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?



The Importance of ZF
Very few mathematicions these days wish to base their mathematics on ZF without the axiom of choice, as your question seems to imply. Yes, there is an intuitionist school about which I don't know a whole lot, but they seem to be quite the minority. So let's consider ZFC. I think the main philosophical argument for it is that the axioms seem obviuosly true, if you are willing to believe that such things as infinite sets exist in some fashion, and that nobody has been able to come up with an equally obviously “true” statement about sets that is not a consequence of these axioms. The independence of the continuum hypothesis doesn't seem to bother people much, since to most of us it doesn't seem either obviously true or obviously false. Though some people are working on settling it by finding other axioms that will at least be considered likely true or at least useful. There was a recent article in the *Notices* about these efforts, in fact. But is there “hard evidence” that ZFC is the best we can come up with? Not by most mathematicians' standards I think, but there seems to be plenty of soft evidence.


I might add that many working mathematicians (I am talking here mostly about analysts, since they are the people I know best) don't care one whit about these questions, but happily go about their business using Zorn's lemma whenever it seems necessary and never let it bother them. And there are some who would rather base all mathematics on category theory rather than ZFC set theory, but that is not my cup of tea, so I will leave it unstirred.


