On 15 Jun., 22:45, Virgil <Vir...@home.esc> wrote: > In article > <f78b53d6-24d1-42e2-86bd-1dd0893b8...@q12g2000yqj.googlegroups.com>, > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 15 Jun., 16:06, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > > > WM says... > > > > >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? > > > > r is computable *relative* to the list L of all computable reals. > > > That is, there is an algorithm which, given an enumeration of computable > > > reals, returns a real that is not on that list. > > > > In the theory of Turing machines, one can formalize the notion > > > of computability relative to an "oracle", where the oracle is an > > > infinite tape representing a possibly noncomputable function of > > > the naturals. > > > We should not use oracles in mathematics. > > WM would prohibit others from doing precisely what he does himself so > often? > > > A real is computable or not. My list contains all computable numbers: > > > 0 > > 1 > > 00 > > ... > > > This list can be enumerated and then contains all computable reals. > > If that list is .0, .1, .00, ..., then it contains no naturals greater > than 1.
This list is the list of all words possible in any language based upon any finite alphabet. The list is given in binary. All alphabets, all languages and all dictionaries are contained in later, rather long but finite lines. Therefore this is a list of everything (that can meaningfully be expressed).
This list does not allow for a diagonal, because that is a meaningless concept. (That is proved in my list, in a later line.)