Yes they are independent of ZFC but they still exist. From the original article talking about CH which is also independent:
> This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception.
They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms.
I believe the same applies for BBs. They have precise values outside of ZFC but we still cannot ever know them because they are incomputable (??)
> but we still cannot ever know them because they are incomputable (??)
That doesn't mean you can't know them. We know BB(2). It means there is no single algorithm which is capable of yielding them all. But there could, theoretically, be an algorithm for BB(10), a different algorithm for BB(11), etc.
In fact, those individualized algorithms don't exist either.
> This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception. They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms.
I believe the same applies for BBs. They have precise values outside of ZFC but we still cannot ever know them because they are incomputable (??)