An uncountable countable set
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From: Tony Orlow on 24 Oct 2006 12:31 Virgil wrote: > In article <453d52d1(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: >>>>>>>> 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 should have said "each"... >>> Each ball is removed before noon. Every ball is removed >>> before noon. All the balls are removed before noon. Given >>> any ball, that 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. >>>> Do you disagree with the statement that, at every time -1/n, when ball n >>>> is removed, for every n e N, there remain balls n+1 through 10n, or 9n >>>> balls, in the vase? >>> At every time before noon, there are not only finitely many balls >>> in the vase, there are still infinitely many balls yet to be put in >>> the vase. Of course, every one of those balls will be removed >>> before noon. Without exception. >>> >>>>>> 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. >>>> There is no ball, in all of N, for which the vase is empty at its >>>> departure from the vase. >>> Yes. There is no last ball. >>> >>> But there is no ball which fails to be removed. >>> >>>>> 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. >>>> I was saying it is non-empty at every one of the finite times before >>>> noon where any ball is inserted or removed. Do you argue against THAT >>>> statement? >>> I agree with that statement. For t<0, the vase is non-empty. >>> >>> At t=0, the vase is empty. >>> >>>>> 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? >>>> No, you misread. >>> You didn't ask me "does it make sense that the vase is >>> non-empty at noon"? Ah well, I'll answer that question anyway. >>> No, it doesn't make sense to me that the vase would be >>> non-empty at noon. Of course it's empty. >>> >>> - Randy >>> >> Even though it didn't become empty at noon, nor before... > > > For it to "become" anything implies a gradual and continuous change > spread over some time interval of positive length, which is not the case > here. Not if removals occur infinitely quickly. It can happen in a moment. But, there has to be at least one moment involved in the event, or it didn't happen. That moment can't be before noon. It can't be noon. Is it after noon, and then time-travels back to happen after all the moments before noon, but just in time to beat noon? You have pretzel time, Virgil. > > The vase is non-empty at every time before noon and empty at noon in > much the same discontinuous way that Sign(x) is non-zero at ever real x > except x = 0. Sign(x)? Whatever. y=9x diverges as x->oo. Sorry.
From: Lester Zick on 24 Oct 2006 12:34 On Mon, 23 Oct 2006 15:00:58 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: [. . .] >>> Unfortunately, transfinitology exists, despite the fact that it makes no >>> sense underneath the hood. When it comes to arithmetic on them, it's one >>> big kludge. But, there are forms of infinite numbers upon which one can >>> define arithmetic. They just have nothing whatsoever to do with omega or >>> the alephs. >> >> Not sure what you're talking about here, Tony. Lots of things exist in >> the sense of having been defined. That doesn't make them true and >> doesn't mean they form any basis for the truth of other things defined >> on them. There's no shortage of things other than infinity on which to >> define arithmetic. >> >What is log2(0)? -00? Not sure what this is in aid of, Tony. What is log0(0)? For that matter what is log0(00) or log-00(00) or x!=0? There are all kinds of restrictions around 0 and 00 precisely because 0 is not a natural number. >>>> Yet I've also been considering what it looks like you're trying to do >>>> with trans finite arithmetic.In particular it occurs to me that if one >>>> takes +00 to be larger than any positive finite -00 correspondingly >>>> must be smaller than any negative finite such that your concept of >>>> circularity among arithmetic numbers might be combined in the >>>> following way: [-00, . . . 3, 2, 1, 0, 1, 2, 3 . . . +00]. The only >>>> difference would be that whereas +00 represents the number of >>>> infinitesimals, -00 would represent the size of infinitesimals. Thus >>>> we'd have a positive axis with the number of infinitesimals and a >>>> negative axis with the size of infinitesimals. At least that's the >>>> best I can make of the situation. >>> Well, I rather think of 1/oo as the size of infinitesimals, or more >>> precisely, for any specific infinite n, 1/n is a specific infinitesimal >>> value. When it comes to the number circle, in some ways oo and -oo can >>> be considered the same so the number line forms an infinite circle, but >>> in others, such as lim(n->oo) as opposed to lim(n->-oo), there is a very >>> clear difference between the two. I think it's a bit like the >>> wave-particle dualism for physical objects, and may actually be directly >>> connected. >> >> Well now you're just back to the idea of arithmetic as some kind of a >> TOE, Tony. It's very simple. The only mechanical definition for 00 is >> any finite/0. And if that product can't be defined than neither can >> 00. There is no specific size to infinitesimals because they're an >> process not a static thing. Any series of infinitesimals varies in >> size continuously. There's a reciprocity between number and size for >> any infinite series but no "circle" between them. > >Technically, the number of reals in the unit interval (0,1] is Big'un. But the point is that they're within the interval. There is no infinite set 1, 2, 3 . . . 00 outside of some interval. >That's also the infinite length of the real number line, in unit >intervals. The unit infinitesimal is Lil'un, or 1/Big'un. Now they're >all specific and related to spatial measure and quantity. :) > >> >> On the other hand if you want to do transfinite arithmetic you might >> ask yourself what the results of 00-00 or 00/00 are. The latter can be >> addressed through application of L'Hospital's rule but I don't know >> any way to address the former. >> >> ~v~~ >The formulas that lend themselves to L'Hospital's Rule usually cannot be >simplified any further to resolve that problem. Subtracting one simple >formula from another is just a matter of combining like terms and >finding the most significant to see if you get a finite result through >mutual cancellations. But L'Hospital's rule applies to ratios, Tony. It only gives the finite ratio between infinities. If you subtract 1/0 from 2/0 what do you get? They both already have common denominators so the answer would seem to be 1/0 which still remains infinite. Kluge is the right word for transfinite arithmetic. ~v~~
From: Lester Zick on 24 Oct 2006 12:37 On Mon, 23 Oct 2006 12:55:19 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <453cb03a(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> 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. > > >And where do these allegedly infinitely numbered balls come from? >I do not recall any of them being mentioned in the original gedanken, so >that TO is creating his own separate gedanken. > >Note that whatever TO may require in his version of the gedanken, his >requirements do not alter that no such thing occurs in the original. > >TO takes the childish position that if he cannot have things his own >way playing by the rules, he will change the rules to get his own way. Of course no respectable modern mathematiker ever did the same. ~v~~
From: Lester Zick on 24 Oct 2006 12:43 On 23 Oct 2006 11:47:05 -0700, imaginatorium(a)despammed.com wrote: > >MoeBlee wrote: >> Lester Zick wrote: >> > You and he just have different perspectives on the problem and nothing >> > in what you have to say has any relevance to what Tony believes any >> > more than what Tony believes has any relevance to what you believe. >> >> Differing perspectives are welcome. But that is different from simply >> saying incorrect things about the technical points and also from giving >> the kind of woozy arguments he gives for his non-axiomatic mathematics. > >Uh, do you have a technical definition for "woozy" at this point? >Sounds rather fascinating... The zen of wooziness is "The set of woozy(x) is when x=Lester". Perfectly recursive bijective well ordered robust set. ~v~~
From: Tony Orlow on 24 Oct 2006 12:39
Virgil wrote: > In article <453d5565(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. > > Then is TO claiming that 1/n = 0 for some number, n, reachable from 1 > by repeated successorship, as these are the only n's allowed. Only if anything occurs at noon in the vase. >>>>> 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? > > The process of inserting and removing balls has noon as lub. So the > process is over and completed at noon but not before. Noon is an artificial LUB imposed on the unbounded set N using the Zeno machine. Noon is not included in the experiment by the very fact that the ball numbers are limited to the standard naturals. Thus t=f(n)<0 for all n e N. >>> 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? > > TO may mean that , but no sensible person will try to claim that for any > n in N, ball n is in the vase after noon - (1/n minuts) > I'm not either. I am saying that noon never happens, and if it does, then infinitely-numbered balls exist in the vase. > > 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, > > The only "upper bound" is on the times at which balls are to be moved, > and that has least upper bound of noon. > No, you are using that to pretend the set is completed, that you have counted the last countable finite natural. That's bull. All your arguments end up being some "largest finite" argument of one sort or another. Omega is a phantom. Learn Non-Standard Analysis. > >> The fact remains that it doesn't become empty before noon, and >> nothing happens at noon, so it doesn't empty. > > It is emptying, in a sense, at each t = -1/n, in such a way that it is > empty at noon. It's emptying with a net gain of 9 balls, accelerating exponentially? Curious. > > Let those who disagree name the natural number on any ball remaining in > the vase at noon. I don't claim any natural number remains at noon. No one does. So, go knock down your straw man while you try to say when the vase empties. |