Why can mathematicians pick and choose axioms?

Obviously, there need to be some 'self-evident' truths that we can't prove, but on which we base certain theorems; e.g. the axiom of choice lead to the well-ordering theorem (which could well be an axiom).

Essentially, what I'm asking is how do mathematicians know what is "so trivially true, that it is assumed to be true"?

For example, Playfair's axiom/the parallel postulate does not seem to be trivially true (at least to me, and from what I gather, many mathematicians tried to prove it), so why did was this axiomatised?

I understand that we can construct useful things from these unproved assumptions, like, for example, the reals, from the axiom of completeness, but, unless it is a defining property, how can one say for certain "statement $X$ is unequivocally true"?

Finally, are there axioms yet to be discovered, and, if so, who determines their validity?

• To answer your title, mathematicians pick and choose axioms to simplify and define concepts elegantly. Also, it would be helpful to define "trivially true". – user122283 Mar 26 '14 at 19:44
• An axiom is not something which is true, but something which is assumed to be true and from which you derive conclusions. For a consistent theory you usually need several axioms. As soon as you need more than one you are faced with the question whether they are independent of each other and do not contradict each other. The challenge is to formulate a set of axioms as small as possible which implies results which correspond to our intuition. – Thomas Mar 26 '14 at 19:52