Date: Jan 29, 2013 6:47 PM
Author: Virgil
Subject: Re: Matheology � 203
In article

<f5702bc4-f905-4e60-94c5-a503f3d9d887@n2g2000yqg.googlegroups.com>,

WM <mueckenh@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.

D) For all natural numbers n , P(n) is true

But A does imply D.

>

> Examples for A:

> 1) For every n in N, there is m in N with n < m.

> 2) For every n in N, the set (1, 2, ..., n) is finite.

> 3) For every n in N, the construction of the first n nodes of the

> Binary Tree adds n paths to the Bibary Tree.

False! At least for binary non-Bibary trees.

The first 3 nodes produce only 2 paths in a finite biNary tree.

The first 7 nodes produce only 4 paths in a finite binary tree.

The first 15 nodes produce only 8 paths in a finite binary tree.

> 4) For every n in N, the anti-diagonal of a Cantor-list is not in the

> lines L_1 to L_n.

Thus not anywhere in the list!

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