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.