On 22 Dez., 22:06, William Hughes <wpihug...@gmail.com> wrote: > On Dec 22, 8:55 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > The bijection of all finite words with all natural numbers has been > > defined in binary: > > However, the bijection you need is all definitons with a subset > of the natural numbers.
Even such a bijection would not support your case because the diagonal of the infinite list of finite definitions is not a finite word. It does not even exist!
> And there is no way to define this subset. > If you put restrictions on the 0/1 sequences you allow to exist you > put restrictions on the subsets you allow to exist. Note, that > subcountable > does not mean countable.
Then you give up the essence of set theory. Then you give up the theorem of Schröder-Bernstein. Then set theory and the hierarchy of infinities need not be contradicted. They simply do not exist. Set theory demands: Every subset of a countable set is countable. To prove that does not require a bijection.