A different viewpoint: "adding a new axiom" is synonymous with "further restricting the things under discussion". Imagine biologists decide that "DAG" is the

A different viewpoint: "adding a new axiom" is synonymous with "further restricting the things under discussion". Imagine biologists decide that "DAG" is the correct abstraction for parent-child relationships. Then they spend years trying to prove the obviously-true fact that parents can't have infinitely many children. Of course, they never can, because DAG nodes can have infinitely many children. Finally they realize they chose the wrong abstraction, they should've chosen "DAGs where all nodes have finitely many children". To suggest adding a new axiom to ZFC is to suggest that ZFC was an over-broad abstraction for capturing what things are supposed to be "sets".