Re: Cantor and the binary tree

In article <MPG.1d0d8f073444225e98968b@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> I have recently reviewed Virgil's proof of the
> uncountability of paths and countability of nodes. He uses a tree with
> paths that represent numbers in the first

I use each node in the max binary tree to correspond to a binary natural
number, depending on the nodes location in that tree.

> and paths that represent sets
> of numbers in the second.

That part TO got right.

> This use of representations without regard to
> the properties of the representations themselves is akin to using
> mapping functions to draw bijections and then disregarding the
> properties of those functions when declaring the sets equivalent.

The only properties one needs to worry about is whether the first
defined correspondence, between nodes and naturals, is actually a
bijection and whether the second defined correspondence, between maximal
paths and sets of naturals, is a bijection, and they both are.
.

Relevant Pages

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