I think axioms are very useful, but i am not at all sure you can prove something with them, in a way that is as logically sound as in math. I mean that, two people may well be able, with the same axioms, to reach a different conclusion. The axioms do not seem mathematically strict that way.
Regardless, it is probably a good idea to state "axioms" because i think it will still make your views much clearer, and although the logic to its conclusion is probably still subjective, it does restrict much what is still reasonable.

I do not think i can give a set of axioms i will find satisfying for longer then this moment, right now. What i still miss is axioms of what is considered "a good thing" and a "bad thing", basically, what values morals are based on.
My axiom my morals are based on:
-Could not find a good definition of want. :-/, basically it is rather arbitrary, and based on a beings instincts, culture, i guess.
-Define good as the collective of what conscious creatures want, weighed by their "level of consciousness".
-Axiom: I want is a weight of getting "what i want" and increasing good.
Not that "level of consciousness" is rather vague too :-/, need it to deal with animals, of course.