In article <3b84e611-0e94-4d16-af3a-a568d149191f@n7g2000prc.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 14 Jun., 20:48, Virgil <virg...@nowhere.com> wrote: > > In article > > <ed7cfe4c-c486-43c0-bb1f-8ceb2652c...@y17g2000yqn.googlegroups.com>, > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 14 Jun., 14:01, William Hughes <wpihug...@hotmail.com> wrote: > > > > > > Assuming infinite paths exist: > > > > > > The path p is in the tree > > > > > > The path p can be distinguished from > > > > every element of P. > > > > > For every path p_n of P there is a digit such that p differs from p_n. > > > But there is no digit of p that differs from every path p_n of P > > > because, if p exists as binary representation, then p is the union of > > > all p_n and as such cannot differ from the union. > > > > Paths are not made up of digits, but of nodes. But in a binary tree, one > > can represent a path as an infinite sequence of binary digits with, say, > > 0 for each left child node and 1 for each right child node. > > > > But no such path p is then a union of other paths, as WM claims, as no > > union of a set of more than one path can be a path itself. > > In an actually infinite binary tree constructed of terminatings paths > only there is/would be the non-terminating path 0.111... > > How does it come into being, if it comes into being?
If one has all finite complete (all path of the same length and all non-leaf nodes having exactly two children) binary trees, then the union of the set of all their node sets is the node set of a maximal infinite binary tree having uncountably many actually infinite paths.
> > > > If p_n refers to any finite or even countably infinite set of paths then > > every non-member of p_n, is distingishable from every member of p_n and > > every subset of p_n. > > Not from every path however that belongs to the binbary tree.
But no one has claimed that. But since your p_n cannot simultaneously include all paths of the resulting maximal infinite binary tree and be countable, that is irrelevant.
> Therefore your claim is not based upon digits or nodes but on > superstition.
The only "superstition" involved is the definition of a maximal infinite binary tree, together with the definition of a path in such a tree.
Which involves much less superstition than 'potentially infinite sets'. > > > > > > > > > If you don't believe me, then play the game. Minimum stake 10^6 > > > dollars or (if you come from Europe) Euros or Pounds. > > > > Then I challenge WM to present his list of paths from which he claims no > > other path can be distinguished. > > You must be able to to distinguish yout path from the complete binary > tree constructed by my list of path by means of nodes only.
WM wants to change the game now. The original proposal was for WM to present a list of infinite binary paths and challenge me to find path not listed.
Which game he would inevitably lose.
But if WM wants to present an unlisted set of paths, I do not guarantee to find another path not in it.
> What is a > name? that which we call a rose by any other name would smell as > sweet.
The quote properly begins "What's in a name?"
> Paths afre not identified by names in Cantor's lists.
So we > will not do it here. > > > > Note that since any counterexample may be dependent on his choice of > > list, as Cantor's depends on the list being given, WM is constrained to > > publish his list first. > > I published the completed binary tree. It contains every sequence of > nodes.
You have done no such thing. You may have referred to it, rather then your incomplete versions of it, but 'publishing' it would require at least explicitly listing ALL its nodes, which being infinite in number, cannot be done via newsnet.
