Re: Possible proof of Gabriel's Theorem?

From: Jason (logamath_at_yahoo.com)
Date: 03/10/05 

Date: 10 Mar 2005 07:50:24 -0800

William Hughes wrote:
> Jason wrote:
> > No. That's where the definition is a load of non-sense because the
> > function is integrated
> > over the entire interval. This means that delta must eventually be
> > zero.
>
> Poppy***. There is nothing in the definition that says
> that delta must eventually become zero, however, according to the
> definition constant functions are integrable.
>
> "the function is integrated over the entire interval"
> is nonsense. What is the difference between integrating
> over an interval and integrating over an "entire interval" ?
>
>
> -William Hughes

When you integrate over an interval, there are *no holes* in the
interval, therefor delta is eventually zero regardless of what
the definition says. In fact we are only interested in the result
once delta is zero, else we do not obtain the correct result.

The problem with *epsilon-delta* logic is that its not supposed
to use *infinitesimals* but every epsilon-delta argument does
exactly this: "Give me any delta/epsilon and I can always find a
smaller delta/epsilon greater than zero..." But this is what an
infinitesimal was initially defined to be: a number greater than zero
but less than every positive number. This is rubbish of course. Modern
Analysis was unfortunately born because of this problem. The problem
started when Newton and others tried to explain what they had done
with dx in:

          x^2 +2xdx + dx^2 - x^2
          ----------------------
                    dx

If we let dx run through zero in the above quotient without any
simplification, we arrive at the form 0/0. Now there is instant
alarm when someone tries to say that 0/0 = 1. However, after the
above has been simplified, this is exactly what has happened, i.e
0/0 is assumed to be 1. I am about to introduce new terminology now
so please hold on to your chair: 0/0 is the *zero unitary quotient*
(zuq from now on) to which the normal laws of arithmetic do not apply.
The law of commutation does not apply. It can only be used as *1* in
arithmetic.

Now dx does in fact become *zero* for otherwise the result of any
differentiation or integration is *incorrect*. You cannot just
decide that the derivative of x^2 is 2x + 0.0000000000000000000001
and use it this way for there will no doubt be an arithmetical
error. The real derivative can be found precisely by using 2^x. Thus,
dx *must* eventually be zero. There is no crap about it getting
infinitely close or sufficiently small or its limit is zero. It is
*zero* in the final result otherwise the derivative is *not* x^2. It
is something else.

What Weierstrass did with his arguments amounts to nothing more than
*deception*. He proved nothing at all. In fact his work leads to
theorems in real analysis which can be proved using epsilon/delta
arguments and are in reality impossible. eg. you can show using real
analysis that a square is a union of balls. However in practice this is
impossible. Now a ball can be a union of squares both in theory
and actual reality.

Sad truth is that this hooligan Weierstrass (drunk, thug - call him
whatever you want) asserted this non-sense theory on the world of math.
I have spoken to so many students who have passed real analysis courses
and still don't understand anything. Any student who is honest will
attest to this. I passed my real analysis course before most on this
forum were even born. Till this day I still have infinitely many
unanswered questions regarding its validity. You have to remember real
analysis came into existence long after calculus was discovered. It is
an attempt by mathematicians of today to cover their asses and to
create job security. I don't believe any poor student should be
subjected to such utter crap or even have to learn how to apply it when
it works because it does not work all the time.
Oh please, most college math professors are not fully competent with
this either! They become *good* at it because this is all they spend
their time thinking about and teaching. Most students never nother with
real analysis after the course ends. They shake their heads in
disbelief and decide to move on. Bull*** baffles the brains!

Jason Wells

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