Re: Two results of set geometry

In article <1190632327.357863.49790@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
WM <mueckenh@xxxxxxxxxxxxxxxxx> wrote:

On 20 Sep., 04:39, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1190035747.850345.62...@xxxxxxxxxxxxxxxxxxxxxxxxxxx> WM
<mueck...@xxxxxxxxxxxxxxxxx> writes:
> On 17 Sep., 02:57, "*** T. Winter" <***.Win...@xxxxxx> wrote:
> > In article <1189945589.699206.305...@xxxxxxxxxxxxxxxxxxxxxxxxxxx> WM
> > <mueck...@xxxxxxxxxxxxxxxxx> writes:
> > ...
> > > My claim is the existence of the bijection.
> >
> > By showing an injection from nodes to paths. But that one is trivial.
> > For
> > a bijection you also need an injection from paths to nodes, or prove
> > that
> > your injection is also a surjection. Your injection is not a
> > surjection,
> > because there is no node that maps on the path 0.10101010..., so you
> > have
> > either to show an injection that is also a bijection or an injection
> > from
> > paths to nodes.
>
> Every "index" of 0.101010... and every initial segment of 0.101010...
> is in the bijection.

*finite initial segment*.

> So what is missing?

The complete path. Which node maps to the complete path?

> But even if you claim that
> 0.101010... does exist "in some other form",

I do not claim anything like that at all. I only claim that your
mapping does not map a node to that path, so that your mapping is not
a surjection.

My mapping maps includes every node of every paths. Therefore it
includes every initial segments of every path - except such which do
not consist of nodes.

So that it leaves out every real which does not have a finite binary
representation, which is most of them.





Do you think that 0.101010... consists of more
than of the complete set of its initial segments?

Yes! Unless 1/3 = m/2^n, for some natural numbers m and n, 0.101010...
does consist of more than of the complete set of its initial segments.

If so, then it is perhaps some mysterious limit of the sequence of its
initial segments? Ok, but the set of limits of paths cannot be larger
than the set of really existing paths --- and that set is countable.

WM's "really existing" paths must then all be finite, in which case his
numbers must ALL be of form z/2^n for integer z and natural n.


Or does every sequence of initial segments have uncountably many
limits?

The "sequence" of all initial segments has uncountably many accumulation
points.

> then it is the supremum
> of the path leading to it (compare the triangular matrix and the
> missing row 111... which exist only as supremum, not taken). The
> number of paths, however, is countable. This implies the countability
> of he number of suprema.

This is nonsense. I ask you to show that it is a surjection but you do
not do so.

There is a bijection between all really existing paths and N.

Then "really existing" means finite, and WM is presuming his conclusions
again. As usual.
.

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