Re: Possible proof of Gabriel's Theorem?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 03/10/05 

Date: Thu, 10 Mar 2005 10:20:58 -0600

On 10 Mar 2005 07:50:24 -0800, "Jason" <logamath@yahoo.com> wrote:

>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..."

I've never seen the statement "Give me any delta/epsilon and I
can always find a smaller delta/epsilon greater than zero..."
in a proof. As always you're showing that you simply do not
know what you're talking about.

Gabriel's theorem is wrong, by the way. You seem to have missed
the posts where this is pointed out.

>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.

Luckily dx = 0 has nothing to do with any of this.

>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.

You have a short memory. We saw some time ago that if we say
that 0/0 = 1 we get the wrong answer for many derivatives.

>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.

Of course you do. That's because you don't understand any of it.

(As you show over and over, when you explain that constant
functions are not differentiable or integrable, etc.)

>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

************************

David C. Ullrich