In article <9496b0d7-70ee-4c48-b574-85e443a99dec@x21g2000yqb.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 3 Nov., 21:35, Virgil <vir...@ligriv.com> wrote: > > In article > > <7dbfb197-e5cc-4fbd-9679-4577c5a83...@du8g2000vbb.googlegroups.com>, > > > > > > > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 3 Nov., 18:12, William Hughes <wpihug...@gmail.com> wrote: > > > > On Nov 3, 1:14 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > On 3 Nov., 15:28, William Hughes <wpihug...@gmail.com> wrote: > > > > > > > > No nodes. Above we have C the finite trees, indexed by the natural > > > > > > numbers > > > > > > and K the union over N of C_n > > > > > > > Correct. That's what I call the old set.(We never talked about a > > > > > single finite Binary Tree.) > > > > > > K is not a finite tree. It does not matter how many finite > > > > trees we talked about, K is a new tree. > > > > > Can you distinguish K by nodes from the complete infinite Binary Tree > > > containing all infinite path that K does not contain? > > > > By any reasonable definition of "path" in K, K must contain uncountably > > many paths. > > All finite subsets of |N form a countable set.
If K is, as it would seem that it must be, the union of node sets of infinitely many finite binary trees, you comment is irrelevant. > > > > One such reasonable definition would be to take the union of a maximal > > nested sequence of paths in nested complete finite trees to be a path in > > the union of those finite trees. In this way every initial segment of > > such a path in K will be a path in one of K's finite subtrees. > > > > And it is easily shown that any such union path must > > (1) be infinite and that > > (2) there are as many such paths as there are subsets of |N. > > and (3) that there are not more than countably many such subsets, > namely all finite subsets.
How you get that infinite sets must be finite is not clear to me, nor how the set of all subsets of an infinite set must be countable.
For any COMPLETE Infinite Binary Tree, any partition of N , the set of positive naturals, defines a path, with one of the sets being the set of node-levels of a path at which that path branches left, and the other being the set of node-levels at which it ranches right.
And that set of partitions of N, with 1 path for each one, is uncountable. --
