An uncountable countable set
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From: Tony Orlow on 28 Oct 2006 15:44 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 popped > into OUT. Simultaneously (poetically speaking). > > No, at every moment, elements are added, and pretending you have reached the Twilight Zone, where finitude meets infinity, and you've finished counting all the finite naturals, is just silly and unproductive. It clearly clouds all measure in this case, which serves as a counterexample to this theory. It's as bad as Banach-Tarsk
From: Lester Zick on 28 Oct 2006 15:44 On 27 Oct 2006 16:09:59 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> Ah, Moe, truth is often a nuisance. > >Then you're as much of a nuisance as is a cool breeze on a sunny spring >day, as a cleansing and quenching rain that ends a drought, as a >magnificent symphony orchestra heard in an amphitheatre of impeccable >acoustics. Just wish you could get your story straight for a change, Moe. ~v~~
From: Tony Orlow on 28 Oct 2006 15:46 Lester Zick wrote: > On Fri, 27 Oct 2006 21:04:07 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> MoeBlee wrote: >>> Lester Zick wrote: >>>> Ah, Moe, truth is often a nuisance. >>> Then you're as much of a nuisance as is a cool breeze on a sunny spring >>> day, as a cleansing and quenching rain that ends a drought, as a >>> magnificent symphony orchestra heard in an amphitheatre of impeccable >>> acoustics. >>> >>> Moe Blee >>> >> As a kitty cat on your lap, on a chilly night... :) >> >> Lesterrrrrrrr...... >> >> >> Actually, I named my last cat Lester, before I met Lester here online. I >> had to scrape his pieces off the road.... > > LOL, Tony! Did you try to put the set of pieces back together? > > ~v~~ Uh, no Lester, I buried him out back, in a pillow case, next to Nelie and Freddy. I was going to make a whole talk show around him, too..... 01oo
From: Virgil on 28 Oct 2006 15:47 In article <4543561c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > > > 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. And which members of a completed N does TO claim are not in the vase by noon? According to ZF, or NBG, if the first numbered ball is in the vase by noon and the successor numbered ball is for each numbered ball, as will be the case under David's scheme, is in the vase by noon, then the completed set of N balls will are already in the vase by noon.
From: Tony Orlow on 28 Oct 2006 15:48
Lester Zick wrote: > On Fri, 27 Oct 2006 21:23:44 -0600, Virgil <virgil(a)comcast.net> wrote: > >> In article <4542aa5c(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: >> >> >>> Stop confusing me with facts, Lester. >> TO is easily confused by facts. >> >> But Lester rarely provides any. >> >> A marriage made in Heaven? Or Hell? > > You're too fickle for me, Virgil. Aatu calls you "pathetic" for your > dogged faithfulness to me so you just jump over to Tony instead. > > ~v~~ It's okay. Aatu says he's pathetic for wasting his time with me too. Kerberos gots some snarlin' to do. Woof! 01oo |