Re: Cantor Confusion


Virgil schrieb:

In article <1159710187.186119.102420@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

*** T. Winter schrieb:

In article <1159611066.767146.101490@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
mueckenh@xxxxxxxxxxxxxxxxx writes:
> cbrown@xxxxxxxxxxxxxxxxx schrieb:
...
> > Therefore, the assertion "M is a complete list of reals" is only true
> > if the assertion "M is complete, and M is not complete" is true.
> >
> > (A and ~A) = false.
>
> A system has the property W, if it can be proved that the reals can be
> well-ordered. A system has the property ~W if it can be proved that the
> reals cannot be well-ordered. A system is self-contradictive, if W and
> ~W can be proved. Therefore the system does not exist.

The situation is slightly different. Neither W nor ~W can be proven, at
least, so mathematicians think.

Zermelo was not a mathematician? He proved by what today is known as
ZFC:

Zermelo, E., "Beweis, daß jede Menge wohlgeordnet werden kann", Math.
Ann. 59 (1904) 514 - 516
Zermelo, E., "Neuer Beweis für die Möglichkeit einer Wohlordnung",
Math. Ann. 65 (1908) 107 - 128

So either W or ~W can be taken as a new
axiom, leading to different branches of set theory. The case is similar
to the parallel postulate which can not be proven from the other
postulates,
so either that postulate or its negation can be taken as an axiom, leading
to different branches of geometry.

By forcing it can be proved that, even including AC, the reals cannot
be well ordered.

That is not in accord with the following:
http://en.wikipedia.org/wiki/ZFC#The_axioms
Axiom of choice: For any set X there is a binary relation R which
well-orders X. This means that R is a linear order on X and every
nonempty subset of X has an element which is minimal under R.

The clue is: There is a well-order (proven by Zermelo) but we can never
know how it looks like (proven by forcing).
What is a thing worth, which cannot exist other than in our mind but
which provably does not exist in our mind?

Regards, WM

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