Re: Cantor and the binary tree



mueckenh@xxxxxxxxxxxxxxxxx wrote:

> We have a law yielding all possible infinite strings of bits.
> We have a law connecting each spring-off, i.e., the source of a newly
> separated path with a node B:
> /
> B
> /\

I don't understand this. As far as I can see, each node 'B' has two
branches from it, to left and to right, and neither identifies a single
path. Rather, each one leads to another (unending, infinite) subtree.

> This is the basic element of the tree and does nowhere change. Hence it
> holds all over the tree. It yields a one-to-one relation between nodes
> and paths. Nobody can reasonably even r[a]ise the question whether there
> could be more paths than nodes.

On the contrary, I raise it. In maths, if the answer to any question is
obvious, it can be dealt with quickly. Anyway, we could identify with
this node the path going right, and subsequently always going left. In
this way we can identify every node with a single path, showing that
there cannot be more nodes than paths, but not showing the reverse
(because we haven't mapped every path to a node in this way). This does
not prove that there is no 1-1 mapping, but if there is one, you still
have work to do to show it. (Other considerations of course show that
this is indeed impossible.)

> ... And if it is meaningful to talk about
> infinity and to count infinity, then it is absolutely clear that nodes
> and paths have same number.

So false. It certainly isn't clear.

> Gottfried Helms wrote: [if your quoting levels are reliable]
> > The concept "number of" may be assigned to describe an (important)
> > property of a finite set, but if we deal with mathematical objects,
> > which are constructed to be infinite, it is better to switch to the
> > terminus "cardinality" (or maybe a better one), to not apriori throw
> > away the ability to include uncountable(*1) infinities in our
> > considerations.
>
> The idea of a bijection of a set with N is a convincing one, in many
> respects. But the idea of considering the basic element
> /
> B
> /\
> of a tree is at least as convincing. To do the last step, we could even
> neglect the terminus "node". Then I assert: There are not more
> separated paths in the tree than paths which separate themselves
> somewhere in the tree. Set theorists say: There are more separated
> paths in the tree than are separated in the tree.

Do you have any proper definitions for this "separating" stuff?

Brian Chandler
http://imaginatorium.org

.

Relevant Pages

  • Re: Cantor Confusion
    ... at least for any level in an infinite tree. ... So WM wants leaf nodes in CIBTs. ... Any two paths separate at some node, ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... Rather, each one leads to another (unending, infinite) subtree. ... Further, you are right, a whole new tree as large as he old one is ... >>> considerations. ... >> separated paths in the tree than paths which separate themselves ...
    (sci.math)
  • Re: Cantor Confusion
    ... >> As there is no last term of the geometric series this makes no sense. ... Since in a tree, each node may be a geometric point in some plane, such ... have a leaf node which is separate from all other paths, but infinite ...
    (sci.math)
  • Re: Cantor Confusion
    ... at least for any level in an infinite tree. ... mathematically defined complete infinite binary tree. ... Any two paths separate at some node, ...
    (sci.math)
  • Re: Cantor Confusion
    ... To object to that is to misunderstand mathematics. ... No path in an infinite tree can be isolated by one node either, ... To "separate" one such path from all others by nodes ...
    (sci.math)