Re: Cantor Confusion

In article <1169113948.875219.279820@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

Virgil schrieb:

In article <1169030073.463923.305630@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

Franziska Neugebauer schrieb:


This is obviously not the union in the sense of set theory. Set
theoretically a tree is a directed graph which is defined as an ordered
pair (V, E). The union of two trees is in general not a tree at all.

What I defined a sthe union of finite trees is the union in the sense
of set theory.

The set theory union of trees is not a tree at all, but with suitable
modifications one can construct something that works somewhat in the way
WM intends. Unfortunately for WM, it disproves his claims:

The union of two finite trees T(m) and T(n) with m and n levels,
respectively, where m < n, is the tree with n levels. This definition
unites sets of nodes (and sets of edges, respectively) and it is valid
for Cut Trees (CT) as well as for trees of type Weeping Willow (WWT).

The union of any finite number of such trees is a finite tree.

In a finite binary tree of order n, if it contains the maximal number of
edges for an order n binary tree, it must contain a terminal edge for
each n-th level node and so it will be a complete tree.

This is not true for an infinite tree, and is not true for infinite
unions of finite trees.

The union of all finite trees is the union of all trees with n levels
where n is a natural number:
UT = T(1) U T(2) U T(3) U ...

The paths in a tree are completely defined by the sequences of nodes
(or edges) which can be followed to an end in CT or without an end in
the union of all CTs as well as in the WWT and the union of all WWTs.

But only by the full sequence, which is, in essence the path itself.

So that paths define paths.
.

Relevant Pages

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