Date: Dec 3, 2012 5:44 PM Author: Virgil Subject: Re: Cantor's first proof in DETAILS In article

<f40c3a4a-0954-451b-bb0e-58e14bf7b6bc@q5g2000pbk.googlegroups.com>,

"Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:

> On Dec 2, 10:39 pm, Virgil <vir...@ligriv.com> wrote:

> > In article

> > <768a7f47-2e23-40c2-a27a-1483f5b65...@qi10g2000pbb.googlegroups.com>,

> > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

> >

> > > Card(N) isn't a real quantity.

> >

> > Neither, in a very primitive sense, are 1, 2 and 3. The naturals are

> > precursors of both the integers and the positive rationals, both of

> > which are precursor to the rational field which is itself a precursor

> > to the reals.

> >

> > But Card(N) is real property, i.e., one which really exists.

> >

> > > So what you say there is wrong, though

> > > it's wrong twice.

> >

> > Only when I copy what you say.

> >

> > > Though if you're interested in the real point at

> > > infinity, well, you should be able to find description of number-

> > > theory's point at infinity or one- or two-point compactification of

> > > the reals and integers (in the projectively extended real numbers).

> > > Cardinals are defined by themselves, don't be putting them where they

> > > don't go, those aren't compatible types.

> >

> > Are you claiming that one cannot speak of the cardinality of the

> > naturals, or rationals, or reals? That you may not know how does not

> > mean others are all equally ignorant.

> >

> >

> >

> > > There are only and everywhere real numbers between zero and one.

> >

> > There cannot be 'only and everywhere' numbers that are constrained not

> > to be anywhere except between zero and one.

> >

> > > Here, 0 < EF(1) < 1.

> >

> > But according to Ross very own definition of his alleged EF, one can

> > show that for every n in |N and every positive epsilon, that EF(n) <

> > epsilon.

> >

> > > The arithmetic of iota-values, representing values

> > > from the continuum, of real numbers, is different for the operations

> > > as addition, and multiplication, simply as repeated addition.

> >

> > Repeated addition, adding two numbers then adding the third , then the

> > fourth, etc., works fine, though a bit tediously, for all number

> > systems contained in the reals. Or even contained in the complexes or

> > quaternions or octonions, or vector spaces, for that mater..

> >

> >

> >

> > > Dirac's delta is regularly used in real analysis, for example in the

> > > solutions of differential equations.

> >

> > When it is used, the analysis is not quite real as it is not a real

> > function.

> >

> > > Heaviside's step can be seen as continuous

> >

> > By what definition of continuous? One may chose to ignore its

> > discontinuity, but that does not make it continuous.

> >

> > , it just is horizontal from the left, vertical at the

> >

> > > origin

> >

> > But how is function which takes the value 0 at 0 and the value 1 at

> > every positive real argument and -1 at every negative real argument

> > satisfy the INTERMEDIATE VALUE theorem on any interval containing 0 as

> > an interior point? or even the mean value theorem?

> >

> > (the INTERMEDIATE VALUE theorem says that a function continuous on any

> > interval [a,b] must assume every value between f(a) and f(b) at some

> > point of that interval)

> >

> > > No, these are considerations of the plain mathematical universe shared

> > > among us, using standard definitions and working toward conciliation

> > > of intuition and rigor, thank you.

> >

> > But Ross' mathematical universe does not use, much less conform to,

> > standard definitions, or even allow them, and he opposes anything

> > resembling rigor.

> > --

>

>

> Hancher writes to our public forum on mathematics: "But Ross'

> mathematical universe does not use, much less conform to, standard

> definitions, or even allow them, and he opposes anything resembling

> rigor. "

>

> I dispute that

Of course your do! WM also disputes his critic's views.

> and it's false on the face of it.

If it were, you should be able to prove it, using standard mathematics.

That you haven't guts your claim.

> That's your bald

> lie, Hancher, and typical of them.

To call it a lie is easy, to prove it is a lie is apparently enough

beyond Ross' powers that he does not even try.

> Do you treat your other colleagues

> that way or do they have nothing to do with you?

I treat colleagues with the respect they deserve, but merely posting to

sci.math does not establish collegiality.

> I find it offensive

> for you to tell or repeat lies about me, or others.

I find it AT LEAST equally offensive when you lie about me, or others.

>

> So, "no", think you.

>

> No, "thunk".

>

> (Shrug.)

>

> And Dirac's delta is modeled by real functions

While the Dirac Delta is certainly expressible as a limit of a sequence

of continuous real functions defined on all of R, and specifically

continuous function at 0, it is not a uniform limit and thus need not

be, and is not, even a function at all on all of R.

A real function defined at 0 has a real number as its value at 0, but

the Dirac Delta does not have a real number value at 0.

> And Heaviside's step is continuous where it is so defined

>, and it's the same function with

> regards to analysis, placing for the point discontinuity a connection,

> obviously enough continuous and here satisfying the IVT.

The Heaviside function has values -1 at -1 and +1 at +1, so if it

satisfies the intermediate value theorem, there must be some x between

-1 and +1 at which the function has the value 1/pi.

But it does not!

>

> And EF goes to one.

Your EF should go to the garbage heap.

> Here you mention delta-epsilonics and I'm quite

> happy to work it up in that, re: density in R_[0,1], then,

> continuity: "Standardly modeled by standard real functions."

You claim that there is some positive real value y such that EF(n) = y

for all n in |N, and such that Sum_{n in |N} EF(n) = 1.

Clearly any such y must be greater than zero, so there must be a

smallest n in |N such that 1/2^n < EF(n), and then also infinitely many

larger n's for which 1/2^n < EF(n) is true.

So I challenge Ross to find any explicit natural number n, of which

there must be infinitely many, for which 1/2^n < EF(n) is true.

1/10^n < EF(m) for any m in |N.

Ross failure to do so will be justifiably taken as evidence of his, and

everyone's, inability to do so, and thus the falsity of hi clims.

>

> There are only and everywhere real numbers, of the continuum, of real

> numbers, between zero and one.

I can find lots of real numbers other than those between 0 an 1.

>

> And the real numbers: they're not yours to keep.

It is their rules that I keeps, and Ross does not.

>

> No, thank you,

>

> Ross Finlayson

--