An uncountable countable set
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From: Randy Poe on 30 Oct 2006 11:34 Lester Zick wrote: > On 28 Oct 2006 12:54:51 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> > wrote: > > > > >Lester Zick wrote: > >> On Fri, 27 Oct 2006 14:23:58 -0400, David Marcus > >> <DavidMarcus(a)alumdotmit.edu> wrote: > >> > >> >Lester Zick wrote: > >> >> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > >> >> >A very simple example is that there exists a smallest positive > >> >> >non-zero integer, but there does not exist a smallest positive > >> >> >non-zero real. > >> >> > >> >> So non zero integers are not real? > >> > > >> >That's a pretty impressive leap of illogic. > >> > >> "Smallest integer" versus "no smallest real"? Seems pretty clear cut. > > > >You must be joking. I can't believe even you can be this dense. > > Oh I dunno. I can be pretty dense. Just not as dense as you, Randy, > but that's nothing new. > > >Is 1 the smallest positive non-zero integer? Yes. > > > >Is it the smallest positive non-zero real? No. 1/10 is smaller. > >Ah well, then is 1/10 the smallest positive non-zero real? No, > >1/100 is smaller. Is that the smallest? No, 1/1000 is smaller. > > > >Does that second sequence have an end? Can I eventually > >find a smallest positive non-zero real? > > > >How about the first? Is there something smaller than 1 which > >is a positive non-zero integer? > > See the problem here, Randy, is that you're explaining an issue I > didn't raise then pretending you're addressing the issue I raised. I'm providing explicit descriptions of why, as the original quote said, there is a least positive non-zero integer, but not a least positive non-zero real. > I > don't doubt there is no smallest real except in the case of integers. > But that is not what was said originally. What was said is that there > is a least integer but no least real. Since the original text is above, we can actually see whether that is what was said. "A very simple example is that there exists a smallest positive non-zero integer, but there does not exist a smallest positive non-zero real." Are you equating "least integer" with "smallest positive non-zero integer" and "least real" with "smallest positive non-zero real"? > Now these strike me as mutually > exclusive predicates. "There is a least integer" and "there is a least real" are both false. But neither of those was what was said. You have provided the original quote above. Reread it. - Randy
From: Tony Orlow on 30 Oct 2006 11:43 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. > > 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??? There is no spoon. There is no God. There is no mind. There is no n. No no no no. No. >>>>> Given that for every positive integer -1/(2^(floor(n/10))) < 0 >>>>> and -1/(2^n) < 0, both sets are in fact the same set, namely N. >>>>> >>>>> Do you agree, or not? Or is it the case that the >>>>> "formulaic between the sets is lost." >>>>> ? >>>>> >>>>> Stephen >>>> The formulaic relationship is lost in that statement. When you state the >>>> relationship given any n, then the answer is obvious. >>> What relationship? For a given n, -1/(2^(floor(n/10))) < 0 >>> if and only if -1/(2^n) < 0. The two conditions are logically >>> equivalent for positive integers. If n is a member of IN, >>> n is a member of OUT, and vice versa. >>> >>> What other relationship do you think there is between >>> -1/(2^(floor(n/10))) < 0 >>> and >>> -1/(2^n) < 0 >>> ?? > >> 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. > > 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. > > In any case, as Brian pointed out, these two sets can be "constructed" > at the same rate: > > int n=1; > while (n>=1) > { > if (-1/2^(floor(n/10)) < 0) > IN.add(n); > if (-1/2^n < 0) > OUT.add(n); > } > > One element is added to each set each time through the loop. You forgot to increment n in the loop, so this doesn't produce anything of interest. If you had included it, first of all, your second 'if' statement is totally superfluous. Just add n to OUT each time. The first if statement and resulting addition only adds the same element which is subsequently removed, not the ten as specified in the problem. Which vase are you talking about? > >>> Do you think there exists a positive integer n such that >>> -1/(2^(floor(n/10))) < 0 >>> and >>> -1/(2^n) >= 0 >>> >>> Stephen > >> Hell no! > > 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. That's why I keep "babbling" about rates and variables. I'm trying to sprinkle some math in the cauldron. Tony
From: Tony Orlow on 30 Oct 2006 11:48 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 removed?" > > - Randy > I have seen and understood your argument. It "makes sense". It seems logical. All balls are inserted and removed before noon, the same set, it would seem. But the method of proof is not correct. This is where IFR applies. There are problems in the theory, which consider interesting paradoxes and cool tricks, and others consider violations of logic. So, that's what we're discussing, I think.
From: Randy Poe on 30 Oct 2006 11:51 Tony Orlow 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. > > > > > 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??? The ORIGINAL problem. This is a new one, inspired by the original, but it is one with no balls, no vases, no time steps, no iterations. Just a definition of two subsets of the natural numbers, one called IN and one called OUT. The definition of the set IN does not include a definition of something called IN(n). You are being asked to characterize these two subsets. - Randy
From: Tony Orlow on 30 Oct 2006 11:58
stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> imaginatorium(a)despammed.com wrote: >>> Tony Orlow wrote: > > <snip> > >>>> The formulaic relationship is lost in that statement. When you state the >>>> relationship given any n, then the answer is obvious. >>> Do "state the relationship given any n"... I mean, what is it, exactly? >>> > >> Uh, here it is again. in(n)=10n. out(n)=n. contains(n)=in(n)-out(n)=9n. >> lim(n->oo: contains(n))=oo. Basta cosi? > > > What is in(n)? The sets I and everyone but you are talking about are > IN = { n | -1/2^(floor(n/10)) < 0 } > OUT = { n | -1/2^n < 0 } > Noone has ever mentioned or defined in(n) > > What is the definition of in(n)? Is is a set? > > Stephen > out(n) is the number of balls removed upon completion of iteration n, and is equal to n. in(n) is the number of balls inserted upon completion of iteration n, and is equal to 10n. 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 |