```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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