Re: Cantor and the binary tree



*** T. Winter wrote:
> In article <MPG.1cfe8ee5a9650528989d2a@xxxxxxxxxxxxxxxxxxxxxxxxx> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> ...
> > So what? As you pointed out, when you add a node, you add a node, a branch,
> > and a path. Actually, you need to add two nodes for each path, but that is
> > irrelevant. They are finitely proportional, so if one is finite, the other
> > is, and if one is infinite, so is the other. Infinite trees have infinite
> > nodes and paths. Finite tree only have finite numbers of nodes and paths.
> > It's a fact.
>
> How do you jump from finite nodes to infinite nodes? An infinite path
> never ends.

That is a bit naive. Every node can be enumerated by a finite natural
number. You have intermingles infinite sets and infinite numbers. The
set of natural numbers is infinite whereas every natural number is
finite. So it is with the nodes. Looking at every node enumerated by a
finite natural number means looking at all nodes without leaving out
one. Therefore we can apply te reasoning which is valid for one node to
every node.

Regards, WM

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