
Re: Cantor's first proof in DETAILS
Posted:
Dec 4, 2012 11:46 AM


On Dec 4, 1:24 am, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Dec 3, 10:47 pm, Virgil <vir...@ligriv.com> wrote: > > > > > > > > > > > In article > > <dac80fce82534320aa884ce8ebe40...@b4g2000pby.googlegroups.com>, > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > Yes, Dirac's delta as it is: is extra the standard, encompassing it, > > > AND beyond it. (And of course it's rigorously modeled by standard > > > real functions and relevant to the pure AND applied in mathematics.) > > > If that were really the case, Ross should be able to give a real value > > of the Dirac delta for any real argument, such as its value at zero. > > > Can you do that, Ross? > > > > And Heaviside's step simply has each value between [1,1] in y on the > > > vertical line through the origin. > > > Graphs of real functions in the xyplane do not have any more than one > > point on any vertical line, so anything like Heaviside's step which > > does, cannot be a real function. > > > > And EF goes to one. So though none of the _finite_ naturals have that > > > n/d > eps, for each positive eps < 1 there is an element of the range > > > y thus that y > eps, or y /> 1, as it does: y = EF(n) > 1. So, as > > > is typical, you didn't complete the statement. > > > Ross claims that there are a sequence of constant functions EF_n(x), 1 > > for for each n in N, with domain {1...n} and constant value 1/n. > > So far so good. > > But now he also claims a constant function, EF(x), with Domain N , with > > EF(m) = EF(N) for all m and n in N and such that the sum > > Sigma_( n in N) EF(n) = 1. > > > Those what are more familiar with standard number systems that Ross > > appears to be will easily see that this cannot be true in those standard > > systems > > > > Would you have that > > > lim Sum 1/2^n < 2, n e N? 0 e N > > > Irrelevant. Ross has to produce a STANDARD real number, x, such that > > Sum_(n= 1...oo) x = 1. > > > Note that for every natural number n, 1/10^n is too large to be that x, > > and lim_(n > oo) 1/10^n = 0, so Ross' alleged x cannot be any larger > > than 0, and any number of zeroes, even infinitely many, still adds up to > > only 0. > > > At least in standard mathematics. > > > But possibly Ross is emulating WM by creating a private world of his own > > where he can command impossibilities. > > > : lim Sum_n=0^\infty 1/2^n = 2. It's > > > > not that there's no smallest n that it is, it's that there's no > > > largest n that it can't be. > > > > So, it is false for you to (m)utter that d/d =/= 1, or that your false > > > challenge applies to any. > > > Your EF function, Ross, would require a real number to exist which is > > larger than zero but smaller than every positive real number. > > > In any standard model of the reals, such numbers do not exist, so that > > before Ross can use his EF for anything, he will have to invent a wols > > new number system in which it can operate witout contradiction. > > > > It's not a failure to note that your > > > "challenge" is to confirm a falsehood, nor is it much of a challenge. > > > But it is a lie to note any such thing, since the only falsehood > > involved is Ross' claim that his EF limit function is compatible with e > > standard real numbers. > > > If EF were actual, why can't Ross give a decimal or fractional value for > > EF(1)? > > > > It's not that there's no smallest n that it is, it's that there's no > > > largest n that it can't be. > > > > There are only, and everywhere, elements of the "real" continuum, as > > > we well know them as real numbers, between zero and one, and so are > > > those from infinity back to infinity fore, for some including > > > infinity, and for none including transfinite cardinals nor your > > > imaginary numbers (which are elements of the Argand plane and in > > > extension to the hypercomplex). > > > But there is no number between zero and every positive real, which > > number is what the value of Ross' EF function would have to be if it > > were to exist. > > > > No, it is our rules, of the reals. > > > Your rules are not the rules of the real reals, only of your imaginary > > ones. > > > . And, those are discoveries, not > > > > inventions, for the real, to the concrete, in mathematics, with > > > rigor. We are simply curators, not owners, of these eternal truths. > > > Then Ross should be fired for incompetence as he sickens then not cures > > them. > > > > And, application already has and uses the projectively extended real > > > numbers, in mathematics, for, for example, physics. > > > Citations? > > > Recall that Ross claims his EF function, the limit one, has the > > properties > > (1) that for all naturals n and m, EF(m) = EF(n), and simultaneosly > > (2) SUM_(N in N) EF(n) = 1 > > > Now if, for some n, EF(n) > 0, the there is some m in N such that > > EF(n) > 1/m. but then SUM_(n = 1..m+1) EF(n) >1, > > > Thus EM(n) = 0 for all n and any sum of EM()s is still 0. > > > The mathematics of the standard set N and the standard ordered field of > > reals, R, are incompatible with existence of Ross' alleged EF function. > >  > > They say if you don't open your mouth you can't say something stupid, > but, they say that. > > Heh, got your goat, there, eh, Hancher. Huff and puff, you. Quit > with your ad hominem attacks. > > You ignoramus, I just noted Dirac's delta is notarealfunction. Or, > that's _not_ really the case. And Heaviside's step is continuous, > now. For that matter it's a real function. > > http://mathworld.wolfram.com/RealFunction.html > > You're slippin'. I think you mean Heaviside's step isn't a C^\infty > or smooth function, it's not even C^1. Though of course there was a > time when step wasn't "modernly" a function: when all functions were > C^\infty: hundreds of years ago. You're slippin', backward, and quit > dragging us with you. > > Then, no, I've never claimed that EF(1) has a standard value as > Eudoxus/Cauchy/Dedekind, an element of the complete ordered field can > be halved, again and again ad infinitum. However, it is the > properties of the function and its range that are relevant, in the > standard, and to the standard. So, the construction is modeled with > real functions, and obviously all the elements of the range are in the > reals, and they go from zero to one. Of course, I went further than > that and describe the real numbers as at once complete ordered field > and partially ordered ring with rather restricted transfer principle. > > And, m > n implies EF(m) > EF(n), because here m > n. It is true, > that. The functions there modeling it are all constant monotone > increasing, and they all go to one. > > Do you or do you not have that lim Sum 1/2^n = 2? Because as above > you would have just proved that false, were you not wrong. > > Quit trying to segregate me you bastard, that's a typical fallacious > argument which we're all quite familiar with from you. And don't > change the discussion subject to simple derogation. This is about > "Cantor's first proof in DETAILS" not your latest slander. And, it's > about Finlayson's proofs in detail, who is me. > > Yeah, I do have a new theory that includes the standard: I have "A > Theory", that includes the standard. Bring your A game. Yeah I > already discovered a new system: same as the old system, and then > some. > > I don't expect you're familiar with Riemann, Lobachevsky, Connes, etc > geometry or the meromorphological (I'm not very well either), in plain > old pure mathematics and their applications, though of course some > erudite readers are. They're in letters. (Here, Physics Letters.) > Then, for nilpotent infinitesimals, Leibniz invented methods of the > integral calculus, and we still use today the "d" for differential and > integral S for summation, of raw differentials, cited in _every_ > calculus text as the mode and universal notation. Of course, our > rigorous development of the fundamental theorems of calculus, Cauchy/ > Weierstrass, and etcetera, keep people from going overboard, but > moreso develop and maintain the general curriculum for application. > Leibniz' notation survives, and it speaks most directly to our > mathematical intuition. > > The standard naturals and reals are insufficient for all that EF > entails, while it's quite perfectly modeled in them: standardly. > And, Goedel proves there is more to the standard than the standard. > Of course, it would be fallacious to claim that because Goedel proves > there are true statements about the elements of discourse extra the > standard that these are thus true: instead, it is defined: > constructively. Goedel accommodates that. >
One fix: there is a smallest n e N s.t. 2  Sum 1/2^n < eps for each eps > 0. So, it wouldn't be proven false that lim Sum 1/2^n = 2 regardless. Here eps is from the complete ordered field or simply enough R in the context.
Yes, that's all, thank you.
Then, the universe: that's everything, isn't it?
Thanks,
Ross Finlayson

