An uncountable countable set
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From: Tony Orlow on 27 Oct 2006 20:25 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>>> cbrown(a)cbrownsystems.com wrote: >>>>> Tony Orlow wrote: >>>>>> Mike Kelly wrote: >>>>> <snip> >>>>> >>>>>>> My question : what do you think is in the vase at noon? >>>>>>> >>>>>> A countable infinity of balls. >>>>>> >>>>>> This is very simple. Everything that occurs is either an addition of ten >>>>>> balls or a removal of 1, and occurs a finite amount of time before noon. >>>>>> At the time of each event, balls remain. At noon, no balls are inserted >>>>>> or removed. >>>>> No one disagrees with the above statements. >>>>> >>>>>> The vase can only become empty through the removal of balls, >>>>> Note that this is not identical to saying "the vase can only become >>>>> empty /at time t/, if there are balls removed /at time t/"; which is >>>>> what it seems you actually mean. >>>>> >>>>> This doesn't follow from (1)..(8), which lack any explicit mention of >>>>> what "becomes empty" means. However, we can easily make it an >>>>> assumption: >>>>> >>>>> (T1) If, for some time t1 < t0, it is the case that the number of balls >>>>> in the vase at any time t with t1 <= t < t0 is different than the >>>>> number of balls at time t0, then balls are removed at time t0, or balls >>>>> are added at time t0. >>>>> >>>>>> so if no balls are removed, the vase cannot become empty at noon. It was >>>>>> not empty before noon, therefore it is not empty at noon. Nothing can >>>>>> happen at noon, since that would involve a ball n such that 1/n=0. >>>>> Now your logical argument is complete, assuming we also accept >>>>> (1)..(8): If the number of balls at time t = 0, then by (7), (5) and >>>>> (6), the number of balls changes at time 0; and therefore by (T1), >>>>> balls are either placed or removed at time 0, implying by (5) and (6) >>>>> that there is a natural number n such that -1/n = 0; which is absurd. >>>>> Therefore, by reductio ad absurdum, the number of balls at time 0 >>>>> cannot be 0. >>>>> >>>>> However, it does not follow that the number of balls in the vase is >>>>> therefore any other natural number n, or even infinite, at time 0; >>>>> because that would /equally/ require that the number of balls changes >>>>> at time 0, and that in turn requires by (T1) that balls are either >>>>> added or removed at time 0; and again by (5) or (6) this implies that >>>>> there is a natural number n with -1/n = 0; which is absurd. So again, >>>>> we get that any statement of the form "the number of balls at time 0 is >>>>> (anything") must be false by reductio absurdum. >>>>> >>>>> So if we include (T1) as an assumption as well as (1)..(8), it follows >>>>> logically that the number of balls in the vase at time 0 is not >>>>> well-defined. >>>>> >>>>> Of course, we also find that by (1)..(8) and (T1), it /still/ follows >>>>> logically that the number of balls in the vase at time t is 0; and this >>>>> is a problem: we can prove two different and incompatible statements >>>>> from the same set of assumptions >>>>> >>>>> So at least one of the assumptions (1)..(8) and (T1) must be discarded >>>>> if we are to resolve this. What do you suggest? Which of (1)..(8) do >>>>> you want discard to maintain (T1)? >>>>> >>>>> Cheers - Chas >>>>> >>>> This is a very good question, Chas. Thanks. I'll have to think about it, >>>> and I'm rather tired right now, but at first glance it seems like it >>>> could be a sound analysis. I've cut and pasted for perusal when I'm >>>> sharper tomorrow. >>>> >>> Here's some of my thoughts: >>> >>> When you say "noon doesn't occur"; I think "he doesn't accept (1): by a >>> time t, we mean a real number t" >> That doesn't mean t has to be able to assume ALL real numbers. The times >> in [-1,0) are all real numbers. > > And I would say that assuming that by a time t, we mean a real number > in [-1,0) is a different assumption than (1). > >>> When you say "if we always add more balls than we remove, the number of >>> balls in the vase at time 0 is not 0", I think "he doesn't accept (8): >>> if the numbers of balls in the vase is not 0, then there is a ball in >>> the vase." >> No, I accept that. There is no time after t=-1 where there is no ball in >> the vase. >> > > I.e., there is a ball in the vase. But then by the argument I > previously gave, there is then a ball in the vase which is not in the > vase. Your reference to "unspecified" balls in the vase at noon I > interpret to be a way of saying that (8) should instead state something > like "if the number of balls in the vase is not 0, then there may be no > /specific/ ball in the vase (because there is instead an /unspecific/ > ball in the vase)". > Specify the largest natural, and I'll grant you that. >>> When you say "an infinite number of balls are removed at time 0", I >>> think "he does not agree with (6) if balls are removed at some time t, >>> they are removed in accordance with the problem statement: i.e. there >>> exists some natural number n s.t. n = -1/t and (some other stuff)". >> I didn't say that exactly. If 0 occurs, then all finite balls are gone, >> but infinite balls have been inserted such that 1/n=0 for those balls. >> So, at noon the vase is not empty, even if it occurs in the problem, >> which it doesn't. >> > > If infinite balls are inserted at some time t = -1/n = 0, then by (5) > each of them are inserted at time t; and at that time exactly 10 balls > are inserted. 10 is not infinite. 10 balls per iteration times oo iterations per second is oo (or 10*oo) balls per second. > >>> All these assertions follow a simgle theme: "If I require that my >>> statemnents be /logically/ consistent, does the given problem make >>> sense; and if so, what is a reasonable resonse?". >>> >>> Cheers - Chas >>> >> That there is a contradiction in your conclusion if you assume that all >> events must occur at some time... > > The "occurence" of these events (ball insertions and removals at > particular times) is described by (1), (5), (6), and (7). There is the event of becoming empty, i.e., in(t)-out(t)=0. Oh, except, that never happens. > >> ... and that becoming empty is the result of >> events that happen in the vase. > > There is no "becoming" empty described in (1)..(8). There is only > "being" empty; which is described by (1), (2), (3), and
From: Tony Orlow on 27 Oct 2006 20:26 Virgil wrote: > In article <1161936661.828422.190110(a)e3g2000cwe.googlegroups.com>, > imaginatorium(a)despammed.com wrote: > >> David Marcus wrote: >>> imaginatorium(a)despammed.com wrote: >>>> Virgil wrote: >>>>> In article <45417528$1(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> <snip> >>>> >>>>>> For what it's worth, and I know this doesn't add a lot of credibility >>>>>> to >>>>>> Ross in your eyes, coming from me, but I think Ross has a genuine >>>>>> intuition that isn't far off with respect to what's controversial in >>>>>> modern math. Sure, he gets repetitive and I don't agree with >>>>>> everything >>>>>> he says, but his cryptic "Well order the reals", which I actually >>>>>> haven't seen too much of lately, is a direct reference to his EF >>>>>> (Equivalence Function, yes?) between the naturals and the reals in >>>>>> [0,1). The reals viewed as discrete infinitesimals map to the >>>>>> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >>>>>> answer your question, I think Ross makes some sense. But, of course, >>>>>> coming from me, that probably doesn't mean much. :) >>>>> Coming from TO it damns Ross. >>>> Even by your standards, Virgil, this is egregiously silly. TO skips the >>>> basic exposition in Robinson's book, but finds a sentence he likes. So >>>> this "damns" Robinson's non-standard analysis, does it? >>> Virgil said "Ross", not "Robinson", I believe. >> Yes, of course. But Virgil's implication is that "TO says person P is >> right about something" implies P is wrong. This may, contingently, be >> true about Ross, but the argument could equally be applied to Robinson, >> in which case the conclusion is obviously not true. > > Since Ross has, by his own posts, shown himself to be as far out of > touch with reality as TO, TO's approval is only piling Pelion on Ossa. > > If TO were to support someone reasonably in touch with mathematical > reality, I should not have regarded it as a "last straw" situation. Like Robinson?
From: Tony Orlow on 27 Oct 2006 20:26 Virgil wrote: > In article <1161936661.828422.190110(a)e3g2000cwe.googlegroups.com>, > imaginatorium(a)despammed.com wrote: > >> David Marcus wrote: >>> imaginatorium(a)despammed.com wrote: >>>> Virgil wrote: >>>>> In article <45417528$1(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> <snip> >>>> >>>>>> For what it's worth, and I know this doesn't add a lot of credibility >>>>>> to >>>>>> Ross in your eyes, coming from me, but I think Ross has a genuine >>>>>> intuition that isn't far off with respect to what's controversial in >>>>>> modern math. Sure, he gets repetitive and I don't agree with >>>>>> everything >>>>>> he says, but his cryptic "Well order the reals", which I actually >>>>>> haven't seen too much of lately, is a direct reference to his EF >>>>>> (Equivalence Function, yes?) between the naturals and the reals in >>>>>> [0,1). The reals viewed as discrete infinitesimals map to the >>>>>> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >>>>>> answer your question, I think Ross makes some sense. But, of course, >>>>>> coming from me, that probably doesn't mean much. :) >>>>> Coming from TO it damns Ross. >>>> Even by your standards, Virgil, this is egregiously silly. TO skips the >>>> basic exposition in Robinson's book, but finds a sentence he likes. So >>>> this "damns" Robinson's non-standard analysis, does it? >>> Virgil said "Ross", not "Robinson", I believe. >> Yes, of course. But Virgil's implication is that "TO says person P is >> right about something" implies P is wrong. This may, contingently, be >> true about Ross, but the argument could equally be applied to Robinson, >> in which case the conclusion is obviously not true. > > Since Ross has, by his own posts, shown himself to be as far out of > touch with reality as TO, TO's approval is only piling Pelion on Ossa. > > If TO were to support someone reasonably in touch with mathematical > reality, I should not have regarded it as a "last straw" situation. Like Boole?
From: Tony Orlow on 27 Oct 2006 20:28 Virgil wrote: > In article <4542164c(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >>>> Please specify the moment when the vase becomes empty. >>> It IS empty at noon, but not before. But I do not know what TO means by >>> "becomes". >> Become: To assume a state not previously assumed. > > Since the vase was empty to start with, it cannot later "become" empty > after once having been empty, at least according to that definition. > > >>>>> The only relevant question is "According to the rules set up in the >>>>> problem, is each ball inserted at a time before noon also removed at a >>>>> time before noon?" >>>>> >>>>> An affirmative answer confirms that the vase is empty at noon. >>>> Not if noon is proscribed the the problem itself, which it is. >>> How so? I see nothing in the statement of the problem which "proscribes" >>> noon. >> Nothing can occur at noon because that implies 1/n=0, false for all >> natural numbers. > > Where in the gedankenexperiment is that required? >>>>> A negative answer directly violates the conditions of the problem. >>>>> >>>>> How does TO answer this question? >>>>> >>>>> As usual, he avoids such relevant questions in his dogged pursuit of the >>>>> irrelevant. >>>>> >>>> Noon does not exist in the experiment, or else you have infinitely >>>> numbered balls. > > Two assumptions both at variance with the original gedankenexperiment. >>> It is specifically mentioned in the experiment as the base time from >>> which all actions are determined, so that if it does not exist then none >>> of the actions can occur. >> No, time begins at -1, such that t(n)=-1/n. n never becomes infinite, so >> t never becomes 0. > > -1 is not a time unless there is a 0 from which to measure it. >>> If there is no noon then there can be no one minute before noon at which >>> the first ball is inserted, so the vase is frozen in a state of >>> emptiness. >> At t=-1=-1/n, n=1. Are you saying 1 is not a natural number? > > I am saying that there are no negative real numbers without a 0 from > which to to mark them. > >> I thought >> the labels were the most important aspect of this for you. Now you want >> to ignore them? Huh! > > Non sequitur. That I want time to be properly measured does not mean > require I want to ignore other things. >>>>>>> Obviously, time is an independent variable in this experiment and the >>>>>>> insertion or removal or location of balls is a function of time. That's >>>>>>> what the problem statement says: we have this thing called "time" which >>>>>>> is a real number and it "goes from" before noon to after noon and, at >>>>>>> certain specified times, things happen. There are only >>>>>>> naturally-numbered balls inserted and removed, always before noon. >>>>>>> Every ball is removed before noon. Therefore, the vase is empty. >>>>>> No, you have the concept of the independent variable bent. The number of >>>>>> balls is related to the time by a formula which works in both directions. >>> Where does the problem say that the numbers on balls being moved >>> determines the time? >> Of each event? Where it says that ball n is inserted at time -1/n and >> removed at time -1/10n. That was a dumb question. > > So the moving of the balls is determined by time, not the other way > around. >>>>> As time is a continuum and the numbers of balls in the vase is not, >>>>> there is no way of inverting the realtionship in the way that TO claims. >>>> Your times are as discontinuous as the number of balls, if no events can >>>> happen at any other moments than those specified. >>> That hardly means that there are no other times in between. >>> >>> Time is a continuum. Or does TO claim that time is quantized? >> Where real time is continuous, there is always something happening. >> That's not the case here. > > Sure there is, we are watching continuously! > >> The moments during events are a countable >> subset of the uncountable interval. > > > The moments of change are countable, but in between these moments the > vase and the balls do not disappear, they still exist, they just don't > move. > >>> So how is the vase be not empty after every ball is removed? >> There is no "after". > > So TO wants to stop the clock? Is his face enough to do it? > > >> You are hiding your largest finite in a moment of >> infinite processing, but it's leaving a hole in your logic. > > TO may want a largest finite in his universe, in which clocks stop for > no reason, but no one else need have one in theirs. > > > > >>>> Like something occurring in time without at least a moment in which it >>>> occurred. >>> In the physical world, nothing happens instantaneously. In the >>> mathematical world, pretty much everything does. >> Uh, yeah, at specific times. But there is no non-self-contradictory >> moment in the problem at which the emptying can occur. > > A bit of it happens each time a ball is removed, and it is completed > when every ball has been removed, i.e., at noon. > > > Or does TO's face stop the clock again? >>> In the mathematical world of the experiment, the balls move in and out >>> of the vase instantaneously, and must be allowed to do so or the >>> experiment cannot be performed at all. >> Of course, each event at a specific time. >> >>> So either things can happen instantaneously or the experiment impossible. >> So what? Happening instantaneously doesn't mean there is no moment in >> the event. It means there's only one. Without at least occupying one >> moment, an event does not happen in time. It's like trying to pretend >> you have a geometric figure that contains no points. The set of times >> that the vase becomes empty is null. > > But there is a noon, if there are any times at all, and at noon the vase > holds no balls >>> If TO allows a finite change of number of balls in the vase to occur >>> instantaneously, what is so difficult about allowing an "infinite" >>> change in the number of balls to occur instantaneously? >>> >>> TO seems to swallow camels and strain at gnats. >>> >> There's nothing difficult about that. I agreed that, without removing >> balls, one gets an uncountable rate of increase at t=0. The problem is >> that there is no time when it can become empty in the original >> expe
From: Tony Orlow on 27 Oct 2006 20:34
Virgil wrote: > In article <45421735(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >>> In order to have emptiness at noon, all removals must take place no >>> later than noon, which they are forced to do by the rules of the problem. >> That means either before noon, or at noon. No balls are removed at noon. >> Balls remain at every time before noon. You're busted. > > Except that every ball inserted before noon has been removed before > noon according to the specifications of the gedankenexperiment. > > So it is TO who is busted for violating the rules of the > gedankenexperiment. At least you're not drenching my face with pepper spray. But, then again, I'm not the one who squealed like a piggy. :) Chuckles with blood splatters, Tony |