In article <email@example.com>, firstname.lastname@example.org wrote:
> On Sunday, 21 July 2013 21:45:30 UTC+2, Virgil wrote: > > > > In order to have all rational paths in a Complete Infinite Binary Tree, one > > must also have all irrational paths as well. > > But in a list of all rational numbers this is not the case.
> There we have not > uncountably many entries. Since the object described by 0.111... is identical > with the object described by 0-1-1-1-... , Virgil claims that the presence of > irrationals depends on how we write the rationals.
Wm, as usual, misunderstands, or at least misrepresents, what I say. It is the structure of the Complete Infinite Binary Tree which cannot exist with only binary rational paths, since the existence of all the binary rational approximations to an irrational in such a tree requires the existence of the path for that irrational as well. > > > WW deludes himself in claiming that any converging sequence of binary > > rational paths in [0,1] does NOT have a limit path in [0,1] > > The sequence of FIS > 0.1 > 0.11 > 0.111 > ... > > has certainly the limit 0.111... . But either this limit is not present > within the sequence of FIS, or it is the sequence itself.
But in any Complete Infinite Binary Tree, the existence of all those infinite paths 0.1(0), 0,11(0), 0.111(0),..., where the (0) indicates an infinite string of 0's, requires the existence of the path 0.111..., or 0,(1), to exist in the tree also.
If that does not happen then whatever structure WM has in his pseudotree fails to be a Complete Infinite Binary Tree at all.
> In the latter case > it does not increse the cardinality of the FIS. Same holds for the Binary > Tree: Irrational paths are either not present or not increasing the > cardinality of the rational paths.
WRONG and WRONG! They ARE present and are collectively uncountable.
At least outside of WM's wild weird world of WMytheology.
Note that the set of paths in a Complete Infinite Binary Tree is easily bijectable with the set of all subsets of |N:
For each path, include n in the subset of |N if and only if the child node of the nth node in that path is a lift-child.
This association clearly bijects the set of all paths with the set of all subsets of |N.
At least everywhere outside of WM's wild weird world of WMytheology , --