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. 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. But I understand that, in order to escape contradicitons, you must distinguish between "computable" and "computable relative" of first class and "coputable relative" of second class and so on, because the idea of uncomputable reals cannot be used to avoid the contradiction of Cantor's theory.