An uncountable countable set
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From: David Marcus on 27 Oct 2006 16:23 imaginatorium(a)despammed.com wrote: > > David Marcus wrote: > > Lester Zick wrote: > > > On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > > > >A very simple example is that there exists a smallest positive > > > >non-zero integer, but there does not exist a smallest positive > > > >non-zero real. > > > > > > So non zero integers are not real? > > > > That's a pretty impressive leap of illogic. > > Gosh, you obviously haven't seen Lester when he's in full swing. (Have > _you_ searched sci.math for "Zick transcendental"?) I did briefly, but there are so many posts, I didn't read them all. -- David Marcus
From: Virgil on 27 Oct 2006 16:33 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 > experiment, and there
From: Virgil on 27 Oct 2006 16:34 In article <45421333(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > 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. > > That is correct. Noon is incompatible with the problem statement. Noon may be incompatible with TO's understanding, but without it, the original gedankenexperiment is totally undefined. > > 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)? > > I don't believe any of those assumptions are the problem. (2) should > state that t<t0, not t<=t0, at any event. But, that's irrelevant. The > unspoken assumption on your part which causes the problem is that noon > is part of the problem. Noon may be incompatible with TO's understanding, but without it, the original gedankenexperiment is totally undefined. > Clearly, it cannot be, because anything that > happened at t=0 would involve n s.t. 1/n=t. That assumption by TO is nowhere justified by the actual statement of the gedankenexperiment. > Essentially, the problem > produces a paradox by asking a question which contradicts the situation. > Nothing happens at noon. The process never completes the unending set. Like God halting the sun in the sky so the Hebrews could keep fighting?
From: Virgil on 27 Oct 2006 16:37 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.
From: Virgil on 27 Oct 2006 16:40
In article <45421835(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Virgil wrote: > >> In article <4540d217(a)news2.lightlink.com>,Tony Orlow <tony(a)lightlink.com> > >> wrote: > >> > >>> Noon does not exist in the experiment, or else you have infinitely > >>> numbered balls. > >> 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. > >> > >> 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. > > > > That's a good point. > > Not particularly. Noon may be used as a time origin, but if all events > happen such that n e N, and all events happen such that t=-1/n, then > t<>0. Nothing can occur at noon, and the vase is not empty before noon, > so it cannot be empty at noon. That's obvious. WRONG! When every ball inserted before noon must be removed before noon, then to claim any balls still remain at noon is silly. But that's TO. > > > > >>> 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. > >> > >> 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. > >> > >> So either things can happen instantaneously or the experiment impossible. > >> > >> 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? > > > > That's another good point. > > > > Yeah, sure, except that I never objected to an infinite number of events > happening in one moment. I objected to them happening in NO moment. So, > that's not a response. |