In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 1 Apr., 17:05, William Hughes <wpihug...@gmail.com> wrote: > > On Mar 27, 8:55 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > Matheology § 233 > > > > > The set of all termination decimals is a subset of Q. If the set of > > > all terminating decimals of the unit interval is arranged as set of > > > all terminating paths of the decimal tree, > > > > It is, of course, impossible to write this out > > (there number of terminating decimals is infinite). > > > But it is possible to construct the Binary Tree according to this > method.
I am not at all sure that it is even possible to build any binary tree this way but it is clearly impossible to built a COMPLETE INFINITE BINARY TREE, this way.
For one thing, in a CIBT, every path is by definition maximal in the sense that no additional node can be added to a path without making the result not a path, and is also minimal in the sense that no node can be removed from it without making the result not a path.
In WM's "trees", every FISON (Finite Initial Sequncee Of Nodes) appears to be a path, which is quite differnt notion of path.
> It is impossible to write them out. Yes. But they are constructed like > the finita paths. It is impossible to prohibit infinite paths (of > rationals and of irrationals) to be constructed when the complete set > of nodes of the Binary Tree is constructed by means of all finite > paths. Therefore it is impossible to distinguish infinite paths by > nodes other than be naming infinite sets of nodes. Alas there are only > countably many names available.
Thus not all CIBT-paths are nameable, just like not all real numbers are nameable.
Nothing in the vast extent of mathematics outide Wolkenmuekenheim requires that all things be nameable. --