It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's generally accepted that there will never be a proof of the theorem one way or another. My question is, why is this? It seems as though ZF is flawed in a number of ways, since propositions like the axiom of choice and the continuum hypothesis are independent of it. Shouldn't our axioms of set theory be able to give us firm answers to questions like these? Yes, the incompleteness theorems say that we'll never develop a perfect set of axioms, and many of the theorems independent of our axioms will probably be quite interesting, but is ZF really the best we can do? Is there hard evidence that ZF is the "best" set theory we can come up with, or is it merely a philosophical argument that ZF is what set theory "should" look like?