In article <e1027530-bcc6-4a92-a3ea-f5e2bce14736@37g2000yqp.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 16 Jun., 20:09, Virgil <virg...@nowhere.com> wrote: > > In article > > <f59dc977-e9eb-4bbc-bc62-42501f504...@r33g2000yqn.googlegroups.com>, > > > > > > > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 15 Jun., 22:04, William Hughes <wpihug...@hotmail.com> wrote: > > > > > > > > You are trying a proof by contradiction of > > > > > > " Actually infinite paths exist" > > > > > > > > We have > > > > > > If actually infinite paths exist > > > > > > there is a path p that can be distinguished > > > > > > from every path of P. > > > > > > > > We need > > > > > > > > If actually infinite paths exist > > > > > > there is no path p that can be distinguished > > > > > > from every path of P. > > > > > You know modus tollens? > > > ((A ==> B) & ~B) ==> ~A > > > > > A: actually infinite paths exist. > > > B: p can be distinguished from every path of P. > > > > But you have not established your " ~"B > > Try to find out where the path 1/3 = 0.010101... or any other path > differs from the complete, maximal, infinite binary tree.
Your countable set of paths, P, does not contain as members all paths of any maximal infinite binary tree, though it may contain as members of members all the nodes.
> The tree is constructed by all terminating paths with tails 000...
Which is a countable set of paths, and therefore, by Cantor, does not include all paths.
> same tree could be constructed by all terminating paths with tails > 0.010101... > There is no difference.
Any set of paths, or other sets of nodes, whose union contains as members all nodes of a prospective maximal infinite binary tree tree can be used to construct that tree, whether that set of sets of nodes contains as members all the paths of that tree or not.
Once one has the set of all its nodes, regardless of how they are come by, one has the tree, and all its paths derive from the node set and tha 'parent of' relation, not from any a priori set of paths.
