I don't think it's possible to avoid having any axioms at all.
I don't know how to justify one's chosen axioms, because any justification is based on some other axiom... maybe some meta-axioms are required?
Try these.
Metaxioms:
- Axioms should be useful
- Axioms should be minimal
Taken together, I think these form the essence of Ockham's razor at a fundamental level. If someone implicitly uses some different set of meta-axioms, then I don't think it's possible to argue that they're wrong... all one can do is disagree.
Based on those meta-axioms, I like the following axioms:
- Objective reality exists
- My senses give a generally reliable impression of reality
- Past experience is a generally reliable indicator of future occurences (inductive reasoning is valid)
Again if someone chooses not to accept those axioms, I can't argue that they're wrong.