You need to distinguish between arbitrary foundations and unfounded foundations. By definition, the most basic foundations of a theory are not going to be rendered non arbitrary by deeper foundations, but that does not mean they are arbitrary....arbitrariness may be removed by other means, such as by choosing your axioms to lead to the results you want.
You need to distinguish between arbitrary foundations and unfounded foundations. By definition, the most basic foundations of a theory are not going to be rendered non arbitrary by deeper foundations, but that does not mean they are arbitrary....arbitrariness may be removed by other means, such as by choosing your axioms to lead to the results you want.