In mathematics labels are _not_ important, definitions are.
One simple example that everybody can relate to: do natural numbers include 0 or not? Who cares? Some definitions include it, some do not. There's even a convention of using N for N with 0, and N+ for excluding it, but even the convention is just a convention, not a definition. You could call them "funky numbers, the set of whole positive numbers including 0", and you're fine. Funky, natural, those are just names, labels, as long as you define them, it doesn't matter.
Same applies to set theory, there's many, many set theories, and they differ between properties, and this has never caused problems, because in mathematical discussion or literature...you provide or point to a definition. So you'll never discuss about "set theory" without providing one or pointing to one.
This is very, very different from how normal people waste their time.
E.g. arguing whether AI "thinks" or not, but never defining what thinking means, thus you can't even conclude whether you think or not, because it's never been defined.