Re: Cantor and the binary tree

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.
> > > >
> > > >
> > > > > > 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.
.

Relevant Pages

  • Re: Cantor and the binary tree
    ... >>> each of these two paths is the root of another maximal binary tree, ... > If we call the set of all paths continuing a particular node the "bunch" ... >>> How does one write down an infinite digit? ... >> discussed the correlation between the paths aand strings of digits. ...
    (sci.math)
  • Re: Cantor Confusion
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  • Re: Binary Tree and Pairs of Nodes
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    ... but the nodes of an infinitely deep binary tree can ... And digits of what? ... Infinite strings ... it's still an enumeration of the reals. ...
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  • Re: Cantor Confusion
    ... And as 10^-k is never zero, that sum is never an irrational number. ... The infinite binary tree contains this limit. ... Therefore we can calculate the union of all ...
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