On Wednesday, January 30, 2013 1:03:49 PM UTC-8, WM wrote: > Now I > construct the complete infinite Binary Tree by means of countably many > paths,
You have never done such a thing.
> such that every node and every possible combination of nodes is > covered by at least one path (in fact by infinitely many).
Yes, infinitely many paths. Uncountably infinite many paths. You have never given any means to count them all. You just assert you have done so. No path in the complete infinite Binary Tree is finite. You can count the finite initial segments of the paths but every one of them has an infinite set of paths associated with them. Some infinite sets are countable and some are not. As it happens the set of initinte paths in the complete infinite binary tree is uncountable while their initial segements are countable.
> But I don't > tell what paths I have used. If it was possible to distinguish > uncountably many paths purely by the nodes, then it could not be a > problem to find further paths, because all must differ somehow from > each other. But, of course, nobody can find such a path without > knowing my choice.
You haven't made a choice. Consider this: "What did you give your son for his birthday?" "I gave him an X-Box." "I asked him and he said that wasn't true." "He must be lying." "I doubt it. He showed me his present and the picture you took when he openned it." "Then it must not have been this birthday." "I never said it was." "How am I to know what I gave my son for his birthday if I don't know what year?" "You don't know what you gave your son for his birthday?"
> This shows that there are further pieces of > information required to distinguish the paths. But as these pieces of > information are necessarily finite words
> (otherwise they could not be communicated)
No, I can talk about he set of real numbers. You're asking for each of the members to be uniquely identified. That can't even be done with the natural numbers because the set is infinite. Even though the set is infinite every member has *A* successor. No member has more than one successor.
> their number is countable. So they cannot be used to > distinguish uncountably many paths.
Yes, exactly. This alone should tell you there's a problem with your conceptualization of the set of infinite paths being countable.
> By this proof by ignorance the usual belief in the presence of > uncountably many infinitely distinguishable paths in the Binary Tree > or infinite sequences of digits is contradicted.
No, it show the problem with a proof by ignorance.
Every pair of distinct real numbers differ at some place in their decimal or binary expansion, however that place cannot be identified without identifyig which pair of real numbers one is talking about. That I can't tell you where they differ until you tell me the numbers doesn't mean the numbers cannot be distinguished. The rational 1/3 doesn't have a finite decimal expansion. None the less it is distinguishable from every rational other than 1/3 at some place in the expansion and only the infinite expansion can be calculated to be 1/3.
> Of course it will > last some time until convinced set theorists will have realized the > power of this method.
Ignorance has no power other than to fuel its eradication. Ignorance is infinite; so, like King Sisyphus we can only get pleasure from the journey not from the destination.