My impression (as a dilettante programmer without relevant credentials) is that there isn't really any question about whether mathematical structures can be rooted in set theory, or can be expressed as extensions of set theory. Disputes about foundations of mathematics are more about how easy or elegant it is to do so. (And in fact my impression is they're mostly about subjective, aesthetic considerations of elegance rather than practical considerations of how hard it is to do something in practice, even though the discussion tends to be nominally about the practical side. Quite similar to disputes about programming languages in that respect.)