Reality Within the Axiomatic Boundaries

Axiom set A could mandate that (X*Y)+Z is not equal to X*(Y+Z) but axiom set B could make mandate that (X*Y)+Z is equal to X*(Y+Z).

I would think that by definition each is true, within each axiom's "universe." The two are not contradictory, unless it's stipulated further that the two sets represent a union ... then we have a problem.

But I think there are certain "universal" axioms that apply to subordinate axioms, right? Can a mathematician stipulate that two parallel lines on a plane do in fact cross?

Note: From Wikipedia definition of "Euclidian" -- "The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions from those axioms." I find it interesting that it doesn't say something like "irrefutable" or "obvious." I like that phrase: intuitively appealing.

I think that they can in fact stipulate whatever they please. But intuitively unappealing axioms cast a suspect light on the results, right?