[general_re]

If the premises you use lead you to conclusions that are contradictory, if your reason (logic) is otherwise correct, one or more of your premises is incorrect.

True conclusions don't mean your premises were necessarily true. False premises, true conclusion:

All cats are birds.
All birds are mammals.
Therefore, all cats are mammals.

Axioms are not assumptions

At their core, they are - or more accurately, they are assertions without proof. Even if you think you can inductively reason your way into axioms, you're just pushing the assumption back a step by assuming that the inductive principle is itself true, when it is definitely unproven, and probably unprovable.

[Roscoe]

Even if you think you can inductively reason your way into axioms, you're just pushing the assumption back a step by assuming that the inductive principle is itself true, when it is definitely unproven, and probably unprovable.

No problem. Just call the assumptions self-evident.