Re: Cantor and the binary tree

In article <ITSnetNOTcom#virgil-C8C4A7.19592607062005
@comcast.dca.giganews.com>, ITSnetNOTcom#virgil@xxxxxxxxxxx says...
> In article <MPG.1d0fd70a2aca34399896a7@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>
> > In article <ITSnetNOTcom#virgil-7F6FBB.13541107062005
> > @comcast.dca.giganews.com>, ITSnetNOTcom#virgil@xxxxxxxxxxx says...
> > > In article <1118166450.883263.12840@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > mueckenh@xxxxxxxxxxxxxxxxx wrote:
> > >
> > > > Virgil wrote:
> > > > > In article <1118049962.009645.111100@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > > > mueckenh@xxxxxxxxxxxxxxxxx wrote:
> > > > >
> > > > > > Daniel W. Johnson wrote:
> > > > > > > <mueckenh@xxxxxxxxxxxxxxxxx> wrote:
> > > > > > >
> > > > > > > > You have shown that set theory is inconsistent. Congratulations.
> > > > > > >
> > > > > > > No.
> > > > > > >
> > > > > > > He has shown that the combination of set theory with your
> > > > > > > assertions is
> > > > > > > inconsistent.
> > > > > >
> > > > > > The basic element of the binary tree is the branching where a path is
> > > > > > separated.
> > > > > > /
> > > > > > B
> > > > > > /\
> > > > > > If set theory cannot describe this fact then it is useless.
> > > > >
> > > > > And if this path separation cannot do contour integration is it equally
> > > > > useless? Set theory is not obligated to analyse binary trees.
> > > >
> > > > Does it forbid my way of analysing the tree?
> > >
> > > Only logic is needed to do that. In a maximal binary tree, every maximal
> > > path can only be "separated" from ALL other paths by the set of ALL
> > > nodes that it passes through (except for the root node which is common
> > > to all) so the set of paths associates with a set of infinite subsets of
> > > the set of nodes.
> > >
> > And yet, one can say each node in the tree is identified by the infinite
> > set of unique paths which pass through it, so by that logic, there are
> > infinitely many more nodes than paths. That chain of thought doesn't
> > work.
>
> Actually, any node is uniquely identified by any pair of paths which
> split at that node, so on that argument the number of nodes cannot
> exceed half the number of paths.
>
No, each of those paths splits into infintiely many paths, so the two
branches that emanate from that node represent as infinite a set of
paths as emanate from the root node. The logic is simple. The is one
node for every branch. It takes two children to split the path leading
to the parent node into two. Adding one child does not produce a new
path. You must have two. There are two branches or nodes for every path
in the tree.
--
Have a nice day

Tony
.

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