An uncountable countable set
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From: Virgil on 28 Oct 2006 19:16 In article <4543b0b3(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. Then let us change the experiment to include the insertion into the vase of a cube at one minute after noon. The experiment now ranges over [-1,1]. What are the contents of the vase at times in [0,1), TO? > > Why, oh why, do the constraints of the problem have to matter? Why, oh > why, must we alway mean a continuum when we call something continuous, > when I want to declare a discontinuity at convenient places like 0? Why > can't we specify what happens at every moment y between x and z, but > that something totally different is the case at z, without anything > changing it? Why, oh why, why don't my labels matter? > > I'm not really sure how to answer this, again..... TO is confused! Still or again? probably still, as there doesn't seem to be much time at which he is not.
From: Virgil on 28 Oct 2006 19:21 In article <4543b321(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: > > 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: > >>>>>>> Tony Orlow <tony(a)lightlink.com> 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. > >>>>>>> What "formulaic relationship"? There are two sets. The members > >>>>>>> of each set are identified by a predicate. > >>>>>> OOoooOOoooohhhh a predicate! > >>>>> This is a non answer. > >>>>> > >>>> That's because it followed a non question. :) > >>> How is "formulaic relationship?" a non question? I do not know > >>> what you mean by that phrase, so I asked a question about. > >>> Presumably you do know what it means, but your refusal to > >>> answer suggests otherwise. > >>> > >>> > >>>>>> If an element satifies > >>>>>>> the predicate, it is in the set. If it does not, it is not in > >>>>>>> the set. > >>>>>>> > >>>>>> Ever heard of algebra or formulas? Ever seen a mapping between two > >>>>>> sets > >>>>>> of numbers? > >>>>> This is a lame insult and irrelevant comment. It says nothing > >>>>> about what a "forumulaic relationship" between sets is. > >>>>> > >>>> What is there to say? You know what a formula is. > >>> Yes, but I do not know what a "formulaic relationship" is. > >>> > >>>>>>> I could define "different" sets with different predicates. > >>>>>>> For example, > >>>>>>> A = { n | 1+n > 0 } > >>>>>>> B = { n | 2*n >= n } > >>>>>>> C = { n | sin(n*pi)=0 } > >>>>>>> Are these sets "formulaically related"? Assuming that n is > >>>>>>> restricted to non-negative integers, does A differ from B, > >>>>>>> C, IN, or OUT? > >>>>>>> > >>>>>>> Stephen > >>>>>> Do 1+n, 2*n and sin(n*pi) look like formulas to you? They do to me. > >>>>>> Maybe they're just the names of your cats? > >>>>> Sure they are formulas. But I am interested in your phrase > >>>>> "formulaic relationship", the explanation of which you seem to be > >>>>> avoiding. > >>>>> > >>>> It's the mapping between set using a quantitative formula. Observe... > >>>>>> A can be expressed 1+n>=1, or n>=0, and is the set mapped from the > >>>>>> naturals neN (starting from 1) by the formula f(n)=n-1. The inverse of > >>>>>> n-1 is n+1, indicating that over all values, this set has one more > >>>>>> element than N, namely, 0. > >>>>> I said that n was restricted to non-negative integers, so this > >>>>> set equals N. > >>>>> > >>>> Ooops, missed that. Sorry. n is restricted to nonnegative integers, but > >>>> f(n) isn't. What you mean is that, in this case, f(n) is restricted to > >>>> nonnegative integers, which means n>=2, and f(n)>=1. So, yes, the set is > >>>> size N, from 1 through N. > >>>>>> B can be simplified by subtracting n from both sides, without any > >>>>>> worry > >>>>>> of changing the inequality, so we get n>=0, neN. That's the same set, > >>>>>> again, mapped from the naturals by f(n)=n-1. > >>>>> Also N. > >>>> Yes, by the same reasoning. > >>>>>> C is simply the set of all integers, which we can consider twice the > >>>>>> size of N. There's really nothing to formulate about that. > >>>>> Once again N. > >>>>> > >>>> Sure. > >>>>> So all three sets are N. So in fact, there is only one set. > >>>>> A, B, and C are all the same set. A, B, C, IN and OUT are all > >>>>> the same set, namely N. You still have not answered what > >>>>> a "formulaic relationship" is. > >>>>> > >>>>> Stephen > >>>> Take the set of evens. It's mapped from the naturals by f(x)=2x. Right. > >>>> Many feel that there are half as many evens as naturals, and this is > >>>> reflected in the inverse of the mapping formula, g(x)=x/2. Over the > >>>> range of N, we have N/2 as many evens as naturals. Over the range of N, > >>>> we have sqrt(N) as many squares as naturals, and log2(N) as many powers > >>>> of 2 in N. That's IFR, using formulaic relationships between infinite > >>>> sets. Byt he way, it works for finite sets, too. :) > >>> 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. > > > > You've lost a capital 'N'; but anyway - IN and OUT are sets. They are > > not numbers. Yes, they are sets of numbers, but in mathematics numbers > > and sets of numbers are different things. > > > > No kidding. And sets have sizes, which are numbers. And the sizes of > sets of numbers are related to how those sets of numbers are defined, > over the number range under consideration. > > >> That is given by out(in) = in/10. > > > > Well, you've lost a capital 'I' now. Or is "in" supposed to be > > something else? > > I didn't really see the significance in yelling about it. YAY!! > > > > > Look, the two sets above are "produced" at exactly the same "rate" > > (insofar as I speak poetry). For each natural number n, we check the > > condition -1/(2^(floor(n/10))) < 0 to determine if it belongs to IN, > > and the condition -1/(2^n) < 0 to see if it belongs to OUT. For all n > > greater than zero, it turns out that both conditions are true, and > > therefore each of these positive naturals is popped into IN and
From: Virgil on 28 Oct 2006 19:25 In article <4543b56f(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: > > > > <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.... Those sets do not have natural number sizes, but do have a difference set IN\OUT = {}. > > 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 Since there are no "moments" involved in the definitions on IN and OUT, what happens at various "moments" is irrelevant.
From: Tony Orlow on 28 Oct 2006 19:40 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Let me recap the discussion: Stephen suggested the following problem >>> (which may or may not have some relationship to any other problem that >>> anyone has ever considered): >>> >>> Define the following sets of natural numbers. >>> >>> IN = { n | -1/(2^floor(n/10)) < 0 }, >>> OUT = { n | -1/(2^n) < 0 }. >>> >>> What is |IN\OUT|? >>> >>> Stephen suggested that this problem would "not cause any fuss at all", >>> i.e., everyone would agree what the answer is. In reply, you wrote, "It >>> would still be inductively provable in my system that IN=OUT*10." We all >>> took this to mean that you disagreed that |IN\OUT| = 0. Now, you seem to >>> be saying that you agree that |IN\OUT| = 0. >>> >>> Care to clear up this confusion? >> No, I don't care, but I'll do it anyway. :) Just kidding. Of course I >> care, or I wouldn't waste my time. >> >> I am beginning to realize just how much trouble the axiom of >> extensionality is causing here. That is what you're using, here, no? The >> sets are "equal" because they contain the same elements. That gives no >> measure of how the sets compare at any given point in their production. >> Sets as sets are considered static and complete. However, when talking >> about processes of adding and removing elements, the sets are not >> static, but changing with each event. When speaking about what is in the >> set at time t, use a function for that sum on t, assume t is continuous, >> and check the limit as t->0. Then you won't run into silly paradoxes and >> unicorns. > > There is a lot of stuff in there. Let's go one step at a time. I believe > that one thing you are saying is this: > > |IN\OUT| = 0, but defining IN and OUT and looking at |IN\OUT| is not the > correct translation of the balls and vase problem into Mathematics. > > Do you agree with this statement? > Yes.
From: cbrown on 28 Oct 2006 19:44
Tony Orlow wrote: > stephen(a)nomail.com wrote: <snip> > > 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 > You may or may not realize it, but this is /exactly/ how your arguments come across in a mathematical context. Spark up that bowl! Party on, Garth! Cheers - Chas |