An uncountable countable set
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From: Tony Orlow on 23 Oct 2006 15:03 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, and yet, when every ball is removed before noon, balls remain in the vase? How do you explain that?
From: Tony Orlow on 23 Oct 2006 15:13 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. Now, technically, we look at the formulas regarding the schedule. If we want to allow infinite n, but not reach noon, we can allow for infinitesimal differences between the times of removal and 0. As for finite n, we will continue to have a difference between time of insertion vs removal, but it will be infinitesimal. However, this is only possible if specific infinite n and infinitesimal intervals are allowed. Without them, yes, you have whatever uncountable number of balls being inserted and removed all apparently at the same time, spending apparently no time in the vase, but apparently growing in number exponentially faster even though it's already uncountably fast. The Zeno machine is a little, shall we say, unrealistic. I think that Ross is right, and Maxwell's Demon would easily destroy the vase before noon with friction alone, if not thermonuclear reaction. Tony
From: Tony Orlow on 23 Oct 2006 15:21 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. All points in the iteration-quantity (x-y) continuum where anything occurs lie on this line. (oo,0) is not one of them. > > >> 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.
From: Tony Orlow on 23 Oct 2006 15:24 MoeBlee wrote: > Tony Orlow wrote: >> In Chapter III, section 3.1.1, he states: >> >> "There is no smallest infinite number. For if a is infinite then a<>0, >> hence a=b+1 (the corresponding fact being true in N). But b cannot be >> finite, for then a would be finite. Hence, there exists an infinite >> numbers [sic] which is smaller than a." > > I'll try to take a look at the book soon, but I very strongly suspect > that he's using 'infinite' not to refer to cardinality, but rather to > position in certain orderings. That is fine, as long as we understand > the terminology in the context. As far as I know, it doesn't contradict > that there is a least infinite cardinality. > > MoeBlee > It has absolutely nothing to do with "cardinality" that I can see. He defines an infinite element of *N as being "larger than any finite number", such that x e N ^ y e *N and ~y e N -> y>x. N is a subset of *N. ToeKnee
From: Randy Poe on 23 Oct 2006 15:27
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 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. > 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. 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. 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 |