I think I get your point, except that nobody ever does this. Nobody goes, "Here's the Pythagorean theorem, but when we're doing physics we don't allow noncomputable numbers as its inputs or outputs." NOBODY DOES THIS. Why are you fixated on noncomputable numbers? I agree they're interesting, but they have no relevance to physics unless someone is claiming the world is a computer or a computation. In which case computability is the least of the issues on the table. But you're not claiming that. So why do you think anyone either is or should be deleting noncomputable numbers from the inputs and outputs of mathematical functions and equations used in physics? Why do you think this? If all you care about is "accuracy and precision" as you put it, then rational numbers are fine. That's what all physical measurements are. Rational numbers with error bars.