In article <email@example.com>, WM <firstname.lastname@example.org> 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.