There are multiple set theories using different axioms. Is the Axiom of Choice based on an observation of nature or do mathematicians keep it around because it's useful? It's a statement about infinities that absolutely have no physical reality. You can do mathematics without it, and the question of whether to do math relying on it is a matter of opinion.
(Yes, proofs relying on AC are arguably true even if you don't accept AC, but as a social reality some sets of axioms are considered valid bases for work and some aren't, you can keep adding stronger axioms to ZFC to prove more things more easily, but how far you go with that before it stops being interesting is a matter of opinion)
