Date: Dec 4, 2012 11:46 AM Author: ross.finlayson@gmail.com Subject: Re: Cantor's first proof in DETAILS 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:

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> > In article

> > <dac80fce-8253-4320-aa88-4ce8ebe40...@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 xy-plane 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 not-a-real-function. 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