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.
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.