Date: Mar 25, 2013 10:48 AM
Subject: Re: Matheology § 224
On Mar 25, 4:45 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 24 Mrz., 23:13, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> > On Mar 24, 2:51 pm, fom <fomJ...@nyms.net> wrote:
> > > On 3/24/2013 4:34 PM, WM wrote:
> > > > On 24 Mrz., 21:29, Virgil <vir...@ligriv.com> wrote:
> > > >> In article
> > > >> <729f073f-8948-4eb9-991a-2bd249ac5...@c6g2000yqh.googlegroups.com>,
> > > >> A binary tree that contains one path of each positive natural number
> > > >> length will necessarily also contain exactly one path of infinite length.
> > > > Like the sequence
> > > > 0.1
> > > > 0.11
> > > > 0.111
> > > > ...
> > > > that necessarily also contains its limit?
> > A binary tree that contains one path, of all zero-branches, of each
> > finite length, will necessarily contain a path of 0-branches of
> > infinite length.
> If actual infinity exists. Otherwise it contains nothing more than all
> finite paths. But here is a question that is easier to answer and to
> decide: Does the Binary Tree that contains all rational paths also
> contain all irrational paths?
> Regards, WM
Given that, for R[0,1]:
a) each irrational has a unique infinite expansion as path
b) each initial segment of the expansion is the initial segment of a
c) every rational's path is in the tree
d) the union of finite initial segments of the expansion as tree
contains the expansion as path
e) thus each irrational's expansion is a path in the tree of rationals
then, yes, that appears to be so.