Date: Jan 29, 2013 7:23 PM
Author: Virgil
Subject: Re: Matheology � 203

In article 
<2c0d6d3b-4b27-488a-a006-7db8b50e685a@u1g2000yql.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 29 Jan., 10:18, William Hughes <wpihug...@gmail.com> wrote:
> > On Jan 29, 10:09 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> >
> >
> >
> >

> > > On 29 Jan., 09:54, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > On Jan 29, 9:33 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > "All" and "every" in impredicative statements about infinite sets.
> >
> > > > > Consider the following statements:
> >
> > > > > A) For every natural number n, P(n) is true.
> > > > > B) There does not exist a natural number n such that P(n) is false.
> > > > > C) For all natural numbers P is true.

> >
> > > > > A implies B but A does not imply C.
> >
> > > > Which is the point.  Even though A
> > > > does not imply C we still have
> > > > A implies B.

> >
> > > > Let  L be a list
> > > >      d the antidiagonal of L
> > > >      P(n),  d does not equal the nth line of L

> >
> > > > We have (A)
> >
> > > >    For every natural number n, P(n) is true.
> >
> > > > This implies (B)
> >
> > > >   There does not exist a natural number n
> > > >   such that P(n) is false.

> >
> > > > In other words, there is no line of L that
> > > > is equal to d.

> >
> > > And how can C be correct nevertheless? Because "For all" is
> > > contradictory.

> >
> >    B: There is no line of L that is equal to d
> >
> > does not imply
> >
> >    C: For all n, line n is not equal to d.
> >
> > B correct does not mean "C correct nevertheless"-

>
> But we know of cases where C is correct nevertheless. I quoted four of
> them in the § 203. Or do you disagree to one of them?
>
> In case you have forgotten the old discussion concerning the
> configurations of the Binary Tree construction, here it is repeated:
>
> The complete infinite binary tree is the limit of the sequence of its
> initial segments B_k:


How is this any different from |N being the the "limit' (or union) of
all its finite initial segments but then having properties that none of
its finite initial segments have, like infinitely many elements.?
>
> The structure of the Binary Tree excludes that there are any two
> initial segments, B_k and B_(k+1), such that B_(k+1) contains two
> complete infinite paths both of which are not contained in B_k.


In fact, no finite initial segment can contains any "Path" at all.


> Nevertheless the limit of all B_k is the complete binary tree
> including all (uncountably many) infinite paths. Contradiction. There
> cannot exist more than countably many infinite paths.


WRONG!

There can be uncountably many infinite binary sequences representing
different real numbers in the unit interval, [0,1], even allowing for
the countably many duplicate representations of binary rationals in that
interval, and an easy bijection between them and the infinite paths in
an infinite binary tree.
--