The problem is that some concepts might not have an "isomorphism" to the real world (as Hofstadter calls it). In fact, finitists believe that the concept of infinity possibly introduces some discrepancy, making the "casting" a lossy transformation.
Presumably the axiom of choice is a good example of something unlikely to be applicable in a real world context. Unless you're trying to anticipate when a rogue mathematician is about to attack, of course.
