TITLE: The Importance of ZF
QUESTION [17 upvotes]: 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?

REPLY [3 votes]: This answer is essentially a Joel's version by another route.
ZF(C), possibly with appropriate large cardinal axioms, is one of the three most important formal axiomatisations in the foundations of mathematics, because it is foundationally complete (Friedman 1997):

The usual set theoretic foundations is very powerful, coherent, concise,
  successful, explanatory, impressive, and totally dominating at this time.
  Taken as a whole, with the major supporting classical developments, it is
  certainly one of the few greatest acheivments of the human mind of all time.
However, it also does not come close to doing everything one might demand
  of a foundation for mathematics. At the present time, there is no full
  blown proposal for scrapping it and replacing it with anything
  substantially different that isn't far more trouble than its worth. Present
  cures are far far far worse than any perceived disease.

...

Now before I remind everybody of some of the most vital features of the
  usual set theoretic foundations for mathematics, let me state a great,
  great, great, theorem in the foundations of mathematics:
THEOREM. Sets under membership form the simplest foundationally complete
  system.
There is one trouble with this result: I don't know how to properly
  formulate it. In particular, I don't know how to properly formulate
  "foundational completeness" or "simplest."