"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.