Real numbers are defined by properties, usually, and it’s just a “I say these things exist” then. For example, axioms of real numbers state that increasing bou

Real numbers are defined by properties, usually, and it’s just a “I say these things exist” then. For example, axioms of real numbers state that increasing bounded sequences have a limit, and it is a real number. It could be that this leads to some contradiction somewhere which we don’t know about. For example, if you erroneously thought that rational numbers in fact themselves satisfy these axioms, you could construct an increasing sequence whose squares converge to 2, and yet the sequence itself doesn’t converge to a rational (because sqrt(2) is not rational). Perhaps some such contradiction exist for real numbers, and in that respect they could “not exist”. That would mean the axioms are somehow inconsistent. As the article states, we can construct real numbers with Dedekind cuts that only rely on rationals; I didn’t realise there is any nuance to that definition.