Re: Cantor Confusion


Virgil wrote:
In article <1166451538.861033.303300@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

Newberry schrieb:

Sorry that I joined a bit late.
Doesn't matter.

Are you saying that (in an infinite
binary tree) the set of paths is uncountable but the set of edges is
countable?

The set of edges is obviously countable by, e.g.,

1, 2,
3,4,5,6,
7,...

As no path can separate from another one without the existence of two
more edges, the number of edges is an upper bound for the number of
paths.


That is only the case in finite trees.
It fails miserably in infinite binary trees in which no path has a last
node or last edge.

Do you agree that the number of edges is the upper bound of the number
of paths but disagree that the edges are countable?

I guess if the number of edges > number of paths and you accept the
diagonal argument then it follows that the edges are uncountable. I do
not know if there is any way to prove directly that the edges are
countable.

.

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