Date: Mar 25, 2013 10:10 PM Author: ross.finlayson@gmail.com Subject: Re: Matheology § 224 On Mar 25, 8:09 am, WM <mueck...@rz.fh-augsburg.de> wrote:

> On 25 Mrz., 15:48, "Ross A. Finlayson" <ross.finlay...@gmail.com>

> wrote:

>

> > On Mar 25, 4:45 am, WM <mueck...@rz.fh-augsburg.de> wrote:

>

> > > Does the Binary Tree that contains all rational paths also

> > > contain all irrational paths?

> > Given that, for R[0,1]:

>

> > a) each irrational has a unique infinite expansion as path

>

> That is the question. If so, why has never anybody written it using

> digits or bits?

>

>

>

> > b) each initial segment of the expansion is the initial segment of a

> > rational

>

> > c) every rational's path is in the tree

>

> That is the question too. Why has never anybody written the complete

> decimal- or binary expansion of a periodic rational?

>

>

>

> > 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.

>

> I agree with your conclusion but not with the premises.

>

> Remember: Never has anybody written an infinite sequence other than by

> using the symbolic method: "1/9" or "1/pi" or "1/(SUM 1/n!)". These

> however are only names to identify or formulas to construct infinite

> paths - not paths that belong to the Binary Tree.

>

> Regards, WM

Two different paths, finite or infinite, have at least one node not in

common.

And they have at most countably many nodes not in common, there are

only countably many nodes.

And, there are only countably many nodes a distinct path, from a path,

could have. Then, there are only that many distinct paths, as for

each there are only countably many others.

Yet, mapping the paths onto 2^w and thus to P(w), that would be a

contradiction to Cantor's theorem.

So, are there uncountably many nodes? Because, there's a rational for

each node, and the rationals are countable. There are only countably

many different paths, from a path. The paths are rooted.

Seems rather muddled.

Infinite sequences are written out as their specification. No, there

are not infinite digital resources to emit each (element of an

infinite sequence), but, a rule to emit each suffices for many

purposes.

Then in as to whether Eudoxus/Dedekind/Cauchy expansions are

sufficient to represent each element of the linear continuum of what

would be the real numbers, is a separate notion. (And, no, they

aren't.)

Regards,

Ross Finlayson