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