On Dec 22, 8:55 am, WM <mueck...@rz.fh-augsburg.de> wrote: > On 22 Dez., 00:34, William Hughes <wpihug...@gmail.com> wrote: > > > > Yes it is definable. It has been defined. Nevertheless it does not > > > exist, because the sets do not exist. > > > It is definable which means that WM can use it to prove that the > > bijection exists, but it does not exist. > > 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. 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.
> > 0 > 1 > 00 > 01 > 10 > 11 > 000 > and so on. > > From this definition the natural number belonging to any desired > finite word can be obtained. It can easily be translated into any > other language. Or do you need some help? > > Nevertheless there is no set of all natural numbers and no set of all > infinite words. > > It is the same with pi. The (potentially) infinite string of digits of > pi can be defined. In fact there are (potentially) infinitely many > definitions. Nevertheless there is no actually infinite string > expressing pi. > > Yes, I know that is not easy to understand. That's why so many > mathematicians have gone astray. > > Regards, WM