I don't think that works. The rational numbers are a dense subset of the real numbers. Informally this means every real number is either a rational number, or is arbitrarily close to a rational number. This means that at any magnification, if there was a hole that is filled by a real number, then their would also be a rational number that is arbitrarily close to that real number.
Well, I'm not a big fan of this "infinite magnification" idea in the first place, but "arbitrarily close" is typically one of those things that infinity can beat.
(Compare e.g. with the fourier transform of a function. It consists of a sum series which comes "arbitrarily close" to the function, but "at the limit" when the number of terms approaches infinity the function and its fourier transform is one and the same.)
