An uncountable countable set
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From: cbrown on 3 Nov 2006 01:16 David Marcus wrote: > Virgil wrote: > > In article <454ab927(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > > > Or, we say that introducing noon into the situation as the time of an > > > event creates a contradiction, since 1/n cannot be zero for any natural > > > n. Since the vase can only become empty (correct me if I'm wrong) if > > > balls are removed, and no balls are removed at noon, it cannot become > > > empty at noon. > > > > How can the vase be non-empty when every ball has been removed? > > Every ball is removed from the vase before noon, but TO claims that some > > of them must somehow sneak back into the vase when no one is looking to > > make the vase not empty after all balls have been removed.. > > Oh, I know that one: Tony shows the vase is not empty and you show it > is. Therefore ZFC is inconsistent! Ah, Grasshopper! You have indeed learned the Way of the Crank Thread. Cheers - Chas
From: cbrown on 3 Nov 2006 02:34 Tony Orlow wrote: > stephen(a)nomail.com wrote: > > Randy Poe <poespam-trap(a)yahoo.com> wrote: > > > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>>> Mike Kelly wrote: > >>>> Nothing is allowed to happen at noon in either experiment. > >>> Nothing "happens" at noon? I take this to mean that there is no > >>> insertion or removal of balls at noon, yes? Well, I agree with that. > >>> But what relevence does this have to the statement "noon does not > >>> exist"? What does that even *mean*? > >>> > >>> When you've been saying "noon doesn't exist", you actually mean to say > >>> "no insertion or removal of balls occurs at noon"? > >>> > >>> How about this experiment, does noon "exist" in this experiment : > >>> > >>> Insert a ball labelled "1" into the vase at one minute to noon. > >>> > >>> ? > > > >> I think that when Tony and Han say "noon doesn't exist" they > >> really mean "there is no noon on the clock in that experiment", > >> as a way of saying "I have no idea how to answer questions about > >> noon in that experiment, so I'll say that there is no noon and that > >> way I don't have to answer any such questions." > > Or, we say that introducing noon into the situation as the time of an > event creates a contradiction, since 1/n cannot be zero for any natural Suppose we want to consider the possibility that the vase is "full"; by which I mean if the vase is full at time t, then if n is a natural number, there is a ball labelled "n" in the vase at time t. Would you claim that if we elimated the removals from the original problem and kept only the insertions, it would then follow that vase becomes full at noon? Cheers - Chas
From: Tony Orlow on 3 Nov 2006 12:35 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> imaginatorium(a)despammed.com wrote: >>> Virgil wrote: >>>> In article <1162268163.368326.64650(a)m73g2000cwd.googlegroups.com>, >>>> imaginatorium(a)despammed.com wrote: >>>> >>>>> Virgil wrote: >>>>>> In article <45462ba0(a)news2.lightlink.com>, >>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> >>>>>>> stephen(a)nomail.com wrote: >>>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>> stephen(a)nomail.com wrote: >>>>>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>>>> stephen(a)nomail.com wrote: >>>>> <snipola> >>>>> >>>>>>>>> OH. So, sets don't have sizes which are numbers, at least at particular >>>>>>>>> moments. I see.... >>>>>>>> If that is what you meant, then you should have said that. >>>>>>>> And technically speaking, sets do not have sizes which are numbers, >>>>>>>> unless by "size" you mean cardinality, and by "number" you include >>>>>>>> transfinite cardinals. >>>>>>> So, cardinality is the only definition of set size which you will >>>>>>> consider.....your loss. >>>>>> It is the only definition of set size that is known to produce a valid >>>>>> partial ordering on sets. >>>>> Huh? I thought cardinality produced a valid *total* ordering on sets. >>>> The cardinalities are totally ordered, but the sets are not. >>>> A total order on sets would require that when neither of two sets was >>>> "larger than" the other that they must be the same set, not merely the >>>> same size. >>> Oh, right. But - and I'm not quite sure how to say this, but the >>> cardinalities _are_ totally ordered; for any two sets A and B, c(A) < >>> c(B), or c(A) = c(B), or c(A) > c(B). If you "reduce" the sets by the >>> cardinality equivalence relation, they are totally ordered. The subset >>> relation doesn't lead to an equivalence relation, only a partial >>> ordering: so there is no s(A) = s(B) unless A=B; but for most pairs of >>> sets, the subset relation simply says nothing at all. (Until His >>> Master's Voice is heard, telling us something totally arbitrary.) >>> >>> Anyway, your claim was clearly wrong, since the subset relation >>> provides a valid partial ordering on sets. > >> If that's what a partial ordering vs. a total ordering is, Bigulosity is >> a partial ordering on sets, not total ordering. Different sets can have >> the same Bigulosity. > > Virgil was muddling everything up, as usual, but the difference between > a partial and total ordering is basically whether there are pairs of > elements for which the order is undetermined. The subset relation is a > very obvious example, where (for example) the set of reals in [0, 1] > and the set of prime integers cannot be compared, because neither is a > subset of the other. Okay, understood. > > "Bigulosity" has never been sufficiently clearly defined to tell, but > since you get very steamed up about subsets, and since the only known > coherent claim is that A proper subset of B -> b(A) < b(B), it's > extremely unlikely Bigulosity could be extended to become a total > ordering. Bigulosity is not based on the subset relation, but on formulaic mappings and infinite-case induction. The proper-subset equality is only mentioned as being one of the most egregious violations of transfinitology. It's fundamentally wrong, intuitively. So, I'm trying to fix that. > > Please compare the Bigulosities of the set of polygons with vertices on > integral x-y coordinates and the set of topologically distinct > polyhedra. Show your working. (Of course you don't need to come up with > an "answer" like "ratio of 5 pi^2", but you need to show how such a > task would be approached. One of the things you still don't seem to > have realised is that before anything can be "maths" it has to be > teachable to other people. I don't think anyone but you has the > faintest idea what Bigulosity is really supposed to be, except in a > ragbag of specific cases.) > > Brian Chandler > http://imaginatorium.org > It consists of a few ideas, and they cover a lot. I don't know where to start with your topologically distinct polyhedra. Why don't you give me a rundown of how YOU compare those two sets?
From: Tony Orlow on 3 Nov 2006 12:46 Virgil wrote: > In article <45476529(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David R Tribble wrote: >>> David R Tribble wrote: >>>>> Each ball n is placed into the vase at time 2^int(n/10), and then later >>>>> removed at time n. This happens for every ball before noon. So every >>>>> ball is inserted and then later removed from the vase before noon. >>>>> >>>>> At any given time n before noon, ten balls are added to the vase and >>>>> then ball n (which was added to the vase in a previous step) is >>>>> removed. Your entire confusion results from assuming a "last" time >>>>> prior to noon, but there is no such time. >>> Tony Orlow wrote: >>>> At no time prior to noon are all balls removed. Nor are any removed at >>>> noon. It cannot be empty, then. >>> The problem states that every ball (every ball) is added to the vase >>> and then later removed from the vase. >> It states the specific times of those events, which imply that there are >> always more balls added than removed at any time. > > Such a statement need not imply any such thing. > And there is no statement in the problem which denies that each ball > inserted before noon is also removed before noon. I don't deny that either, but the stated schedule of events implies that the vase never empties. >>> We conclude from this that every ball is removed (eventually). >> Yes, you conclude an end to the unending set by compressing events at a >> point in time so they cannot be distinguished and the difference between >> in and out is hidden. Whoopedy doo. It's a parlor trick. > > That TO does not understand something may make it a parlor trick to him, > but need not make it so to anyone of greater comprehension. "Greater". Heh. >>> You conclude that at no time are all balls removed. >> There is no finite t<0 when all balls have been removed. Agree? >> >> There are no balls removed at t=0. Agree? > > Note how TO carefully avoids the point, which is whether there are any > balls which have not been removed by t = 0. > "removed by time t"="removed before time t" or "removed at time t". Agree? >>> Obviously you think that there are balls left in the vase that never >>> got removed. In fact, you say that there are an infinitude of balls >>> left in the vase. Yet somehow you cannot name a single one of them. >>> >> I can, as soon as you tell me how many you inserted to begin with. >> Multiply that by 9/10 and you have an answer. > > Since we claim that every ball inserted before noon has been removed by > noon, and the gedankenexperiment confirms this, the number of balls > inserted is irrelevant. > TO be all removed by noon, they must be all removed before noon, or at least some removed at noon. > >> Except that you can't, >> because what you're doing is not math, but Zeno-esque logic trick. > > To those like TO, who do not understand math, most of math seems like > tricks. But their blissful ignorance is still ignorance. Soothe yourself.
From: Randy Poe on 3 Nov 2006 13:30
Tony Orlow wrote: > Virgil wrote: > > Such a statement need not imply any such thing. > > And there is no statement in the problem which denies that each ball > > inserted before noon is also removed before noon. > > I don't deny that either, but the stated schedule of events implies that > the vase never empties. .... at any time < 0, which is when all the scheduled events occur. The schedule implies no such thing about times >= 0. - Randy |