[nexttk]

About the axioms, not really. Axiom sets is mostly there just as a 'short hand' to quickly describe a context we're talking about, but ultimately you could just do away with them. E.g. if we let A be the set of axioms from some theory (e.g. set theory, number theory etc.) and you have a mathematical statement of the form X => Y within that theory, you could just as well consider the statement "A ^ X => Y" in the purely formal system without any axioms at all, then it is purely a logical question (essentially, if X => Y is a theorem within theory A) and more objectively true than "X => Y" which would be theory-independent.