Re: Cantor Confusion


*** T. Winter schrieb:


> Therefore it is impossible to exchange omega letters in a diagonal.

Wrong. For each element of the list a digit is calculated in the diagonal.
As there are infinitely many (omega) elements in the list, there are
infinitely many (omega) digits in the diagonal.

omega is the supremum, not the maximum. It does not contribute a
diagonal digit.

An infinite diagonal requires not only an infinite length but also an
infinite width of the matrix. Therefore your absurd infinite number of
finite lines does not help you. Here we have the same facts as in our
old problem
0.1
0.11
0.111
....

you remember? Without an infinite number in the list there is no
infinite diagonal defined.


> The
> diaogonal cannot be roader than the list. The length of the diagonal is
> the minimum of width and length. This knowledge is prior to your
> axioms.

Width and length are equal.

Fine. But the width is finite by definition. (We do not put finite
segments together, but we have only finite seqments.)

> > > All finite sequences are countable. They yield another finite sequence.
> > > Hence they are uncountable. Contradiction.
> >
> > The statement "they yield another finite sequence" is wrong in the
> > context of the axiom of infinity.
>
> A matrix with width A and length B has a diagonal which has min(A,B)
> elements. If your axiom contradicts this, then the axiom contradicts
> mathematics and should be abolished.

That is not contradicted. Width and length are equal.

This amounts to say that there are infinite natural numbers or that the
diagonal is longer than any line.
Impossible.

The maximum of a set of finite numbers which has no maximum is simply
not present. It is *not* an infinite number which is larger than any
finite number, because a maximum must belong to the set. And a supremum
not belonging to the elements of the set does not yield a diagonal
digit.



> > > Forget the Turing machines. The diagonal of the list of all finite
> > > sequences (words) of a finite alphabet is a finite sequence because the
> > > diagonal cannot have more places than the words in the list.
> >
> > Forget that arguing with negation of the axiom of infinity. I am talking
> > in the context of that axiom. Due to that axiom there is a set of
> > natural numbers (all finite), that is itself not finite.
>
> And if you can conclude that in this context every straight line
> crosses itself 17 times, then you will also take that as a fact?

You, if that follows from some axiom, it would really be possible, unless
the added axiom leads to an inconsistencey. But I think there might be
surfaces where that is even valid.

Let us stick to Euclidean geometry. But that is unimportant. I see it
is impossible to convince you of the existence of reality.

Regards, WM

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