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:
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> > On Jan 29, 9:33 am, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > "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.

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

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

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> > 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.
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> 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