At first I was thinking about the axiom of choice, but let's keep it general. What motivates the inclusion of new axioms (or change the ones we already have in an already defined axiomatic theory?. It seems that one motivation could be a way to solve problems that couldn't be solved before and are proved to be impossible to solve without the additions of a new axiom.
But this doesn't seem to turn mathematics a little bit upside down?. This is, some axioms seem to be very intuitive -field axioms would be an example- but other axioms -like the ones for topology or some of the ZFC- are instead constructed by having a good idea of the theory that we want as a consequence of the axioms, and even if I agree that axioms should be constructed that way I cannot help to have an uneasy feeling if later the theory is changed, even if this means solving new problems (like if we're changing the rules of the game on the run). So, what conditions would have to satisfy a proposed axiom to be considered, besides consistency with the former theory?, what motivates any change in an already defined theory?
I know this is a somewhat weird question, I just hope having been clear about my question.
Not the same question but maybe relevant: Believing the Axioms by Penelope Maddy.
http://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf http://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms2.pdf