2) Mathematics is the subject in which, regardless of what the authorities tell you is true, you can verify every last iota of truth, with a minimum of equipment.
Although seldom noticed, mathematics is one of the few, maybe the only, subject not open to interpretation.
e.g. Some say the set of natural numbers contain 0, some say it doesn't.
Granted, the kind of math being discussed here isn't open to much interpretation, but I thought I'd note it.
Using game theory to make predictions, expected utility theory, etc the list goes on and on (sorry to pick on economics, I'm sure there are plenty of other fields that suffer the same problems).
Just look at some of the assumptions made by the financial engineers who caused the recent financial crisis if want your head to explode: http://en.wikipedia.org/wiki/Black–Scholes#Model_assumptions
In fact, the problem with the axioms your talking about isn't that the math is wrong, its that the axioms aren't "true" of the real world.
Not that what you are saying isn't important, but you are talking about two completely different things. I mean, compare the axiom of choice to "people maximize expected utility." One of them is about the pillars of mathematics/logic, one is about how people act.
There may be disagreements in matters of taste as to what Choice of axioms (har har, little joke there) one makes, but given a basis of axioms to work from--rarely more than a handful--the rest is, for most purposes, not at all open to interpretation.