Date: Jan 29, 2013 3:33 AM
Subject: Matheology § 203

"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.

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.
4) For every n in N, the anti-diagonal of a Cantor-list is not in the
lines L_1 to L_n.

Regards, WM