Re: The complete infinite binary tree has only countably many infinite paths.
- From: William Elliot <marsh@xxxxxxxxxxxxxxxx>
- Date: Tue, 24 Mar 2009 08:57:35 -0700
On Tue, 24 Mar 2009, WM wrote:
The complete infinite binary tree has only countably many infiniteIt has uncountably many paths, one for each binary sequence.
paths.
0.
/ \
0 1
/ \ / \
0 10 1
...
A) Construct the binary tree starting from a "tree" that has only oneExtreme vagueness. If at each node without a branch, you added another infinite series you will have countable many series. You repeat this again, infinitely.
path, say p_0 = 0.000... 0.
|
0
|
0
|
0
...
Add all paths that end by infinitely many zeros. Every path that you
add must start at a node of p_0 or at a node of a path already
constructed. The number of paths in the tree grows by not more and not
less than 1 when 1 path is added. However, after completing that
procedure all nodes and every infinite sequence of bits (including the
path 0.111...) is present, namely represented in the infinite binary
tree.
B) The complete infinite binary tree can be constructed by an infiniteAre you writing mathematics or science fiction?
agglomeration of elements of the form
|
o
/ \
The number of distinct lines is increased by 1 by 1 node.
(lines going out - lines coming in - nodes) = (2 - 1 - 1) = 0
This procedure, even when applied infinitely often, cannot but yield
the result 0 respectively a countable number of lines. The set of all
lines limits the set of all distinct paths.
The construction is possible in (B) as well as in (A) because the
elements used for construction is countable infinite.
C) Consider the edges of the complete binary tree (an edge connectsYou're writing science fiction.
two subsequent nodes of a path). Take all edges and put them on one
and the same level of the tree, side by side, such that the "tree" now
is an array of parallel edges: |||||||... This array limits the number
of possible paths of the tree. It is an upper limit, because every
path there has only one edge. And there is no further edge remaining
to distinguish any further paths.
Remark: Of course a set of n edges can be put in n! differentRemark: apply at philosophy where such sophistry is philosophical.
sequences. But the edges of the binary tree are not subject to
arbitrary ordering. Each one has one and only one fixed place in the
tree. Therefore the number of edges limits the number of paths.
Remark: The tree contains all possible sequences of bits includingRemark: your application for mathematics has been rejected.
0.111... . Nevertheless the tree contains only a countable number of
paths, as we see by each of the arguments (A - C). This shows that
"most" of the real numbers cannot exist as independent bit sequences.
This shows that most of the real numbers are not subject to being put
in a list or resulting from a list as anti-diagonal number.
.
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