On Dec 30, 12:18 pm, Virgil <vir...@ligriv.com> wrote: > On 12/29/2012 11:22 AM, WM wrote: > > > > > > > > > > > On 28 Dez., 23:04, Virgil <vir...@ligriv.com> wrote: > >> In article > >> <2925a2ea-c16d-483e-91da-3bb0084e7...@b16g2000vbh.googlegroups.com>, > > >> WM <mueck...@rz.fh-augsburg.de> wrote: > >>> On 27 Dez., 21:49, Virgil <vir...@ligriv.com> wrote: > > >>>>> It is the set > >>>>> of all finite paths extending from the root node to a given node. > > >>>> In every Complete Infinite Binary Tree, every finite path ends at > the > >>>> root node of of another Complete Infinite Binary Tree. So for every > >>>> finite path, there are uncountably many extensions of it in every > >>>> Complete Infinite Binary Tree. > > >>> And every extension is contained in the CIBT constructed from all > >>> finite paths. > > >> Claimed but never proven. > > > Definitions need not be proven. > > That they are instantiated must be proven, and that you have not done. > > You have not proven that there is a CIBT constructed or constructible so > as NOT to contain uncountably many paths, whereas many have proven that > any CIBT must have more than any countable set of paths in order to be a > COMPLETE INFINITE BINARY TREE. > > Given any countable set of such paths, any proof that they are countable > requires that they be listable, but one can prove that they are not > listable by showing that no list of them can be complete. > And no mater how vociferously WM tries to argue otherwise, in standard > mathematics that can all be done. > --
Well, not when "standard" was "pre-Cantorian", they didn't have the notion of a relevant structure of what "uncountability" might be. Then, to declaim that modern mathematics as "standard" proves that a diagonal argument as Al-Jofar's for the tree holds, is well true, but, modern mathematics is incomplete, and, as described above, a bread- first traversal of the tree, sees that not hold.
Basically establishing a symmetry through the center of the tree, in the lexicographic ordering of the paths, then from seeing that from left at zero to right at one there is the inexhaustibility of the domain as to course-of-passage, with a "completed" infinity or for that matter, as modeled from the finite, completed symmetry: this non- standard (or not-yet-standard) course, that is well supported by the classical and as well in asymptotics by the modern and concrete, sees different results from that.
Draw the line, Cantor shows is it's point-to-point, not the stippling of the stellation, our complete ordered field as structure above the continuum follows from simpler principles (of points that make space).
Draw the line. Split the tree. Diagonal? Where's the middle? The middle is defined by the ends.