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Topic: Matheology § 203
Replies: 202   Last Post: Feb 10, 2013 3:34 AM

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 JT Posts: 1,448 Registered: 4/7/12
Re: Matheology § 203
Posted: Feb 4, 2013 10:14 PM

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
>
> 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

Date Subject Author
1/29/13 mueckenh@rz.fh-augsburg.de
1/29/13 William Hughes
1/29/13 mueckenh@rz.fh-augsburg.de
1/29/13 William Hughes
1/29/13 mueckenh@rz.fh-augsburg.de
1/29/13 William Hughes
1/29/13 mueckenh@rz.fh-augsburg.de
1/29/13 William Hughes
1/29/13 mueckenh@rz.fh-augsburg.de
1/29/13 William Hughes
1/29/13 mueckenh@rz.fh-augsburg.de
1/29/13 William Hughes
1/29/13 mueckenh@rz.fh-augsburg.de
1/29/13 William Hughes
1/29/13 mueckenh@rz.fh-augsburg.de
1/29/13 William Hughes
1/30/13 mueckenh@rz.fh-augsburg.de
1/30/13 William Hughes
1/30/13 mueckenh@rz.fh-augsburg.de
1/30/13 William Hughes
1/30/13 mueckenh@rz.fh-augsburg.de
1/30/13 William Hughes
1/30/13 mueckenh@rz.fh-augsburg.de
1/30/13 William Hughes
1/30/13 mueckenh@rz.fh-augsburg.de
1/30/13 William Hughes
1/30/13 mueckenh@rz.fh-augsburg.de
1/30/13 William Hughes
1/30/13 mueckenh@rz.fh-augsburg.de
1/30/13 William Hughes
1/30/13 mueckenh@rz.fh-augsburg.de
1/30/13 William Hughes
1/31/13 mueckenh@rz.fh-augsburg.de
1/31/13 William Hughes
1/31/13 mueckenh@rz.fh-augsburg.de
1/31/13 William Hughes
1/31/13 mueckenh@rz.fh-augsburg.de
1/31/13 William Hughes
1/31/13 mueckenh@rz.fh-augsburg.de
1/31/13 William Hughes
1/31/13 mueckenh@rz.fh-augsburg.de
1/31/13 William Hughes
2/1/13 mueckenh@rz.fh-augsburg.de
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2/1/13 William Hughes
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2/1/13 Virgil
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2/2/13 William Hughes
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2/2/13 William Hughes
2/2/13 Virgil
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2/2/13 William Hughes
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2/3/13 mueckenh@rz.fh-augsburg.de
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2/5/13 Virgil
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2/4/13 JT
1/29/13 Scott Berg
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