On 29 Jan, 10:09, WM <mueck...@rz.fh-augsburg.de> wrote: > On 29 Jan., 09:54, William Hughes <wpihug...@gmail.com> wrote: > > > > > > > > > > > On Jan 29, 9:33 am, WM <mueck...@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. > > > Which is the point. Even though A > > does not imply C we still have > > A implies B. > > > Let L be a list > > d the antidiagonal of L > > P(n), d does not equal the nth line of L > > > We have (A) > > > For every natural number n, P(n) is true. > > > This implies (B) > > > There does not exist a natural number n > > such that P(n) is false. > > > In other words, there is no line of L that > > is equal to d. > > And how can C be correct nevertheless? Because "For all" is > contradictory. > > There is no natural number that finishes the set N. > There is no finished set N.
> There is, in the list of all reminating decimals, no anti-diagonal, > that differs from all terminatig decimals at digits belonging to at > least one of these terminating decimals. Reason: The list is complete. > If you don't believe, consider the Binary Tree constructed from all > finite paths only. > Again, the only solution is, there is no complete set Q. > > There is, in the construction of the complete Binary Tree, no node > that adds more than one path to the tree. Nevertheless the completely > constructed tree contains uncountably many paths. No reason to be > taken aback, at least a little bit? > > Nevertheless, the steps of construction can be enumerated and > therefore can be considered as a list. In no line you find any > infinite path. But the complete list contains uncountably many > infinite paths - if such exist in the complete construction. > > Regards, WM