You're right, I misread what you wrote. You're actually only covering a small portion of the rationals. Either way, to order the reals, you have to include the

You're right, I misread what you wrote. You're actually only covering a small portion of the rationals. Either way, to order the reals, you have to include the non-computable ones, which cannot be written down in any systematic way. This also means that ordering and equality relations cannot be systematically defined either.