Re: Cantor Confusion


Dik T. Winter schrieb:

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

And it does not contribute a line. But the number of lines is omega.
And the number of digits is omega.

I think we have obtained agreement, that we disagree in this point and
that further discussion will not lead to a result acceptable for both
parties. The final state is:

1) The number of columns is equal to the number of lines.
2) Both are infinite, according to set theory this means omega or
aleph_0.

But you believe: omega or aleph_0 is the maximum of the set of lines
and the supremum of the set of columns, which, however, is not taken by
any column. You believe the different meaning is not important.

I believe that a taken maximum and a not taken supremum are so
different (although the difference is very tiny) that set theory,
necessarily assuming this, is wrong.

The difference between maximum und supremum is, by the way, the reason
why I insist that an actually infinite set of natural numbers must
contain a non-natural number.

> 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.

I state (you know that) that the diagonal: 0.111...

Yes, again mixing up maximum and supremum.

> > > 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.)

Nope. The width is unlimited, as is the length. And so is not finite.

But the length is realized as a whole number by a set of elements
(lines) while the width is not realized by a set of columns.

> > > 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.

Wrong.

If you map the diagonal on a line by d_kk -> d_k, then the line (d_k)
is longer than any line (a_mk) for any m e N. Therefore the diagonal
has more elements than any line (if omega > m for all m e N).


> The maximum of a set of finite numbers which has no maximum is simply
> not present.

Right.

> It is *not* an infinite number which is larger than any
> finite number, because a maximum must belong to the set.

Right.

> And a supremum
> not belonging to the elements of the set does not yield a diagonal
> digit.

And again right. And still your conclusion is wrong. Obviously the diagonal
has no finite length, because if it had it would have the same length as
one of the numbers, and then the next number would be longer, which is
a contradiction. So the diagonal is infinite in length.

And nevertheless its length is assumed to be described by a
(non-natural) number. (One cannot compare numbers with non-numbers.)

> > > 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.

What is the "existence of reality" in this context?

That what exists, that was really is, contrary to a straight line which
crosses itself.

Regards, WM

.

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