On Oct 30, 3:20 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> CLAIM: GIVEN AN ENUMER-ABLE SET OF REALS > YOU CAN'T PROVE SQUAT!
For sake of simplicity, let's refer to the denumerable binary sequences rather than to reals themselves. (We understand that there is a 1-1 correspondence between the real interval [0 1] and the set of denumerable binary sequences.)
By definition, an enumerable set S of reals is one for which there exists an enumeration f of S.
So let f be an enumeration of S. Consider the denumerable binary sequence g defined by g(n)=0 if f(n)(n)=1 and g(n)=1 if f(n)(n)=0. We see easily that g is not in the range of f.
My claim is not that given an enumerable set S of denumeragble binary sequences we can constructively produce a denumerable binary sequence not in S.
Rather, the claim is that given an enumeration f of a set S of denumerable binary sequences we can constructively produce a denumerable binary sequence not in S.
Again, what we claim is that there is no enumeration of the set of real numbers. And that is proven by proving, as we do (and we do it constructively), that given an enumeration of a set S of denumerable binary sequences there is a denumerable binary sequence not in the range of the enumeration, i.e., not in S.