In most fields of mathematics, it is not necessary to state that ZFC is assumed. Axiom of choice is singled out sometimes, but the other axioms are considered s

In most fields of mathematics, it is not necessary to state that ZFC is assumed. Axiom of choice is singled out sometimes, but the other axioms are considered so self-evident that it's not even worth mentioning. Yet from a logical standpoint, there's nothing special about ZFC to be so universally adopted. We could've had a completely different set of axioms and completely different mathematical concepts as a result. The reason why ZFC is chosen is because it results in concepts that are applicable in the real world. If a new independent axiom could give a rise to some useful concept applicable across many different mathematical fields, eventually ZFC+X might be adopted as the universal axiom system and it wouldn't be necessary to write "assuming X" in every proof.