I see. We actually failed to 'define a system' for what it means to be a real number. What the set of real numbers contains can't be defined in any strict formal way.
Even for the range [0, 1] saying all numbers including rationals and irrationals is an incomplete cop-out. The rationals are defined. Only some irrationals are/can-be. Saying a 'number that is not rational' is not a definition--it is negative space. Prime numbers are the negative space of composite numbers--they are however countable and computable. The negative space within real numbers is different. There are no possible constructions to reach some/all of them.
Does the same problem arise with any uncountably infinite set or only not-well defined ones? Is "The Set of all Subsets of Natural Numbers" (which is uncountable) also non-mathematical in the same sense? A program (requiring infinite storage and computation time) can be constructed.
Even for the range [0, 1] saying all numbers including rationals and irrationals is an incomplete cop-out. The rationals are defined. Only some irrationals are/can-be. Saying a 'number that is not rational' is not a definition--it is negative space. Prime numbers are the negative space of composite numbers--they are however countable and computable. The negative space within real numbers is different. There are no possible constructions to reach some/all of them.
Does the same problem arise with any uncountably infinite set or only not-well defined ones? Is "The Set of all Subsets of Natural Numbers" (which is uncountable) also non-mathematical in the same sense? A program (requiring infinite storage and computation time) can be constructed.