Re: Cantor and the binary tree

Virgil said:
> In article <MPG.1d209cf5218a75aa989e3b@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > Virgil said:
> > > In article <MPG.1d1ce9d98c76af60989e25@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> ...
>
> > > > Therefore, for every two branches there is one path.
> > >
> > > One path each way, so that makes two paths, but the next node on
> > > each of these two paths is the root of another maximal binary tree,
> > > so each of TO's "paths" is, in fact, as many paths as in the whole
> > > tree.
>
> > You are not paying attention. The child branch is the continuation of
> > the parent branch's path, while the sibling node is the divergence of
> > a new path from that path.
>
> If this is a maximal binary tree, then, however arranged, from each node
> there extend two dependent branches reaching two more nodes, and so on
> ad infinitum.
>
> If one takes the part of the tree depending on any one node, wherever in
> the original tree it may occur, then the subtree starting from and
> dependent on that node is tree-isomorphic to the entire tree. And
> whether a particular dependent branch/node is considered a child or a
> sibling does not change this fact.
>
> If we call the set of all paths continuing a particular node the "bunch"
> for that node, then for any node, the child-bunch and the sibling bunch
> are both tree-isomorphisms of the parent bunch.

Yes, but each one is HALF of the "bunch" coming from the parent, even if
infinite. It's like the evens are half the naturals.
> > > > >
> > > > >
> > > > > > > It remains the case that there are easy bijectins between
> > > > > > > the set of naturals and either the set of nodes or the set
> > > > > > > of branches, but at best an injection which is not a
> > > > > > > bijection from the set of naturals to the set of paths.
> > > > >
> > > > >
> > > > > > That is absolutely not true.
> > > > >
> > > > > It absolutely is true!
> > > > >
> > > > > > Each infinite path corresponds to one number with infinite
> > > > > > digits.
> > >
> > > How does one write down an infinite digit?
> > An infinite number of digits, as you well know. We have repeatedly
> > discussed the correlation between the paths aand strings of digits.
> > No one has discussed individual digits that are infinite, except when
> > I have talked about digits at infinite offsets from the digital
> > point, which no one seems to want to discuss anyway. SO, this is
> > justa nother attempt on your part at deliberate onfuscation and
> > derailment of the discussion. Nice try.
>
> The elimination of ambiguity is mathematical discusins is never a waste.
>
The deliberate confusion of the topic is always a waste.
--
Smiles,

Tony
.

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