
Cantor's first proof in DETAILS
Posted:
Dec 4, 2012 4:24 AM


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.
Ross Finlayson United States

