Looking into the open problem section of the book Model theory by Chang and Keisler, I noticed that many problems assumed semi-axioms like GCH. I talk about 'semi-axioms' because these "axioms" are shown (using informal ZFC) to be independent from the usual axioms ZFC.
How can one philosophically justifiy the usage of these axioms? Of course, it's legal to use any axiom one wants to use and one also can argue that in mathematics (being a "mind game", as I recall Asaf Karagila writing in one thread), one doesn't have to justify any axiom.
I have the following thoughts: Note that it depends one the philosophy of set theory one beliefs in. If one beliefs in truth platonism for arithmetic, then the usage of semi-axioms might be justified as follows: We use ZFC to conceptually to find a proof of the independence of a semi-axiom; then we convert this proof into a proof in Peano arithmetic, and since one beliefs in truth platonism of arithmetic, one can now be sure that ZFC + [the semi-axiom] is consistent. If one doesn't believe in determined truth values of arithmetical statements, then it's difficult to justify the use of these semi-axioms, since one then doesn't belief that statements such as "ZFC + [the semi-axiom]" (which is an essentially arithmetical statement) has a definite truth value! If one is a set-theoretical platonist, then the usage of semi-axioms is also problematic, since one then don't know, if one is using a true or a false assumption.