An uncountable countable set
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From: stephen on 28 Oct 2006 16:28 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. 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'. >>>> 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 } Why do you keep babbling about rates? We are talking about an abstract math problem. 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. >> >> 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
From: Virgil on 28 Oct 2006 16:30 In article <45437f07(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> stephen(a)nomail.com wrote: > >>>>>>>>> What are you talking about? I defined two sets. There are no > >>>>>>>>> balls or vases. There are simply the two sets > >>>>>>>>> > >>>>>>>>> IN = { n | -1/(2^floor(n/10)) < 0 } > >>>>>>>>> OUT = { n | -1/(2^n) < 0 } > >>>>>>>> For each n e N, IN(n)=10*OUT(n). > >>>>>>> Stephen defined sets IN and OUT. He didn't define sets "IN(n)" and > >>>>>>> "OUT > >>>>>>> (n)". So, you seem to be answering a question he didn't ask. Given > >>>>>>> Stephen's definitions of IN and OUT, is IN = OUT? > >>>>>> Yes, all elements are the same n, which are finite n. There is a > >>>>>> simple > >>>>>> bijection. But, as in all infinite bijections, the formulaic > >>>>>> relationship between the sets is lost. > >>>>> Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > >>>>> (The vertical lines denote "cardinality".) > >>>> Um, before I answer that question, I think you need to define what you > >>>> mean by "|IN - OUT|" =0. How are you measuring IN and OUT, and how do > >>>> you define '-' on these "numbers"? > >>> IN and OUT are sets, not "numbers". For any two sets A and B, the > >>> difference, denoted by A - B, is defined to be the set of elements in A > >>> that are not in B. Formally, > >>> > >>> A - B := {x| x in A and x not in B} > >>> > >>> Note that the difference of two sets is again a set. For any set, the > >>> notation |A| means the cardinality of A. So, saying that |A| = 0 is > >>> equivalent to saying that A is the empty set. In particular, for any set > >>> A, we have |A - A| = 0. > >>> > > > >> Sure, in the sense of containing the same n's, they are the same set. Which it all that is in question. So TO concedes the issue.
From: Randy Poe on 28 Oct 2006 16:33 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
From: stephen on 28 Oct 2006 16:33 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
From: Virgil on 28 Oct 2006 16:37
In article <45438acc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <454286e8(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> Like, perhaps, the Finlayson Numbers? :) > >>> Any set of numbers whose properties are known. Are the properties of > >>> "Finlayson Numbers" known to anyone except Ross himself? > >> 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? > > > > Do they form a field? > > > > Brian Chandler > > http://imaginatorium.org > > > > Good question. Ross? What says you to this? > > Here's what Wolfram says applies to fields: > http://mathworld.wolfram.com/FieldAxioms.html > > My understanding, looking at each of these axioms, is that they apply to > this system, and that it's a field. I suppose you would want proof of > each such fact, but perhaps you could move the process along by > suggesting which of the ten axioms you think the Finlayson Numbers might > violate? After all, if you find only one, then you've proved your point. > Not that I am necessarily concerned with whether they form a ring or a > field or whatever, until that becomes important. Is it? Why the question? > > Tony TO, as usual, begs the question. It is not up to anyone else to prove that "Finlayson numbers" do not form a field, it is for those who claim that it is a field to prove that it is. And to be a valid substitute for the reals, they would have to form a complete Archimedean field, which I rather doubt they will. |