Yes, but I think that's "accepted" in the sense of "until we run into a contradiction", at which point you have to start over with new axioms. I.e. it turned out your axioms weren't all true.

I think that in the definition you gave "accepted" should be replaced by "assumed".

"John can dig a hole ten feet deep in an hour. How long would it take him to dig a hole five feet deep?"

The first sentence of that problem is an axiom. It is absolutely 100% true, within the context of that story problem. It could contradict another axiom, but it can't turn out to be false, because it's not a statement about the real world.