Date: Mar 25, 2013 10:48 AM
Author: ross.finlayson@gmail.com
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>
> wrote:
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> > On Mar 24, 2:51 pm, fom <fomJ...@nyms.net> wrote:
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> > > On 3/24/2013 4:34 PM, WM wrote:
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> > > > On 24 Mrz., 21:29, Virgil <vir...@ligriv.com> wrote:
> > > >> In article
> > > >> <729f073f-8948-4eb9-991a-2bd249ac5...@c6g2000yqh.googlegroups.com>,

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> > > >> A binary tree that contains one path of each positive natural number
> > > >> length will necessarily also contain exactly one path of infinite length.

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> > > > Like the sequence
> > > > 0.1
> > > > 0.11
> > > > 0.111
> > > > ...
> > > > that necessarily also contains its limit?

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> > 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.

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> 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
rational

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.

Regards,

Ross Finlayson