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.

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