is non-empty at > noon. I never said such a thing. I'm certainly not going to > defend YOUR illogical position. I was saying it is non-empty at every one of the finite times before noon where any ball is inserted or removed. Do you argue against THAT statement? > > So you now agree that it makes no sense that the vase > could be non-empty at noon? That the vase must, in other > words, be empty? > > - Randy > No, you misread.
From: Randy Poe on 23 Oct 2006 15:45
Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>> > >>>>>> At every time before noon there are a growing number of balls in the > >>>>>> vase. The only way to actually remove all naturally numbered balls from > >>>>>> the vase is to actually reach noon, in which case you have extended the > >>>>>> experiment and added infinitely-numbered balls to the vase. All > >>>>>> naturally numbered balls will be gone at that point, but the vase will > >>>>>> be far from empty. > >>>>> By "infinitely-numbered", do you mean the ball will have something other > >>>>> than a natural number written on it? E.g., it will have "infinity" > >>>>> written on it? > >>>>> > >>>> Yes, that is precisely what I mean. If the experiment is continued until > >>>> noon, > >>> The clock ticks till noon and beyond. However, the explicitly- > >>> stated insertion times are all before noon. > >>> > >>>> so that all naturally numbered balls are actually removed (for at > >>>> no finite time before noon is this the case), > >>> There is no finite time before noon when all balls have > >>> been removed. > >>> > >>> However, any particular ball is removed at a finite > >>> time before noon. > >>> > >>>> then any ball inserted at > >>>> noon must have a number n such that 1/n=0, > >>> However, there is no ball inserted at noon. > >>> > >>>> which is only the case for > >>>> infinite n. If the experiment does not go until noon, not all naturaly > >>>> numbered balls are removed. > >>> The experiment goes past noon. No ball is inserted at noon, > >>> or past noon. > >>> > >>> - Randy > >>> > >> Randy, does it not bother you that no ball is removed at noon, > > > > ... I agree with that... > > > >> and yet, when every ball is removed before noon, > > I should have said "each"... Each ball is removed before noon. Every ball is removed before noon. All the balls are removed before noon. Given any ball, that ball is removed before noon. > > ... I agree with that... > > > >> balls remain in the vase? > > > > .. I don't agree with that. > > > > When did I ever say balls remain in the vase? Every ball > > is removed before noon. No balls remain in the vase at > > noon. > > Do you disagree with the statement that, at every time -1/n, when ball n > is removed, for every n e N, there remain balls n+1 through 10n, or 9n > balls, in the vase? At every time before noon, there are not only finitely many balls in the vase, there are still infinitely many balls yet to be put in the vase. Of course, every one of those balls will be removed before noon. Without exception. > >> How do you explain that? > > > > I would certainly have difficulty understanding how the vase > > could be non-empty at noon, given that every ball in the vase > > is removed before noon. > > There is no ball, in all of N, for which the vase is empty at its > departure from the vase. Yes. There is no last ball. But there is no ball which fails to be removed. > > But YOU are the one who says the vase is non-empty at > > noon. I never said such a thing. I'm certainly not going to > > defend YOUR illogical position. > > I was saying it is non-empty at every one of the finite times before > noon where any ball is inserted or removed. Do you argue against THAT > statement? I agree with that statement. For t<0, the vase is non-empty. At t=0, the vase is empty. > > So you now agree that it makes no sense that the vase > > could be non-empty at noon? That the vase must, in other > > words, be empty? > > No, you misread. You didn't ask me "does it make sense that the vase is non-empty at noon"? Ah well, I'll answer that question anyway. No, it doesn't make sense to me that the vase would be non-empty at noon. Of course it's empty. - Randy |