Axioms are subjective (?), and, since propositions are based on axioms, isn't everything subjective? (of course, the answer should be from the perspective of someone who believes in objectivity)
Or am I wrong in the assumption that axioms are always subjective?
You can fix any group of axioms you like and start deducing consequences. You built a theory in that way. Your theory is useful if there are models of the theory that are interesting to someone and you were able to deduce non trivial consequences from the set of axioms you have chosen. I would say that in order to discuss sets of axioms you need to introduce the notion of model. Otherwise things get messy. Looking at things on this way I would say you don't have to worry about subjectivity and objectivity. An axiomatic theory does not tell you how the world is. It gives you tools to deal with the aspects of reality it models.