In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 2 Jul., 03:40, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > > > Sorry, no dreaming involved. You did show that you could cover each node by > > countably many paths. You did *not* show that the set of all paths was > > exhausted by that process. > > Proof by your failure to find a path that is not in the tree.
There are no infinite binary paths which are not in TA maximal infinite binary tree, but there are such paths which are not is any given countable set of paths, as Cantor proved. > > The tree is constructed by terminating paths with tails 000... > appended, but the same tree can also be constructed by terminating > paths with tails 010101.. or pi appeded. If you, by looking at the > completely constructed tree only, can find a path that is not in the > tree and that was not used to construct the tree, then you are right. > But you cannot. Therefore your assertion concerning missing paths is > wrong.
If all one needs is a path not used in the construction, and that construction uses only countably many paths, then it is trivial to find paths not used in that construction, even though they appear in the "constructed" tree. > > > > > I show that the number of paths that can be identified by sequences of > > > nodes or bits is countable. > > > > Not at all, because you did not show a bijection. > > I did use the countable set of terminating paths (extended by a > special kind of tails). And you cannot identify a path that I did not > use, if you don't know from what kind of paths the tree was > constructed.
But as soon as I do know which ones were used, I can find others in the tree which were not used.
WM's version of the problem is like having to guess whether a tossed coin has come up heads or tails without looking at it, but the actual problem allows one to look.
> This proves
WM's arguments only prove that he does not understand what is going on.
In WM's world, maximal infinite binary trees are prohibited from existing, so any arguments he makes about their properties is fatuous.
In a ZFC world, where maximal infinite binary trees can exist, their properties derive from the ZFC axioms.