I've got a post up on the teaching blog on the case against axiomatic rigor in lower level classes, but the best part is probably this anecdote.
A few years ago, when I was teaching math at a big state university, a colleague told me the following.
She was comparing notes with a professor at a nearby school on how their respective real analysis courses were going. She told him that they had just proved that the square root of two was an irrational number. He laughed and said she was way ahead of him; his class had just proved that the square root of two is a number.