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

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 William Hughes Posts: 2,267 Registered: 12/7/10
Re: Matheology § 198
Posted: Jan 25, 2013 2:41 AM

On Jan 25, 8:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 25 Jan., 01:27, William Hughes <wpihug...@gmail.com> wrote:
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> > On Jan 24, 8:52 am, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > The following is copied from Mathematics StackExchange and
> > > MathOverflow. Small wonder that the sources have been deleted already.

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> > > 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?

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> > > 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?

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

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

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> 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.
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> 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.

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.