An uncountable countable set
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From: Virgil on 30 Oct 2006 16:02 In article <454628de(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <454364ae(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Apparently you are not aware of my > >> position on the subject. Bijections alone do not prove equinumerosity > >> for infinite sets. > > > > If they do not then nothing does. > > > > > > > >> Cardinality is a rough measure of equivalence class, > >> not a precise measure of the size of a set. > > > > There is no better measure. > > > > > >> In order to precisely > >> compare such infinite sets of values, one must measure over a common > >> infinite value range formulaically. > > > > Except that TO has never proved that his "formulaic measures" form a > > proper partial order relation on sets the way cardinality does. > > > > For cardinality, one can easily show that if |A| >= |B| then it is false > > that |B| > |A|. > > > > Can TO prove a similar result for his "formulaic measures"? > > > > At any rate he has never done so. > > > > And absent such a proof, and other proofs necessary for a partial > > ordering, his "formulaic measures" are, at best, dubious. > > Since the ordering on the set sizes is done using inductively proven > inequalities between formulas describing them Induction does not ever prove that what is true for finite cases need be true for infinite cases. At least no form of induction in ZF or NBG. And TO does not have a system in which he can establish any other sort of induction. So TO is way off base again.
From: Randy Poe on 30 Oct 2006 16:03 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'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. 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
From: Virgil on 30 Oct 2006 16:22 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: > >>> <snip> > >>> > >>>>> What does that have to do with the sets IN and OUT? IN and OUT are > >>>>> the same set. You claimed I was losing the "formulaic relationship" > >>>>> between the sets. So I still do not know what you meant by that > >>>>> statement. Once again > >>>>> IN = { n | -1/(2^(floor(n/10))) < 0 } > >>>>> OUT = { n | -1/(2^n) < 0 } > >>>>> > >>>> I mean the formula relating the number In to the number OUT for any n. > >>>> That is given by out(in) = in/10. > >>> What number IN? There is one set named IN, and one set named OUT. > >>> There is no number IN. I have no idea what you think out(in) is > >>> supposed to be. OUT and IN are sets, not functions. > >>> > > > >> 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. TO has as yet to prove that his peculiar notions of set sizes are self-consistent. > > > > > In any case, it still does not make any sense. I am not sure > > what |IN| is for any n. IN is a single set. There is only > > one set, and it does not depend on n. In fact, there isn't > > an n specified in the problem. Yes I used the letter n in > > the set description, but that does not define an entity named 'n'. > > > > There most certainly is an 'n'. The problem describes a repeating > process, each repetition of which is indexed with a successive n in n, > and during each repetition of which ball n is removed. What do you mean > there's no n??? He means that while there is a variable named 'n', there is no individual named 'n'. A distinction too fine for TO's coarse thought processes to detect. > >> Like, wow, Man, at, like, each moment, there's, like, 10 going in, and, > >> like, Man, only 1 coming out. Seems kinda weird. There's, like, a rate > >> thing going on.... :D > > > > What rate? There is no rate. There are just two sets > > IN = { n | -1/2^(floor(n/10)) < 0 } > > OUT = { n | -1/2^n < 0 } > > 9 balls/iteration. There are no "iterations" in IN or in OUT. They are sets, not "iterations" > > > > > Why do you keep babbling about rates? We are talking > > about an abstract math problem. > > Which involves a process in time which happens at a certain rate at any > given point. Not so. TO is so hung up on a different problem that he sees ghosts of it even where it dies not exist. Like Scrooge seeing Marley's ghost on the door knocker. Perhaps it is a penance for TO's past sins, as it was Scrooge's. > > So you must believe that IN is a subset of OUT, as every > > integer n that satisfies > > -1/(2^(floor(n/10))) < 0 > > also satisfies > > -1/(2^n) >= 0 > > and if IN is subset of OUT then > > | IN - OUT | = 0 > > > > Stephen > > > > I know you think that logic is valid, as that's what you've been taught, > and it sounds nice and clean, but bijections without regard to their > formulaic mappings do not provide measure over infinite sets. If one has that A is a subset of B, then regardless of how TO chooses to mis-measure things, it will transpire that A\B = {}. > I'm trying to sprinkle some math in the cauldron. TO's sprinklings partake more of "eye of newt and toe of frog" than of anything mathematical.
From: Virgil on 30 Oct 2006 16:25 In article <45462ce1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> David Marcus wrote: > >>>>>>>>> Tony Orlow wrote: > >>>>>>>>>> David Marcus wrote: > >>>>>>>>>>> Tony Orlow wrote: > >>>>>>>>>>>> Mike Kelly wrote: > >>>>>>>>>>>>> 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. > >>>>>>>>>>> So, "noon exists" in this case, even though nothing happens at > >>>>>>>>>>> noon. > >>>>>>>>>> Not really, but there is a big difference between this and the > >>>>>>>>>> original > >>>>>>>>>> experiment. If noon did exist here as the time of any event > >>>>>>>>>> (insertion), > >>>>>>>>>> then you would have an UNcountably infinite set of balls. > >>>>>>>>>> Presumably, > >>>>>>>>>> given only naturals, such that nothing is inserted at noon, by > >>>>>>>>>> noon all > >>>>>>>>>> naturals have been inserted, for the countable infinity. Then > >>>>>>>>>> insertions > >>>>>>>>>> stop, and the vase has what it has. The issue with the original > >>>>>>>>>> problem > >>>>>>>>>> is that, if it empties, it has to have done it before noon, > >>>>>>>>>> because > >>>>>>>>>> nothing happens at noon. You conclude there is a change of state > >>>>>>>>>> when > >>>>>>>>>> nothing happens. I conclude there is not. > >>>>>>>>> So, noon doesn't exist in this case either? > >>>>>>>> Nothing happens at noon, and as long as there is no claim that > >>>>>>>> anything > >>>>>>>> happens at noon, then there is no problem. Before noon there was an > >>>>>>>> unboundedly large but finite number of balls. At noon, it is the > >>>>>>>> same. > >>>>>>> So, noon does exist in this case? > >>>>>> Since the existence of noon does not require any further events, it's > >>>>>> a > >>>>>> moot point. As I think about it, no, noon does not exist in this > >>>>>> problem > >>>>>> either, as the time of any event, since nothing is removed at noon. It > >>>>>> is also not required for any conclusion, except perhaps that there are > >>>>>> uncountably many balls, rather than only countably many. But, there > >>>>>> are > >>>>>> only countably many balls, so, no, noon is not part of the problem > >>>>>> here. > >>>>>> As we approach noon, the limit is 0. We don't reach noon. > >>>>> To recap, we add ball n at time -1/n. We don't remove any balls. With > >>>>> this setup, you conclude that noon does not exist. Is this correct? > >>>> I conclude that nothing occurs at noon in the vase, and there are > >>>> countably, that is, potentially but not actually, infinitely many balls > >>>> in the vase. No n in N completes N. > >>> Sorry, but I'm not sure what you are saying. Are you saying that what I > >>> wrote is correct or are you saying it is not correct? I'll repeat the > >>> question: > >>> > >>> We add ball n at time -1/n. We don't remove any balls. With > >>> this setup, you conclude that noon does not exist. Is this correct? > >>> Please answer "yes" or "no". > >>> > >> What do YOU mean by "exist"? Does anything happen which is proscribed if > >> noon DOES arrive? No, not in this case. So, noon case "exist" or not. > > > > There is no event at noon. There is no "noon case". But you > > seem to be saying that arrival of the actual time of noon, everywhere > > in the world, is somehow controlled by how we define a certain set > > of events. > > > > If you mean is there an event at noon, then say so. Don't say > > "noon doesn't happen". > > > > There's an event at -60 seconds. The next event is at -30 seconds. > > There's no event at -50 seconds. But would you really say > > "-50 is proscribed in this experiment" or "-50 doesn't exist"? > > > >> In > >> the other case, the vase also does not empty before noon, and nothing > >> happens at noon. So, then, why do you conjecture that it's empty AT noon? > > > > In the absence of any events happening at noon, we need to > > define what is meant by "number of balls in the vase at noon". > > > > We define that as "number of balls which have been inserted > > at t<=noon and not removed". > > > > Forget calling this the "number of balls in the vase at noon". That > > bothers you. Will you allow us to discuss "the set of balls which > > have been inserted but not
From: Virgil on 30 Oct 2006 16:29
In article <45462f29(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > contains(n) is the number of balls in the vase upon completion of > iteration n, and is equal to in(n)-out(n)=9n. > > n(t) is the number of iterations completed at time t, equal to floor(-1/t). > > contains(t) is the number of balls in the vase at time t, and is equal > to contains(n(t))=contains(floor(-1/t))=9*floor(-1/t). > > Lim(t->-0: 9*floor(-1/t)))=oo. The sum diverges in the limit. > > See how that all fits together? Its almost like physics, eh? > > Tony Tony's tales here are like the story of the twin baby skunks named In and Out. Which while quite funny, is irelevant to the issue of whether every ball inserted before noon is removed before noon. |