Here is a view of axioms (and mathematicians) from the poet, Katherine O'Brien: (Please pardon the sexist language!)

Mathematician

In midair somewhere
he lays an axiomatic floor.
On it he sets a hypothetical plank
on which he raises a logical ladder
which he proceeds to climb.
There is risk, suspense and drama:
any loose rung, any misstep fatal.
At the proper confluence of space and time
he steps off onto a higher platform
with a broader panorama.

The whole thing is fabrication.
But so was creation.

In the 1999 spring term Geometry Seminar, now called Explorations in Geometry, and in the fall term 2002 Exp's in Geom students listed several qualities that would be desirable for a collection of axioms for a mathematical system to have. First, the 1999 list:

1. Specific in language and easy to understand

2. General enough to apply to a wide range of situations

3. Simple and most basic

4. Creates one unique idea

5. None is a consequence of the others

6. Thorough

7. Non-contradictory

And, the 2002 students' list:

1. When combined, one can make more complex constructions or deduce more complex results

2. They can apply to different-looking situations

3. They are consistent, non-contradictory

4. They have simplicity and generality

5. They are all clear, concise, and undstandable

6. They shouldn't rely on specific measures

7. There should be no redundancy

8. All terms should be defined, or be clearly labeled as undefined