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Re: Matheology § 038
Posted:
Jun 14, 2012 5:28 AM
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On 14 Jun., 09:57, Jürgen R. <jurg...@arcor.de> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> schrieb im Newsbeitragnews:d84ffa69-66ad-42b2-8904-a268c86d8a24@a16g2000vby.googlegroups.com...
>
> > Matheology § 038
>
> > One of the most often heard arguments in favour of transfinite set
> > theory is the completeness requirement of |R and real functions. It is
> > not true.
>
> > Well, then tell me, Herr Professor Doktor Mueckenheim, how do you
> > solve the equation
> > ih partial(du/dt) = H(u) [JR, Matheology § 022, sci.logic June 13,
> > 2012]
>
> Evidently you misunderstood the question. The question is:
Are there uncountably many real numbers.
The answer is a resounding: No.
>
> Is there in your version of mathematics such a thing as
> "Calculus", i.e. "Mathematical Analysis"?
Of course. Euler, Gauss, Cauchy, Weierstarss: They all knew how to do
analysis.
>
> In particular:
>
> If there are at most countably many real numbers then
> every (Lebesgue) integral vanishes. Is there such a
> thing as an integral in your version of mathematics?
No. It is a gross mistake to assume that the rational numbers could be
counted. This is concluded from the correct observation that counting
up to every rational number is possible. The conclusion is wrong
because after every rational and every natural number, there are
infinitely many others possible. No infinite set can be counted. Not
even the natural numbers. The asserted bijection n - n is invalid,
except for a small fraction of 0 % of possible natural numbers (not
possible in MatheRealism, but possible in classical mathematics).
> If there are no more than countably many real numbers
> it is unclear how the derivative of a function
> can be well-defined.
That depends on the defintion of "well- defined". But obviously there
are not uncountably many real numbers and obviously no plane and no
bridge have crashed for that reason.
> Or are your considerations limited to operations with
> numbers, i.e. to arithmetic?
Mathematics uses numbers and limits, but only those which exist.
We cannot sum all terms of any geometrical series. We can only
determine supremum and infimum.
Cantor postulated an improper limit to be a proper limit - and failed.
Regards, WM
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