WM says... > >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.
On the contrary! Many real numbers in physics are not computable to infinite precision (for example, the fine structure constant). Yet, we can certainly compute other real numbers *relative* to such parameters. We can easily devise an algorithm to compute the square of the fine-structure constant, for example. This algorithm will take as an input an approximation to the fine-structure constant, and will return an approximation to the square of the fine-structure constant.
In this sense, the square of any real number is computable relative to that real number.