Date: Jan 29, 2013 6:47 PM
Subject: Re: Matheology � 203
WM <email@example.com> 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!