In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> My question: > How can we distinguish between that infinite binary tree that contains > only all finite initial segments of the infinite paths and the > complete infinite binary tree that in addition also contains all > infinite paths?
The difference is that the one without all its infinite paths can only exists in Wolkenmuekenheim
In any binary tree, every node is in a finite initial sequence of some path, so once one has all finite initial sequences, one has all nodes.
And once one has all nodes, one also has all sets of nodes including of those uncountably many sets of nodes called paths, even if WM denies it.
WhaT WM claims in his WMytheological dreamland , is to allow an infinite set of nodes but only allow finite subsets of it, but outside of his WMytheology, things do not work like that.
Outside of WMytheology, if we can have all members of a set, we can also have all subsets of that set, even if that set and some of its subsets are actua lly infinite.
So WM's choice is either no infinite sets at all, including no Complete Infinite Binary Tree at all, or the standard Complete Infinite Binary Tree with its actually uncountably many paths.
WM is trying to straddle a fence that is not there to be straddled. --