An uncountable countable set
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From: Virgil on 23 Oct 2006 16:19 In article <453d0c4e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > If I read this correctly, you agree that at all times every ball that is > > in the vase has a natural number on it, but at noon you say that there > > is a ball in the vase that does not have a natural number on it. Is that > > correct? > > No. I am saying that if only finite iterations of the ball process > occur, then noon never occurs in the experiment to begin with. If noon > DOES exist in the experiment, then that can only mean that some ball n > exists such that 1/n=0, which would have to be greater than any finite n. What part of the gedanken experiment statement says anything like that? > > > > Now, please > > explain what "emptying" means. > > > > "Empty" means not having balls. To become empty means there is a change > of state in the vase ("something happens" to the vase), from having > balls to not having balls. Does "emptying" (going from a state with specific balls in the vase to a state with no balls in the vase) occupy a duration of time greater than zero? > > Now, when does this moment, or interval, occur? If it is an instantaneous process, it would have to "happen" at noon. But as every ball is removed strictly before noon, it does not have to happen at all.
From: Virgil on 23 Oct 2006 16:33 In article <453d0ea5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: > > Why are these "infinitely-numbered balls" inserted, then? > > Because the experiment of inserting ten and removing one, repeatedly, > has extended to a time that requires the numbers on the balls to be > infinite. > > The rules > > quite explicitly say that no ball is inserted unless it has a (finite) > > natural number written on it. > > Then every ball is inserted and removed at a finite time before noon, > but noon itself is never reached. Why not? The continuum of times continues despite what functions of time may be doing. > > We've had variations on the rules that > > say that the demon, after going to put a ball in the vase, > > double-checks, and if the ball doesn't have a pofnat written on it, > > throws it away. Why in your version of the experiment are the rules > > just ignored when it seems to give you the answer you want? > > The rules are not being ignored. The rules that say that all balls are numbered with standard naturals is being violated by the creation of non-standardly numbered balls. >The experiment cannot continue until > noon because that requires infinite naturals, but if it stops before > noon it is not completed. Where in the statement of the problem is there any mention, or even hint, that non-naturally numbered balls will be needed, or even allowed? > During all the time that the experiment can > continue without allowing infinite values of n, there is a steadily > growing number of balls in the vase, nine per iteration, each more > quickly than the last. But, noon cannot be reached with the pofnats. So noon exists after the pofnats. > > > > > Also, suppose for the sake of argument, that there _are_ these > > "infinitely numbered" balls. Are you saying that there is a point at > > which all of the "finitely numbered" balls have been removed (leaving > > the vase empty, which isn't what you are hoping for)? > > At noon, all finitely numbered balls have been inserted and removed, but > in order for noon to arrive, an uncountably infinite number of balls > with infinite values have to have been inserted. But since we can only count those finitely numbered balls which existed at the beginning, and not any of those which TO has whipped up from nowhere, the vase is empty of any balls that count. > > Or are you saying > > there comes a point at which a ball with a number "near the end" of the > > pofnats is being removed, and at the same time the balls being put in > > are actually "infinitely numbered"? That appears to imply that there > > exists a pofnat, call it B, such that B is finite, but 10*B is > > infinite. Is that right? How does this square with even your confused > > understanding of the Peano axioms? > > No, that's idiotic. At noon there is a condensation point in time due to > the Zeno machine where a finite process is being executed infinitely > quickly. That's different from Zeno's Paradox, where the distance > covered is equal to the time elapsed, both of which shrink exponential, > so there is a finite limit to a process with an infinite number of > steps. Here, each step has the same finite difference, and the Zeno Time > Machine causes an infinite number of successions in a moment. So, you > get uncountable infinity at t=0. Wrong. TO may get all sorts of things, but his world is a dream world where anything can happen. In a sane mathematical world, when one starts with only naturally numbered balls, unnatural balls do not start popping up out of nowhere just to satisfy TO's delusions. > Finite naturals can't get you there, > and where they do get you, it's obvious the vase is far from empty. But after they are all past, the vase is empty.
From: Virgil on 23 Oct 2006 16:41 In article <453d1210(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > 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, and yet, > when every ball is removed before noon, balls remain in the vase? How do > you explain that? Trivial! There is no last one to be removed before, yet all are removed before noon.
From: Virgil on 23 Oct 2006 16:44 In article <453d1457(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > imaginatorium(a)despammed.com 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, 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. > > > > <snip> > > > >> Also, suppose for the sake of argument, that there _are_ these > >> "infinitely numbered" balls. Are you saying that there is a point at > >> which all of the "finitely numbered" balls have been removed (leaving > >> the vase empty, which isn't what you are hoping for)? Or are you saying > >> there comes a point at which a ball with a number "near the end" of the > >> pofnats is being removed, and at the same time the balls being put in > >> are actually "infinitely numbered"? That appears to imply that there > >> exists a pofnat, call it B, such that B is finite, but 10*B is > >> infinite. Is that right? How does this square with even your confused > >> understanding of the Peano axioms? > > > >> Brian Chandler > >> http://imaginatorium.org > > > > Also, supposing for the sake of argument that there are "infinitely > > number balls", if a ball is added at time -1/(2^floor(n/10)), and removed > > at time -1/(2^n)), then the balls added at time t=0, are those > > where -1/(2^floor(n/10)) = 0. But if -1/(2^floor(n/10)) = 0 > > then -1/(2^n) = 0 (making some reasonable assumptions about how arithmetic > > on these infinite numbers works), so those balls are also removed at noon > > and > > never spend any time in the vase. > > > > Stephen > > Yes, the insertion/removal schedule instantly becomes infinitely fast in > a truly uncountable way. The only way to get a handle on it is to > explicitly state the level of infinity the iterations are allowed to > achieve at noon. When the iterations are restricted to finite values, > noon is never reached, but approached as a limit. That assumes that there are no times other than times of ball movements, which is not the case. There are times before any movements, times between any two movements and times after all movements.
From: Virgil on 23 Oct 2006 16:59
In article <453d1636(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <453cacc8(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <453bc7c9(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <453b326d(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>> Virgil wrote: > >>>>>>> In article <4539000e(a)news2.lightlink.com>, > >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>> Claimed but not justified. TO's usual technique! > >>>>>> You didn't justify yours. It's clearly nonsensical. It pretends > >>>>>> there's > >>>>>> a time between noon and all times before noon. > >>>>> I only claim there is a time between any time before noon and noon. > >>>>> > >>>> When does the vase become empty? > >>> It is empty at noon and is not empty at any time before noon, but I have > >>> no idea what TO means by "When does the vase become empty?", as it seems > >>> to imply a continuity at 0 that does not exist. > >>> > >> You claim that time is crucial to this problem, but you claim that time > >> is discontinuous? > > > > Quite the contrary, time IS continuous, and I have never claimed > > otherwise. > > > > But functions of time need not be, and in the number of balls function > > cannot be. > > > > The function y=9x is continuous, even if you're only interested in the > values at integral values of x. If x is restricted to integer values, as it must be here, f(x) = 9*x is not in the least continuous. > >> Define "time". > > > > Time is a real variable. > > So, there is nothing between time x and the set of all times y<x, right? > > > > >> Everything that occurs in time includes > >> at least one moment. Name one moment when the vase is emptying. > > > > > > The lack of any such moment is one reason that the "number of balls" as > > a function of time cannot be continuous at noon. Another is that noon is > > a cluster point of other discontinuities of that function. > > Only fictional discontinuities that reside in the Twilight Zone. If f(t) = number of balls in the vase at time t, does TO claim the function is continuous at any time at which balls are being moved in or out of the vase? Each of these times is a time of a jump discontinuity of that function, so that it is only in TO's own twilight zone that these discontinuities do not exist. And noon IS a cluster point of such continuities. That function does not have a limit at noon, so cannot be continuous there, but it can have a value at noon, and that value can be 0 without conflicting with any of the rules of the GE but cannot be anything else without conflicting with the rules of the GE. |