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Topic: Matheology § 198
Replies: 40   Last Post: Jan 26, 2013 6:54 PM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Matheology � 198
Posted: Jan 25, 2013 3:26 AM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 25 Jan., 08:41, William Hughes <wpihug...@gmail.com> wrote:
> > On Jan 25, 8:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> >
> >
> >
> >

> > > On 25 Jan., 01:27, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > On Jan 24, 8:52 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > The following is copied from Mathematics StackExchange and
> > > > > MathOverflow. Small wonder that the sources have been deleted already.

> >
> > > > > How can we distinguish between that infinite Binary Tree that contains
> > > > > only all finite initial segments of the infinite paths and that
> > > > > complete infinite Binary Tree that in addition also contains all
> > > > > infinite paths?

> >
> > > > > Let k  denote the L_k th level of the Binary Tree.  The set of all
> > > > > nodes of the Binary Tree defined by the union of all finite initial
> > > > > segments (L_1, L_2, ..., L_k) of the sequence of levels U{0 ... oo}
> > > > > (L_1, L_2, ..., L_k) contains (as subsets) all finite initial segments
> > > > > of all infinite paths. Does it contain (as subsets) the infinite paths
> > > > > too?

> >
> > > > > How could both Binary Trees be distinguished by levels or by nodes?
> >
> > > > They cannot of course. Both have exactly the same levels and the same
> > > > nodes.

> >
> > > > They can of course be distinguished.
> >
> > > > In one case you do not include infinite subsets.
> > > > In the other you do.

> >
> > > My question aimed at the posiibility to distinguish the Binary Trees
> > > by a mathematical criterion, namely that one that is applied in the
> > > diagonal argument. Of course you have understood that.

> >
> > > That does not hinder you to believe in addition in matheological
> > > concepts that cannot be based on mathematical facts like nodes,
> > > levels, or digits.

> >
> > Nope.  The concept is based on nodes, and levels.

>
> How do you express actual infinity by means of nodes or levels?
> Try it.
> Write a sequence like xxxxxxxxxx... and do never stop.

It is the knowledge that it never stops that makes it NOT FINITE.
It is also the knowledge that no finite initial segment of natural
numbers (fison) surjects onto it that makes it infinite.

> Tell me when you have expressed an infinite sequence.

Tell me when you can surject a fison onto it.
> >
> > We can use the same set of nodes to make two collections of
> > sets of nodes.  One collection contains all sets of nodes, X, with
> > the property that there is a node in X with a level greater or
> > equal to that of any other node in X.
> > The other collection contains all sets of nodes, Y, with the property
> > that there is no node in Y with a level greater or equal to that of
> > any other node Y.

>
> No. You cannot express the latter by nodes and levels. If you don't
> believe me, try it.

We have already done so to our own satisfaction, by showing that no
fison surjects onto any a path ( as a sequnce of nodes) in a real
Complete Infinite Binary Tree
>
> Regards, WM

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