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?