An uncountable countable set
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From: Virgil on 31 Oct 2006 15:21 In article <454762d8(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > David R Tribble wrote: > >>> Every member of N has a finite successor. Can you prove that your > >>> "infinite naturals" are members of N? > > > > Virgil wrote: > >>> The property of not being an infinite natural holds for the first > >>> natural, and holds for the successor of each non-infinite natural, so > >>> that it must hold for ALL naturals. > > > > Tony Orlow wrote: > >> It holds for all finite naturals, but if there are an infinite number of > >> naturals generating using increment, then there are naturals which are > >> the result of infinite increments, which must have infinite value. The naturals are naturally well ordered, which means that any non-empty subset of them must have a first element. So either the set of infinite naturals is empty or it has a first element. Which is it TO? > > > > Can you show us one of those infinite naturals? > > > > > > I meant an infinite number of increments, each being a successive > difference of +1 in measure. So what is the first infinite natural, TO? And which finite natural is it the successor of?
From: Virgil on 31 Oct 2006 15:30 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. > > > > > 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. > > > 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. > > 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. > 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.
From: imaginatorium on 31 Oct 2006 15:30 Virgil wrote: > In article <1162276209.968986.110750(a)f16g2000cwb.googlegroups.com>, > 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. > > Since there are sets that the subset relation cannot compare for "size", > since trichotomy does not hold for the subset relation, I do not regard > the subset relation as a valid measure of size in the same sense that > cardinality is, at least assuming the axiom of choice. Nor, obviously, do I. My point was that in your haste to try to beat Tony on quantity, what you actually said was simply wrong. What you said was that "[cardinality] is the only definition of set size that is known to produce a valid partial ordering on sets." You do not, I presume, claim this is actually true? AAMOF, of course cardinality isn't the only definition of set size that produces a total ordering on sets: Let the set {} be "empty" Let the set {x} for any x be "a singleton" Let the set {a, b} for any a,b be "small" Let the set {p, q, r } for any p q r be "medium" And let all other sets be "large" These sizes are totally ordered, are they not? Brian Chandler http://imaginatorium.org
From: Virgil on 31 Oct 2006 15:33 In article <45476d37$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > Virgil wrote: > >>> Are the properties of "Finlayson Numbers" known to anyone except > >>> Ross himself? > > > > Tony Orlow wrote: > >> Uh, yeah, I think I understand what his numbers are. Perhaps you've seen > >> our recent exchange on the matter? They are discrete infinitesimals such > >> that the sequence of them within the unit interval maps to the naturals > >> or integers on the real line. Is that about right, Ross? > > > > Only a countable infinity of them? Then the number of infinitesimals > > in [0,1] is exactly the same as the reciprocals 1/n for every natural > > n>0, right? But there are c reals in [0,1], so are there more reals > > than infinitesimals? > > > > I think Ross has to answer that one. In my book, the naturals are really > *N, the hypernaturals, and so there are an uncountably, actually > infinite, number of them, and then EF works for me as a special case of IFR. TO's book only exists in TO's twilight zone.
From: Virgil on 31 Oct 2006 15:45
In article <45476fcb(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Randy Poe wrote: > >>>>> Tony Orlow wrote: > >>>>>> Virgil wrote: > >>>>>>> In article <4542201a(a)news2.lightlink.com>, > >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>> > >>>>>>>> cbrown(a)cbrownsystems.com wrote: > >>>>>>>>> When you say "noon doesn't occur"; I think "he doesn't accept (1): > >>>>>>>>> by a > >>>>>>>>> time t, we mean a real number t" > >>>>>>>> That doesn't mean t has to be able to assume ALL real numbers. The > >>>>>>>> times > >>>>>>>> in [-1,0) are all real numbers. > >>>>>>> By what mechanism does TO propose to stop time? > >>>>>> By the mechanism of unfinishablility. > >>>>> But that's why I asked you a question about variables labelling > >>>>> times yesterday, when noon clearly occurred. > >>>> The experiment occurred in [-1,0). Talk of time outside that range is > >>>> irrelevant. Times before that are imaginary, and times after that are > >>>> infinite. Only finite times change anything, so if something changes, > >>>> it's at a finite, negative time. > >>>> > >>>>> I can define a list of times t_n = noon yesterday - 1/n seconds, > >>>>> for all n=1, 2, 3, ... > >>>> Are there balls in the vase for t<-1? No. > >>> What balls? What vase? > >>> > >>> I'm naming times. They're just numbers. > >>> > >>>>> Clearly this list of times has no end. But didn't noon happen? > >>>> Nothing happened at noon to empty the vase, \ > >>> What vase? Why are you obsessed with vases? > >>> > >>> Do you deny me the ability to create a set of variables > >>> t_n, n = 1, 2, ...? Why do vases have to come into it? > >> I thought we were trying to formulate the problem. > > > > No, we (some of us) are trying to formulate a completely > > different problem, with balls and vases (possibly even > > times) explicitly removed so that other aspects can be examined. > > > > Yet you keep trying to put balls and vases back in, after being told > > that they are not present in the new problem. > > I am pointing out that your formulation doesn't match the original > problem. You say above you are possibly eliminating times form the > problem. Sorry, but that's a required feature, and if your mathematical > description ignores them, then it's missing the boat. TO has several times himself tried to eliminate times, because keeping the times in requires that he admit that every ball inserted into the vase at a time before noon is also removed at a time before noon. If times are in, then TO is wrong. > > > > > I'm asking a question not having to do with balls and vases > > but that does involve times. I'm trying to get at this "noon doesn't > > happen" concept and what about the problem parameters > > makes you think the experiment can stop time. In another thread, > > even time has been removed and the discussion is simply about > > subsets of the natural numbers. > > That approach does not represent the problem. Set theory without times > cannot describe this unending process. When you include time, this > becomes clear. And it becomes (or remains) clear that TO has the wrong end of the stick. As usual! > > > > > In both cases, the idea is to get AWAY from the balls and > > vases problem and focus on specific, separate properties of > > that problem. Without balls or vases. OK? > > > > - Randy > > > > As long as it's true to the problem. I am not interested in whether you > can concoct some OTHER experiment where I'll agree the vase is empty. > That's a waste of time. The problem is with the original problem, and > changing the problem to suit your answer is non-logical. Then why has TO tried so hard and so often to do precisely that? According to the original problem, every ball inserted before noon is removed before noon. |