Date: Jan 29, 2013 7:23 PM
Subject: Re: Matheology � 203
WM <email@example.com> 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.
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.