Date: Jan 14, 2013 3:53 PM
Author: Virgil
Subject: Re: Matheology � 191

In article 
WM <> wrote:

> On 13 Jan., 21:41, Virgil <> wrote:
> > In article
> > <>,
> >
> >  WM <> wrote:

> > > On 12 Jan., 23:03, Virgil <> wrote:
> >
> > > > > You are invited to "discern" another path from the countable bunch of
> > > > > infinite paths that I used to construct the Binary Tree.

> >
> > > > Until you list the ones that you used, there is no way to "discern"
> > > > another, but any list you provide also provides a nonmember.

> Here are all paths that I used:
> 0.
> 0 1
> 01 01
> ...
> and so on. Every node that you arrive at allows to continue left or
> right. And all that is countable.

The number of nodes involved is countable, but every countable set has
uncountably many subsets, at least outside WMytheology, and so the
number of paths can be, and is, uncountable outside WMytheology.
> > Why should I like to have something which does not exist?
> You like it so much, because otherwise you would never have started to
> believe in uncountability.

I believe the proofs that no set can be surjected to its power set are
valid, and will continue to believe it until I see an EXPLICIT
surjection from some set to its power set.
> >
> > If WM wants to claim such a list exists, then HE is reqponsible for
> > proving it, and the only valid proof is to present us with it, or at
> > lest some unambiguous rule for generating it.

> Follow any desired infinite path of the Binary Tree above, which has
> been constructed from all finite initial segments of all paths.

WM's set of finite initial segments are like the set of binary
rationals, and it takes a set of infinitely many rationals to detremine
each irrational that way.

So WM needs a different infinite set of finite binary sequnces for most
infinite binary sequences. Thus WM needs something like the the power
set of his set of finite initial segments.

> You
> will never gather more than countably many infinite paths.

Then we will never access all of the Complete Infinite Binary Tree, even
though it does exis.

> And you
> will never miss any path representing a real number of the unit
> interval.

While it may be difficult to get an irrational real (by getting a
suitable sequence of binary rational reals), it is trivial to get lots
and lots of binary rational reals.
> Regards, WM