Re: Is continuum completely filled up?

On 1/13/2007 1:21 AM, David Marcus wrote:
Eckard Blumschein wrote:
On 1/4/2007 1:53 PM, Albrecht wrote:

I see such misconceptions related to Cantorian naivity. Refer to
Galilei's clarity, instead:
There is no amount of elements ((that one can quantify)) inside any
piece of continuum.

(>) With 1) math is unable to explain expansion, extent and measure.

Only as long as it follows Dedekind, Cantor, and other trolls.

2) is consistent to our experience that we can found as many points on
a line as we want. But than we must consider that lines consist of
lines, and nothing more. Points are properties of lines but not parts.
Infinitely many points denotes the incapability to have them all. In
this view there is no actual infinity.

Be not stupid, follow Leibniz. Accept infinity and the reals like
valuable fictions. Calculate as if they were rationals if admissible.

The set theory is based on the view 1).

No. Even worse, set theory is based on schizophrenia in re.
Cantor's definition of a set claimed to allow both options at a time.
Therefore its torso has beem mumified into ZFC axioms.

All very poetic. Unfortunately, the mathematical content of what you
wrote is zero.

Because the mutilated text was not understandable, I added in (( )) two
necessary amendments.

If you have anything mathematical to say,

First, I responded to Albrecht. I suggested to him accept infinity and
the reals, etc. This is indeed not a suggestion to you.

When I mentioned that Cantor's definition of a set claimed in a
schizophrene manner that an infinite set can be considered like every
single element of it and at the same time like an entity of all
"elements", then this may be new to you. However, just for this reason
and pertaining resulting paradoxes, this definition has been declared
untenable even by Fraenkel.

please state
your theorem and proof.

Are not these words ridiculous in that case?

Oh, and we are still waiting for your proof that
the functions you were asked about are continuous or not.

What function? Presumably your function does not have much relevance to
anything.

I am still waiting for an answer to a perhaps overly relevant question:
How did Cantor make sure that the numbers he used in DA2 were actually
real numbers and not rational ones like in case of his DA1? Isn't this
the key to he only correct notion of the reals?

I merely found that the numbers in DA2 differ from the numbers in DA1 in
that they must be actually infinite because one has to change _all_
diagonal numerals. Correspondingly, the power set requires _all_ natural
numbers. So I conclude: Real numbers must be based on actual infinity
instead of potential infinity. Isn't this reasoning logical compelling?
Moreover it is obviously plausible. The consequence for mathematics is:

Real numbers must not be defined arbitrarily. DA2 would perhaps not work
with real numbers according to definitions that do not demand actually
all of the infinitely many numerals. The only correct definition of the
same reals as assumed with DA2 was given by Meray and perhaps by Cantor
himself: Belonging reals are _fictitious_ limits.

Eckard Blumschein


.

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