An uncountable countable set
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From: Virgil on 3 Nov 2006 15:34 In article <454b7dd6(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: ... > Bigulosity is not based on the subset relation, but on formulaic > mappings and infinite-case induction. Since TO's versions of both "formulaic mappings" and "infinite-case induction" are outside of both mathematics and logic, they are irrelevant here. > The proper-subset equality is only > mentioned as being one of the most egregious violations of > transfinitology. It's fundamentally wrong, intuitively. An intuition as the sole foundation of mathematics is fundamentally wrong mathematically. Since TO rejects logic, his intuitionism is merely foolishness. > > 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? As polygons and polyhedra are sufficiently dealt with as a part of standard mathematics, why doesn't TO learn enough standard mathematics to compare them himself. TO can hardly claim that "bigulosity" will do a better job of something that standard mathematics already does until he can say with some assurance how each of them does that job. That wold require that TO learn a few things about standard mathemetics on his own, not merely be spoon fed a few particulars when his ignorance shows itself.
From: Virgil on 3 Nov 2006 16:05 In article <454b8064(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. It does not imply any such thing. TO maintains that the given scheduling prevents the vase from emptying, but allows that putting the balls in earlier, but taking them out as originally scheduled, will leave the vase empty. What that implies is that TO has lost his marbles. At least as far as this gedankenexperiment is concerned. > > >>> 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. It is an even weirder parlor trick to claim that if one starts with all the balls in the urn a priori and takes them out according to schedule one will end up with fewer than if one starts with an empty vase and follow the gedankenexperiment. But that is what TO claims. But no sensible person will agree that the times of insertion make any difference at all to the status at noon, as long as they are compatible with the original times of removal. > > > > 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. Indeed. If TO can explain how changing the times to insertion, as long as they remain before the original times of removal, can affect the contents of the vase after al removals are completed, we wold be much obliged. > > >>> 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. According to the gedankenexperiment, each ball is removed before noon. Which ball(s) does TO claim have not been removed before 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. It is TO's overheated imagination that needs soothing. TO imagines that the contents of the vase at noon is somehow related to the pattern of insertions into the vase before noon. But since every insertion of any ball is followed by the removal of that ball ,still before noon, the time of its insertion is irrelevant. And, for all the difference it will make at noon, we may as well start with all of the balls in the vase. Let us look at two models (1) All the balls start in the vase and ball n is removed at 1/(2^n) minutes before noon. (2) the vase starts empty, at 1.1/(2^n) minutes before noon ball n is inserted and at 1/(2^n) minutes before noon ball n is removed. To has conceded that in the first case the vase is empty at noon. In the second case, there is never more than 1 ball in the vase at any time so at noon there is no more that 1 ball in the vase,but as each ball inserted was removed, it is not hard to conclude that the vase is empty at noon in this case also. Now look at TO's argument. TO claims that for some case in which the insertions are arranged to be between those case 1 and those in case 2, the vase is somehow not empty. Exactly where do those insertion patterns causing non-emptiness start and where do they end in between those of case 1 and case 2?
From: Mike Kelly on 3 Nov 2006 16:44 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>>> Mike Kelly wrote: > >>> <snip> > >>>>> Now correct me if I'm wrong, but I think you agreed that every > >>>>> "specific" ball has been removed before noon. And indeed the problem > >>>>> statement doesn't mention any "non-specific" balls, so it seems that > >>>>> the vase must be empty. However, you believe that in order to "reach > >>>>> noon" one must have iterations where "non specific" balls without > >>>>> natural numbers are inserted into the vase and thus, if the problem > >>>>> makes sense and "noon" is meaningful, the vase is non-empty at noon. Is > >>>>> this a fair summary of your position? > >>>>> > >>>>> If so, I'd like to make clear that I have no idea in the world why you > >>>>> hold such a notion. It seems utterly illogical to me and it baffles me > >>>>> why you hold to it so doggedly. So, I'd like to try and understand why > >>>>> you think that it is the case. If you can explain it cogently, maybe > >>>>> I'll be convinced that you make sense. And maybe if you can't explain, > >>>>> you'll admit that you might be wrong? > >>>>> > >>>>> Let's start simply so there is less room for mutual incomprehension. > >>>>> Let's imagine a new experiment. In this experiment, we have the same > >>>>> infinite vase and the same infinite set of balls with natural numbers > >>>>> on them. Let's call the time one minute to noon -1 and noon 0. Note > >>>>> that time is a real-valued variable that can have any real value. At > >>>>> time -1/n we insert ball n into the vase. > >>>>> > >>>>> My question : what do you think is in the vase at noon? > >>>>> > >>>> A countable infinity of balls. > >>> 1) It's not clear to me what you mean by that phrase but I'll assume > >>> the standard definition. Still, the question remains of which balls you > >>> think are in the vase? Does every natural number, n, have a ball in the > >>> vase labelled with that n? > >> Conceptually, sure. > > > > Yes or no? What is the set of balls in the vase at noon? Which balls > > are in the vase and which are not? > > > >>> 2) How come noon "exists" in this experiment but it didn't exist in the > >>> original experiment? Or did you give up on claiming noon doesn't > >>> "exist"? What does that mean, anyway? > >> 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. > > > > ? > > > >> They both end up with countably many balls in the vase at noon. > > > > For now, I am going to try to restrict myself to discussing this new > > experiment, because I want to understand what "noon doesn't exist" is > > supposed to mean. And, again, your answer is ambiguous. I asked which > > balls are in the vase at noon, not the cardinality of the set of balls > > in the vase at noon. I then asked whether "noon exists", not whether > > anything "happens" at noon. Please try answering the questions people > > actually ask; it aids in communication. Why have you ignored everything I said above here? Are you simply unwilling to critically examine the assumptions and inferences you make? Maybe you just missed that part of my post, I'll ask again. What is the set of balls in the vase at noon in the simplified experiment? What does "noon does not exist" mean? > >> The experiment's stated sequence logically precludes that the vase become empty. > > > > It logically precludes that balls without a finite natural number on > > them get added to the vase, but that doesn't seem to bother you. Ho > > hum. > > > > <snip more stuff about original experiment> > > > > The iterations of insertion and removal are specified as such, and their > times specified, such that the number of balls is a function of time, > discontinuous for finite n or t, but constant for n, and constantly > exponential for t. For t < 0, sure. > It is true that: > 1) the vase contains balls, is thus non-empty, at every time before noon. > 2) No removals occur at noon. > 3) The vase can only become empty, after having contained balls, though > removal of balls. As other people have noted, the same "logic" applies if you insert all the balls 1 minute before noon and then remove them as in the original experiment. Yet you say that in that situation the vase is empty at noon. Could you explain this apparent inconsistency in your position? -- mike.
From: Lester Zick on 3 Nov 2006 16:50 On 2 Nov 2006 15:26:22 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> On 2 Nov 2006 11:21:10 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >> >Lester Zick wrote: >> >> I'm perfectly content to stay right here while I refine my definitions >> >> and terminology to bring them into better conformance with standard >> >> mathematical usage and more appropriate neomathematical forensic >> >> modalities. >> >> Evasion noted. > >You just quoted yourself, then commented 'Evasion noted'. Nicely done. Oh well considering the context you dropped somehow I'm not surprized. ~v~~
From: Lester Zick on 3 Nov 2006 16:51
On Thu, 2 Nov 2006 19:21:08 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >MoeBlee wrote: >> Lester Zick wrote: >> > On 2 Nov 2006 11:21:10 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> > >Lester Zick wrote: >> > >> I'm perfectly content to stay right here while I refine my definitions >> > >> and terminology to bring them into better conformance with standard >> > >> mathematical usage and more appropriate neomathematical forensic >> > >> modalities. >> > >> > Evasion noted. >> >> You just quoted yourself, then commented 'Evasion noted'. Nicely done. > >Even Lester has to be right occasionally. You know, David, it occurs to me that even if I'm only right once mathematically you're a dead duck. Moe(x) was DOA. ~v~~ |