On 7 Feb., 19:14, William Hughes <wpihug...@gmail.com> wrote: > On Feb 7, 5:57 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > On 7 Feb., 15:56, William Hughes <wpihug...@gmail.com> wrote: > > > > On Feb 7, 3:25 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > <snip> > > > > >... a subset S of the countable set F of finite words bijects with > > > > the set D of definable numbers > > > by definition. > > > > Nope. Every D corresponds to some finite word. > > > No, D is a set or at least a collection. A definable number is an > > element of D. > > > > However, S, > > > the collection of all the correspondences, may not be a subset > > > of F (subsets must be computable). > > > Need not be a subset. It is sufficient to know that there are not more > > than countably many correspondences, > > There is no set of correspondences thus there is no number > of correspondences. You cannot know anything about > the number of correspondences.-
You are in error again. There is the axiom of power set. For any F, there is P such that D e P if and only if D c F. According to it every subset of the countable set F exists. Will you dispute that the finite definitions of numbers are a subset of F? Not even the axiom of choice is required to prove that.