In article <email@example.com>, firstname.lastname@example.org wrote:
> On Monday, 22 July 2013 21:45:08 UTC+2, Virgil wrote: > > 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. > > Paths of irrationals in mathematics are not defined by "structure". That is > purest matheology.
They may not be in WM's wild weird world of WMytheology, but they are in the wider worlds of mathematics that allow such sets as |N and 2^|N. > > For every finite step n in the construction of the Binary Tree (by nodes or > by FIS or by paths of rationals), there is no irrational path. They sneak in > only after all finite steps.
But a binary tree cannot contain all finite paths without containing all infinite paths just like a unary tree cannot contain all FISONs as subsets without containing |N.
> That is obvious nonsense. But if it was not > nonsense, then the same could happen in every Cantor-list.
Not until lists (which are really just unary trees) become binary trees.