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:

>

> > > "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)

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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.

>

> > 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