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Re: Matheology § 198
Posted:
Jan 25, 2013 10:04 AM
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On 25 Jan., 08:41, William Hughes <wpihug...@gmail.com> wrote:
> On Jan 25, 8:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > 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:
>
> > > > 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.
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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.
>
> > 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.-
And both sets of nodes are completely exhausted by the same paths with
the only difference that one kind is called X and is finite and the
other kind is called Y and is infinite. And, of course, there is a
bijection between both kinds because they contain same nodes. And,
yes, there is a last minor difference, there are far more Y's than
X's. But nobody would notice.
Regards, WM
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