Re: Cantor Confusion

In article <1167110178.666662.149790@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Newberry" <newberry@xxxxxxxxxx> wrote:

Virgil wrote:
In article <1167094162.439384.295810@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Newberry" <newberry@xxxxxxxxxx> wrote:

David Marcus wrote:
Newberry wrote:
Virgil wrote:
In article <1166895046.650593.195620@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Newberry" <newberry@xxxxxxxxxx> wrote:
Virgil wrote:
In article
<1166854303.474151.267360@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Newberry" <newberry@xxxxxxxxxx> wrote:
Is it true that the ratio of edges over paths converges to
two as
we
approach infinity?

lim{n-->oo} (2*2^n - 2)/2^n = 2

It is true that the ratio of terminal nodes to paths converges
to 1
as
the path lengths increase towards infinity.

What about the ratio of all the edges to all paths? Does it
converge
to
2?
lim{n-->oo} (2*2^n - 2)/2^n = 2

It does not matter.

Why does it not matter?
The cardinality of the inexes in the limit is aleph0, and the
cardinality of the nodes in any infinite path is aleph0. It means
that
in calulating the limit
lim{n-->oo} (2*2^n - 2)/2^n = 2
we transversed all the infinite paths.

What does "traversed" mean? And, how is it relevant to determining the
cardinality of the set of paths?

It means that we have taken into account the entire tree and we
determined that the number of edges in said entire tree is twice as
higher as the number of paths.

Then it means that you are producing nonsense.


It is not me who is producing it.

You are making a claim which has as a consequence the same sort of limit
argument that every path in the infinite tree has a terminal node, as
well as a root node.


The same " limit" argument will conclude the in a tree in which no path
has a terminal node, every path has a terminal node.

That is the contradiction we were talking about.


For every finite tree, the number of terminal nodes (leaf nosed) is
exaclty equal to the number of paths, so that if the limit argument is
valid for the edge to path ratio, it must equally be valid for the
terminal node to path ratio.
.

Relevant Pages

  • Re: Cantor Confusion
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  • Re: Cantor Confusion
    ... It is true that the ratio of terminal nodes to paths converges to 1 ... the path lengths increase towards infinity. ... It means that we have taken into account the entire tree and we ... has a terminal node, every path has a terminal node. ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... >> For finite paths (having a root node and a leaf or terminal node) one is ... >> root which procede through infinitely many nodes in this infinite tree, ... The number of paths having leaf nodes equals the number ...
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  • Re: Cantor Confusion
    ... It is true that the ratio of terminal nodes to paths converges to 1 ... the path lengths increase towards infinity. ... It means that we have taken into account the entire tree and we ... has a terminal node, every path has a terminal node. ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... > For finite paths (having a root node and a leaf or terminal node) one is ... > root which procede through infinitely many nodes in this infinite tree, ... > and never have a terminal or leaf node, ... >>> One can hardly imagine a simpler mathematical proof. ...
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