An uncountable countable set
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From: Tony Orlow on 23 Oct 2006 19:51 Virgil wrote: > 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? The part that says that ball n is removed at t=-1/n, combined with t=0, or t=-0. Then 1/n=0, true only for infinite n. > >>> 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? It doesn't even appear to have that single moment to occur, in this experiment, since it can't occur before noon, nor at noon, nor thereafter. Certainly, if the vase starts with some uncountably infinite number of balls which are removed according to the Zeno schedule, it will empty, the vast, infinite majority being removed AT noon. But, if this experiment is to empty, and is an experiment in time, then you should be able to say when that occurs. >> Now, when does this moment, or interval, occur? > > If it is an instantaneous process, it would have to "happen" at noon. So, you are saying that something does occur at noon. But, what causes that? Surely there are no naturally-numbered balls being added or removed at noon? > > But as every ball is removed strictly before noon, it does not have to > happen at all. You mean the vase doesn't have to empty? Then what makes you think it's empty? I know, I know. You have your logic. But, it amounts to artificially creating an upper bound to a boundless set, compressing some infinity of elements into some single moment or real point, where the comparison between what is entering and what's exiting is totally clouded. The fact remains that it doesn't become empty before noon, and nothing happens at noon, so it doesn't empty. :)
From: Tony Orlow on 23 Oct 2006 19:54 Virgil wrote: > 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. We are talking about at noon itself, not at any finite time before noon.
From: Tony Orlow on 23 Oct 2006 19:59 Virgil wrote: > 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. Look, fine, relax. Call it a constant discrete increment within the set. For each iteration, the increase in set size is exactly 9 balls. That does not change. Every point of the graph of events, where x is iterations and y is balls in the vase, lies on the line y=9x.
From: Tony Orlow on 23 Oct 2006 19:59 Virgil wrote: > In article <453d16f3(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. > > But none of Robinson's non-standard numbers are cardinalities. No kidding. They actually make sense.
From: David Marcus on 23 Oct 2006 20:20
cbrown(a)cbrownsystems.com wrote: > David Marcus wrote: > > Your assumptions seem consistent with the following formulation of the > > problem. > > > > For n = 1,2,..., define > > > > A_n = -1/floor((n+9)/10), > > R_n = -1/n. > > > > For n = 1,2,..., define a function B_n by > > > > B_n(t) = 1 if A_n < t < R_n, > > 0 if t < A_n or t > R_n, > > undefined if t = A_n or t = R_n. > > > > Let V(t) = sum{n=1}^infty B_n(t). What is V(0)? > > Yes; and in fact I chose (1)-(8) to be consistent with just about any > sensible interpretation of the problem as given. > > But I'm currently trying with Tony to completely avoid numerical > arguments such as the above, which rely on a complicated definition of > "the number of balls at time t", in favor of the much simpler to agree > with statement "either there is a ball in the vase at time t, or the > number of balls in the vase at time t is 0; and not both". It is worth a try. Although, I think you'd do better to limit your replies to a few connected paragraphs rather than reply to so many individual paragraphs of Tony's post. While carrying on many threads of a discussion in one post saves time (like I'm doing in this post), it only seems to work if both people are basically on the same wavelength. Otherwise, people pick and choose which comments they reply to. > I mean, given his confusion over simple logical arguments like "If > (A->not A), then not A", I shudder to think what subtle > misunderstandings exist in his version of "define a sequence of > functions indexed by n in N". Perhaps. Although, I think he might have taken Calculus at some point. Many people who take Calculus learn something about functions despite being unable to reason logically. > > It is rather amazing. > > It's also sort of fascinating - how can one /not/ understand the > argument, and yet give the impression of understanding /some/ sort of > logic? It's like some sort of mental blind spot. Do you really think he understands any logic? I believe that 98% or more of people don't think conceptually/logically. Instead they rely on the brain's amazing ability to do pattern matching. Pattern matching is extrememly useful, but it dosn't do logic. > > The logic seems to be that the limit of the number > > of balls in the vase as we approach noon is infinity, so the number of > > balls in the vase at noon must be infinity, but all numbered balls have > > been removed, therefore the infinity of balls in the vase at noon aren't > > numbered. It does have a sort of surreal appeal. > > If we assume at the start that the number of balls at t=0 is /anything > but/ 0 (as TO apparantly does, although he has yet to realize it), then > pretty much anything goes. Let your imagination roam! There are a prime > number of cubical balls in the vase at noon! ZFC is inconsistent! > Cantor is alive and living in Brooklyn New York! I am the current King > of France! I don't agree that he is assuming that. I think he isn't reasoning logically at all. The number of balls approaches infinity as time approaches noon. If you imagine a vase filling up with an infinite number of balls, it is rather hard to imagine them suddenly all disappearing. Of course, mathematics isn't constrained by our imagination. It relies on precise definitions and logic. And, functions do not have to be continuous. -- David Marcus |