Virgil
Posts:
8,833
Registered:
1/6/11


Re: Matheology � 233
Posted:
Apr 1, 2013 7:54 PM


In article <c1713766dbe44327b3aea7aff2e80eee@h9g2000vbk.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 1 Apr., 17:05, William Hughes <wpihug...@gmail.com> wrote: > > On Mar 27, 8:55 am, WM <mueck...@rz.fhaugsburg.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 CIBTpaths 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. 

