Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Cantor's first proof in DETAILS
Posted:
Dec 3, 2012 5:44 PM
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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
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