Date: Mar 30, 2013 9:37 PM
Author: Virgil
Subject: Re: Matheology � 224

In article 
<1ca524f9-9057-4949-b68d-fcdda479188f@fn10g2000vbb.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 30 Mrz., 19:03, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <050b4a95-d2b0-433b-98b6-d63c34635...@m9g2000vbc.googlegroups.com>,
> >
> >
> >
> >
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:

> > > On 29 Mrz., 19:40, Virgil <vir...@ligriv.com> wrote:
> > > > In article
> > > > <ce3c22f2-9116-4621-b3b4-e722fe51a...@a14g2000vbm.googlegroups.com>,

> >
> > > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > On 26 Mrz., 22:47, Virgil <vir...@ligriv.com> wrote:
> >
> > > > > > But a tree that contains paths for all binary rationals will
> > > > > > contain a
> > > > > > path for all limits of a sequences of binary rationals.

> >
> > > > > Does a sequence always contain its limit?
> >
> > > > Depends on the sequence, of course. but a sequence of paths in a
> > > > Complete Infinite Binary Tree in which the nth path must share at least
> > > > n nodes with each of its successors will always converge, though not
> > > > neccessarily to a binary rational.

> >
> > > A sequence of numbers may converge, but not necessarily to a limit
> > > that is a term of the sequence.

> >
> > Precisely my point!

>
> Precisely not your point.





It is still my point, even though clearly WM does not understand it.

> The infinite path is not in the infinite
> sequence of finite paths which are used to construct the complete
> tree.


Each node of the infinite limit path is in all but finitely many of the
infinite sequence of infinite but binary-rational paths of which is a
limit.

otherwise that path would not be a limit or the tree would not be a CIBT.
> >
> > > A sequence of paths may converge, but not necessarily to a limit that
> > > is a term of the sequence.

> >
> > So WM acknowledges that A sequence of binary rational paths can converge
> > to a path that is not a binary rational

>
> and that is not in the tree of all binary rationals.


A tree having all binary rationals AS INFINITE PATHS is the only sort
that can be a CIBT. So whatever sort of trees WM is talking about they
cannot be Complete Infinite Binary Trees.
> >
> >
> >

> > > In mathematics more precision is required.
> >
> > Certainly more than WM is capable of producing,.

>
> You intermingle the paths of the tree and the limits which are neither
> paths nor belong to the tree.


In CIBTs all paths are infinite, so WM must be talking about other types
of trees
> >
> >

> > > > In a COMPLETE INFINITE BINARY TREE, all paths are actually infinite
> > > > --

> >
> > > This is again a simple statement of countermathematical  belief
> >
> > It is matter of simple definition.

>
> No.


YES!

WM does not get to decide what can be or cannot be a definition.
> >
> > The definition of a COMPLETE Infinite Binary Tree requires that no path
> > in such a tree can terminate.

>
> An infinite sequence does not terminate. Nevertheless its limit is in
> general not in the sequence. Correct or not?



If they are sequences of paths in a Complete Infinite Binary Tree which

There are a lot of infinite sequences that are "eventually constant" in
a CIBT if one identifies each path with an infinite sequence of 0's and
1's in the binary representations of numbers in [0,1], and those
sequences which are eventually all 0's, and only those, are "eventually
constant".

But most of the sequences in a CIBT, even if convergent, are NOT
eventually constant,
--