Date: Apr 2, 2013 4:45 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 224

On 2 Apr., 00:14, William Hughes <wpihug...@gmail.com> wrote:
> On Apr 1, 10:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
>
>

> > On 1 Apr., 15:19, William Hughes <wpihug...@gmail.com> wrote:
>
> > > On Mar 30, 3:36 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > On 30 Mrz., 10:17, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > On 24 Mrz., 18:09, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > <snip>

>
> > > > > > > The only difference is that in the second case you consider
> > > > > > > some subsets of the nodes to be paths, that are not considered
> > > > > > > to be paths in the first case.

>
> > > > > > Well, that is a correct description. It implies that these additional
> > > > > > subsets cannot be distinguished by nodes from the finite subsets

>
> > > > > Piffle.  It is trivial to distinguish a subset that has a node
> > > > > at a last level from a subset that does not have a node
> > > > > at a last level.

>
> > > > No, that is impossible if an infinite path consists of infinitely many
> > > > finite subsets.

>
> > > Let the subset of nodes in the infinitely many finite subsets
> > > be Q.

>
> > > Q is contained in both trees, is not a path
> > > in the Binary Tree that contains only all
> > > finite paths (Q does not have a node at
> > > a last level)

>
> > The path to which all finite paths
> > 0.1
> > 0.11
> > 0.111
> > ...
> > contribute is a path  too in the Binary Tree that contains all finite
> > paths.

>
> Nope.  This set of nodes has no node at a last level.
> Every path in the Binary tree that contains all finite paths
> has a node at a last level.


Every node is such a node at a last level. This is the nature of the
nodes. Nevertheles there is no last node.
>
> The difference between the trees is not which
> subsets of nodes exist, but which subsets are
> considered to be paths.


The tree of all finite paths and the tree of all paths like every tree
has infinite paths. Therefore there is no tree which has only finite
subsets that are considered paths.

>  Only in one of the trees
> can a subset of nodes with no node at a last
> level be considered a path.


Is this tree

0.
0 1
0 1 0 1
...

that one with infinite subsets not considered paths?

Regards, WM