In article <email@example.com>, firstname.lastname@example.org wrote:
> On Sunday, 21 July 2013 05:06:42 UTC+2, Virgil wrote: > > In article <email@example.com>, > > firstname.lastname@example.org wrote: > On Saturday, 20 July 2013 21:31:03 > > UTC+2, Virgil wrote: > > >>> Any countable set can be listed without inclusion of any non-members. > > > > at least if not > > > Since the definition of "countable" requires the ability to surject |N to > > the the collection of whatever is alleged to be countable, there is no |not > > if". > > The rationals are countable. If written in the Binary Tree just these aleph_0 > elements remain aleph_0 elements, don't they?
Of course, but their paths are not the only paths in any such a tree, the presence of paths for all binary rationals requires the presence of limit paths for all converging sequences of binaries as well. And that produces a path for every member of the uncountable real unit interval!
> Do the rational points of the > unit interval have a cardinality aleph_0? You think so.
I know better. But when one also has all the irrational points of the unit interval, as one does everywhere but in WMytheology, that real interval is an uncountable set.
The paths in a Complete Infinite Binary Tree which correspond to binary rationals are those countably many paths which contain only finitely many branchings in one direction or the other, which leaves all those uncountably many paths with infinitely many branchings in both directions in that tree unaccounted for by WM. > > But the rationals written in the Binary Tree create a set of cardinality > larger than aleph_0.
In order to have all rational paths in a Complete Infinite Binary Tree, one must also have all irrational paths 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] --