> because it asserts the existence of natural numbers that have no "written form"
I don’t see why that should imply it wouldn't be "okay" to add ¬Con(ZFC).
It may be highly counterintuitive, but the history of mathematics is full of counterintuitive results that nowadays are accepted as true in mainstream mathematics.
Well-known examples are the existence of irrational numbers, the claim that the set of natural numbers has the same size as that of the rational numbers, the existence of hyperbolic geometry, and the Banach-Tarski paradox.
I don’t see why that should imply it wouldn't be "okay" to add ¬Con(ZFC).
It may be highly counterintuitive, but the history of mathematics is full of counterintuitive results that nowadays are accepted as true in mainstream mathematics.
Well-known examples are the existence of irrational numbers, the claim that the set of natural numbers has the same size as that of the rational numbers, the existence of hyperbolic geometry, and the Banach-Tarski paradox.