Re: WM and end segments...

Ben Bacarisse <ben@bsb.me.uk> writes:

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>

For some damn reason when I hear end segments from WM I think of a
tree. Take the following infinite 2-ary tree that holds the positive
integers:
___________________________________________
                  0
                 / \
                /   \
               /     \
              /       \
             1         2
            / \       / \
           /   \     /   \
          3     4   5     6
         / \   / \ / \   / \
      .........................
___________________________________________
>
this goes on and on for infinity... We all can see how this can go for
infinity, right WM? Wrt trees there are only leaves in a finite view of
it. However, the "infinite view" of the tree has no leafs because it never
ends... Fair enough? Or too out there?

>
That's a can of worms in WMaths.  WM has written 734,342,120 nonsense
posts about binary trees over the years.  It's one of his favourite
examples to use to bamboozle his poor students.
>
The infinite binary tree -- simply a graph with node set N and edge set
(n, 2n+2) (in your numbering) -- is a particular puzzle for WM because

Correction, there are two such edges of course: (n, 2n+1) and (n, 2n+2).

the node and edge sets are countable but the path set isn't.
>
Can you see a proof that the infinite rooted paths can be mapped, one to
one, with an uncountable subset of R?
>

... The infinite one has no leaves.

>
If you consider graphs in general, they do not have to be infinite to
have no leaves.

--
Ben.