```Date: Feb 4, 2013 10:14 PM
Author: JT
Subject: Re: Matheology § 203

On 29 Jan, 10:09, 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.>> There is no natural number that finishes the set N.> There is no finished set N.Correct> There is, in the list of all reminating decimals, no anti-diagonal,> that differs from all terminatig decimals at digits belonging to at> least one of these terminating decimals. Reason: The list is complete.> If you don't believe, consider the Binary Tree constructed from all> finite paths only.> Again, the only solution is, there is no complete set Q.>> There is, in the construction of the complete Binary Tree, no node> that adds more than one path to the tree. Nevertheless the completely> constructed tree contains uncountably many paths. No reason to be> taken aback, at least a little bit?>> Nevertheless, the steps of construction can be enumerated and> therefore can be considered as a list. In no line you find any> infinite path. But the complete list contains uncountably many> infinite paths - if such exist in the complete construction.>> Regards, WM
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