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Topic: Wm mis-explains what he means by a Binary Tree
Replies: 8   Last Post: Feb 7, 2014 2:54 PM

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Virgil

Posts: 8,833
Registered: 1/6/11
Re: WM mis-explains what he means by a Binary Tree
Posted: Feb 6, 2014 4:46 PM
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In article <a37eee53-513b-4299-9c30-36ec271eaec1@googlegroups.com>,
WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:

> Am Donnerstag, 6. Februar 2014 10:19:54 UTC+1 schrieb Virgil:
>
>

> >
> > The finitely defined real number r = Sum_(n in |N) 1/2^(n!), in base 2,
> >
> > is not anywhere in WM's "rationals-complete list"

>
> No it is not since it cannot be represented by digits.


It IS representable by digits in base 2 by having a 1 in each n! place
and 0's elsewhere.

What exactly does WM mean by "represented by digits"?
Rationals do not all have a representation "by digits" in any one base

> You will never distinguish this
> number *by digits* from all terminating rationals.


I already did!
>
> > or in his
> > pseudo-binary tree, but does represent a path in any Complete Infinite
> > Binary Tree

>
>
> There is no path in the Complete Infinite Binary Tree that defines this
> number r.


There is in any Complete Infinite Binary Tree.
In is only in WM's incomplete trees that it may be overlooked.



> I have proved this by constructing a Binary Tree

WM's trees are all incomplete.

> from a set of
> paths that contain all nodes of r but does not contain that path itself.



One must construct the tree first and then that tree defines its paths.

WM is trying to do things backwards by patching together a bunch of what
he calls paths (which cannot be paths until after the tree is completed)
and calling it a tree.

A standard model for a Complete Infinite Binary Tree , as Ben said, is
the set of naturals as the set of nodes, with (n,2*n), n in |N,
linking n as the parent node with 2*n as its left child node and (n,
2*n+1) linking n with its right child node, 2*n+1.

This tree exists as a tree prior to any definition of pathhood within
it, so that tree-existence is entirely independent of and prior to
path-existence.

Thus, as usual, WM's approach to trees is directly backwards.
--





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