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Virgil
Posts:
8,833
Registered:
1/6/11
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Re: Wm mis-explains what he means by a Binary Tree
Posted:
Feb 5, 2014 3:46 PM
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In article <ae5ead69-4310-4121-b9c0-3f509064dc8e@googlegroups.com>,
WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:
> Am Mittwoch, 5. Februar 2014 20:20:51 UTC+1 schrieb Ben Bacarisse:
> > WM <wolfgang.mueckenheim@hs-augsburg.de> writes:
> > > Am Mittwoch, 5. Februar 2014 17:41:13 UTC+1 schrieb Ben Bacarisse:
> > >>
> > >> If they gave the
> > >> "obvious" construction based on the bijection f: N -> P that the path
> > >> p(n) "goes the other way" to the path f(n)(n) does would you mark them
> > >> down?
> > >
> > > They would know that also the other way is already realized, for every
> > > n, in a rationals-complete list. And they would know that this
> > > rationals-complete liste is realized by the Binary Tree.
But no COMPLETE INFINITE BINARY TREE is "realized" by a "rational
complete " list.
But is realized as follows:
A Complete Infinite Binary Tree can be formed from the actually infinite
set |N as its set of nodes, with the left and right child of node n
being node 2*n+0 and 2*n+1, respectively. Nothing further is needed to
make |N with that given parent-child relationship on its members into a
Complete Infinite Binary Tree.
Note that the tree exists without needing any definition of path at all.
One can then define what it means to be a path in that tree:
Definition of a path in the tree defined above: a subset of |N which
contains 1 and which for each n in it also contains one but not both of
2*n+0 and 2*n+1 is a path.
Note that this does not define individual paths, but only defines how to
tell whether a subset of |N is a path or not.
But it allows us to match any particular path to a unique infinite
sequence of 0's and 1's, corresponding to the sequence of 2n+0 and 2n+1
child nodes in that path.
Note that under the above definition of pathhood only an ACTUALLY
INFINITE sequence of 0's or 1's corresponds to a path. No finite such
sequence can represent a path.
All of which is perfectly straightforward and proper in all standard
versions of mathematics, and is only forbidden by WM in
Wolkenmuekenheim.
--
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