On Wednesday, 26 February 2014 22:12:47 UTC+1, Virgil wrote:
> > That is correct. The set of natural > > numbers however is not every countable set but a very special countable set. > > > > Does WM claim that there are infinite sets in which no subset can be > order-isomprphic to the set of naturals?
Not me. You believe that all real numbers R except a countable set D are undefinable. So you accept the set R \ D which has no definable elements and no definable subset, hence, no countable subset. (Countable means enumerable and that implies definablilty of the elements.) >
> > Some of the many sets produced are > > > 1, 1s, 1ss, ... > > Ambiguous unless one is guaranteed that they are all different
Obviously 1s is different from 1ss. > > > 1, 11, 111, ... > > > 1, 1^1, 1^1^1, ... > > > > Unless WM is using "^" for something other than exponentiation, > one has 1 = 1^1 = 1^1^1, ... , so WM is claiming a one-element set of > all naturals.
The notations differ from each other. Peano did not distinguish between value and notation. Another fault of his misconception.