On 15 Jun., 18:49, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > WM says... > > > > > > > > >On 15 Jun., 16:18, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > >> Peter Webb says... > > >> >"WM" <mueck...@rz.fh-augsburg.de> wrote in message > >> >news:firstname.lastname@example.org... > >> >> On 15 Jun., 12:26, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > > >> >>> (B) There exists a real number r, > >> >>> Forall computable reals r', > >> >>> there exists a natural number n > >> >>> such that r' and r disagree at the nth decimal place. > > >> >> In what form does r exist, unless it is computable too? > > >> >Of course its computable. > > >> No, it's computable *relative* to the list of all computable reals. > >> But that list is not computable. > > >That is nonsense! > > >The list of all definitions is possible and obviously contains all > >definitions of real numbers. > > I was talking about the list of all *computable* reals. There are > definable reals that are not computable, and Cantor's proof shows > how to define one.
No, it does not show that because it is impossible to define a real by an infinite definition. A Cantor-list without a finite definition however is an infinite definition.
Sensible definitions are finite. A definition allows you to communicate the defined. That is impossible for a Cantor-list, except the list has been constructed according to a finite definition. But then also the diagonal has a finite definition - in every specific language.
So you explanation is nonsense in highest degree. > > You can similarly get a list of all definable reals for a specific > language L.
Every real that is definable in a specific language is definable in another specific language. The set does not grow or shrink or change when the language is changed.
No, my list contains all words in all languages, even the definitions of the languages and all the dictionaries. There is nothing else.
> Then Cantor's proof allows us to come up with a new > real that is not definable in language L. (It is definable in a > new language that extends L).
The diagonal procedure does not define new languages. It uses simply the diagonalisation. The language that I use is not to be extended. There a real either has a finite definition or it has not. There does no extension help.
When have you obtained the meaning of a real number from an infinite sequence of digits for the first time?