In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 16 Jun., 12:38, William Hughes <wpihug...@hotmail.com> wrote: > > On Jun 15, 4:44 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 15 Jun., 22:04, William Hughes <wpihug...@hotmail.com> wrote: > > > > > > > > > > Your claim is that "no possibility exists to construct or to > > > > > > > > distinguish by one or many or infinitely many nodes > > > > > > > > of the tree another path." > > > > > > You agree that if actually infinite paths exist > > > > your claim is false. > > > > > No. I see that my claim is correct under any circumstances. That fact > > > is independent of whether or nor actually infinite paths exist. > > > > Nope. You have repeatedly agreed that if actually infinite paths > > exist, then a path p exists that can be distinguished from > > every element of P. > > > > WM: If actually infinite paths exist, then there is a path p that can > > be > > WM: distinguished from every path of P. > > A ==> B > > > > You also claim > > > > WM: And: > > WM: There is no path p that can be distinguished from every path of > > P. > > ~B > > > <snip> > > ((A ==> B) & ~B) ==> ~A
If no infinite sets can exist, as WM claims, then everything that WM claims about them is false.. > > > > > > > Every path of the tree is is from P. > > > > > > Nope. Every *node* of the tree is from P. > > > > > Every path of the tree is from P. I explicitly forbid every other path > > > to enter my tree. > > > > Nope. You cannot forbid every other path to enter the > > tree. You add nodes to the tree. > > I add infinite paths.
Then WM is claiming that infinite paths both exist and do not exist, so that his set theory is self contradictory. > > > When you add a node to the tree you add subsets of nodes > > to the tree as well. You add a subset of nodes that is > > not in a single element of P. > > Wrong.
It is WM who is wrong, as there are all sorts of subsets of the set of nodes which are not paths, so cannot be in P, but are sets of nodes in that tree.
> I add a path p_n like > 0.himpidimpydowadididum000...
Then WM no longer has a binary tree at all, but only an irrelevant figment of his warped imagination.
> That path p_n is different from any other path p_m and there is no > path p_m that together with p_n produces anything different from all > finite subsets that are already contained in p_n or p_m.
No finite subset of the node set of a maximal infinite binary tree is a path in that tree. > > You are dreaming of an ideal set theoretic world - without paradoxes > and antinomies.
Why not? Such ideal worlds are what mathematics is all about.
> But this world has come to an end in 2004 with the > first construction, recognition and careful interpretation of the > binary tree.
Which binary tree or trees would that be? None of WM's quasi-trees can qualify as a maximal infinite binary tree.
And as it is only in WM's misrepresentations of ZF, and not in ZF itself, that any anomalies appear, the maximal infinite binary trees in ZF behave quite nicely.