I don't like axiomatization in finance. I believe you should learn things more sloppily the first time around, and get a visceral feel for the subject. That's certainly the case if you're going to be a practitioner. Axiomatization should be left for a second course.
The trouble is that many students get their first exposure to finance and options pricing through an axiomatic approach. As a result, I interviewed many people at Goldman Sachs who were looking for jobs, and when I asked them to explain in simple terms why you could value options, many of them looked puzzled for a while, and then started talking about Girsanov's theorem. This is a little in flavor like asking people to explain why you can send a rocket to the moon and they begin by talking about Hamilton's equations, or worse, tensor calculus rather than Newton's laws. But it's even worse that: axiomatization is OK in physics because the laws it leads to work so well. In finance, the power of axiomatization is largely wasted.
I think avoidance of axiomatization in first courses is the right thing to do in physics too. In first learning about electricity and magnetism, observation should come first, and Maxwell's equations should be slowly deduced from electrostatics and magnetism and the forces between currents and, finally, Maxwell's leap of intuition about the displacement current. Instead, I've seen junior-level electricity and magnetism texts start with Maxwell's equations in tensor form: D mew F mew new = J new.
Axiomatization is very powerful in mathematics and leads to results you didn't think of before. But in the sciences, I have a sneaky feeling that axiomatization takes over when people run out of good ideas.