Date: Jan 13, 2013 5:05 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 190

On 13 Jan., 22:59, Virgil <vir...@ligriv.com> wrote:
> In article
> <fcbf94db-8b22-4a16-b837-ccd23e176...@k6g2000yqf.googlegroups.com>,
>
>
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 13 Jan., 00:26, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <4bffb7f3-9bfa-4dae-9108-da5e24389...@f4g2000yqh.googlegroups.com>,

>
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > On 12 Jan., 22:00, Virgil <vir...@ligriv.com> wrote:
> > > > > In article
> > > > > <c0615860-6190-4c10-9185-78ed2f6a2...@x10g2000yqx.googlegroups.com>,

>
> > > > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > > Matheology 190
>
> > > > > > The Binary Tree can be constructed by aleph_0 finite paths.
>
> > > > > >         0
> > > > > >       1, 2
> > > > > >   3, 4, 5, 6
> > > > > > 7, ...

>
> > > > > Finite trees can be built having finitely many finite paths.
> > > > > A Complete Infinite Binary Tree cannot be built with only finite paths,
> > > > > as none of its paths can be finite.

>
> > > > Then the complete infinite set |N cannot be built with only finite
> > > > initial segments {1, 2, 3, ..., n} and not with ony finite numbers 1,
> > > > 2, 3, ...? Like Zuhair you are claiming infinite naturals!

>
> > > A finite initial segment of |N is not a path in the unary tree |N.
>
> > > And neither |N  as a unary tree nor any Complete Infinite Binary Tree
> > > has any finite paths.

>
> > >    "A Complete Infinite Binary Tree cannot be built with only
> > >    finite paths, as none of its paths can be finite."

>
> > > Means the same as
>
> > >    "A Complete Infinite Binary Tree cannot be built HAVING only
> > >    finite paths, as none of its paths can be finite."

>
> > > WM has this CRAZY notion that a path in a COMPLETE INFINITE BINARY TREE
> > > can refer to certain finite sets of nodes.

>
> > > And no one is claiming any infinite naturals, only infinitely many
> > > finite naturals.

>
> > So each n belongs to a finite initial segment (1,2,3,...,n).
> > Same is valid for the nodes of the Binary Tree: Each node belongs to a
> > finite initial segment of a path, the natural numbers (1,2,3,...,n)
> > denoting the levels which the nodes belong to.

>
> Since every binary path has a node at every "level" (distance from the
> root), it can only be represented by an infinite set of naturals in this
> way.


Irrelevant. Every distance is finite. There is no distance that is
larger than every finite distance. All finite distances are countable.

Note: There is no natural number larger than every natural number. And
the number of natural numbers is completely irrelevant in this
context.

Regards, WM

Regards, WM