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?
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.

3$\begingroup$ I would disagree with the statement that "nobody has been able to come up with an equally obviously “true” statement about sets that is not a consequence of these axioms". I would argue that the statement "all sets of reals are Lebesgue measurable" could be considered "obviously true" in the same way as the axiom of choice. I would also imagine that working in a model of set theory where all sets of reals had Lebesgue measure would be something very appeal to analysists (although I have no empirical evidence for this fact). $\endgroup$ – Nate Ackerman Aug 16 '12 at 2:51

2$\begingroup$ I also heard (although it is folklore) that when Solovay first constructed his (here's an inline link) model in which all sets of reals are Lebesgue measurable and the axiom of dependent choice holds, he believed that at least some analysts would prefer to work with it than with the axiom of choice. $\endgroup$ – Nate Ackerman Aug 16 '12 at 2:54

2$\begingroup$ Nate, Solovay himself told me precisely that, so I don't consider it merely folklore. To my way of thinking, the fact that most people did not give up AC and adopt that theory illustrates the enormous intuitive pull of the axiom of choice: it is far stronger than people sometimes describe. $\endgroup$ – Joel David Hamkins Nov 16 '12 at 13:53
There is a very active ongoing debate within set theory about whether mathematics needs new axioms, and philosophers of mathematics are weighing in on all sides. Relevant considerations include many very deep topics in set theory, including independence, forcing and the large cardinal hierarchy. Some of these topics are at once highly technical and philosophical at the same time. It is fair to say that there is an emerging field called the philosophy of set theory that is grappling precisely with these issues.
Let me try to mention just a few of the considerations. First, the historical fact remains that the ZFC axioms are sufficiently powerful to carry out almost all of the construction methods that arise in mathematics outside set theory. Indeed, the ZFC axioms are provably far more powerful than necessary for the vast majority of ordinary mathematics. This is proved by the stunning results of the field of Reverse Mathematics (see Steve Simpson, Harvey Friedman etc.), which calculates for a huge collection of classical mathematical theorems exactly which axioms are needed to prove them. Reverse Mathematics proceeds by proving the axioms from the theorem as well as the theorem from the axioms (over a very weak base theory), thereby showing the necessity of those axioms, and it turns out that most all of the classical theorems of mathematics can be proved in relatively weak theories.
Nevertheless, within set theory, set theorists have discovered the ubiquitous independence phenomenon, by which an enormous number of settheoretical assertions turn out to be independent of the ZFC axioms. This means that they are neither provable nor refutable in ZFC. We now have thousands of instances of fundamental set theoretic propositions that are known to be independent of ZFC. This includes almost any nontrivial statement of infinite cardinal arithmetic (such as the Continuum Hypothesis), as well as an enormous number of statements in infinite combinatorics, and so on. This phenomenon supports the view that ZFC is a weak theory, unable to decide these questions.
But of course, by the Incompleteness Theorem we know that any theory we can write down will exhibit this independence phenomenon. It is impossible in principle to avoid it.
Large cardinals are strong axioms of infinity, some of which go back to the time of Cantor (so they are not new), which are not provable in ZFC and which transcend ZFC in consistency strength, forming a vast hierarchy of consistency strength above it. Thus, they tend to make up for the weakness of ZFC (although there remains extensive independence even with large cardinals). Some set theorists make the case that the existence of large cardinals have numerous attractive regularity consequences, even for low down for sets of reals, that they seem to point the way towards the finally true set theory, which must remain elusively hidden from us because of the Incompleteness theorem. Making sense (or nonsense) of this view is a central concern of the emerging Philosophy of Set Theory.
People use whatever is most useful. ZFC just happens to be a fairly simple formalization of the way people think of sets such that we can eliminate imprecision sufficiently to do good mathematics. There are no "theorems" independent of ZFC in ZFC. CH is not a theorem in ZFC. Choice isn't either, but on the other hand choice is useful so everyone uses it. If you don't accept the axiom of choice then you can't have things like arbitrary products, and for some strange reason studying infinite products is actually very useful to practicing mathematicians.
That's not to say that studying things like CH isn't useful. I think it is, but right now it's just not pertinent to most mathematicians.

1$\begingroup$ Well, the axiom of choice, being an axiom of ZFC, is a theorem of ZFC. (Yeah, that's nitpicking …) $\endgroup$ – Harald HancheOlsen Nov 14 '09 at 23:23

3$\begingroup$ I'd dispute that ZFC is a formalization of the way people think of sets. The axiom of foundation, for instance, says that every nonempty set X has an element whose intersection with X is empty. For instance, it says that there is some real number none of whose elements are real numbers. What does that even mean? Does 6 have that property? Does pi? Most people wouldn't dream of talking about "elements of a real number". Those that would, only would because they've taken a course on ZFC. $\endgroup$ – Tom Leinster Nov 21 '09 at 1:00

2$\begingroup$ @Tom: I can understand objecting to an axiom when one has what might be a counterexample, but you seem in contrast to object to the Foundation axiom on the grounds that it is obviously true for sets of reals. After all, we all agree that no real number is an element of another. The idea that sets are built up in a wellfounded cumulative hieararchy has been present in set theory since the earliest days of Cantor, and was part of the preaxiomatic picture. $\endgroup$ – Joel David Hamkins Dec 29 '09 at 21:52

3$\begingroup$ Joel: my objection isn't that Foundation is "obviously true" for sets of reals, it's that it doesn't make sense for sets of reals. That is, it contradicts how most people think about sets, because it involves considering "elements of a real number". This is an objection not to Foundation specifically, but to the basic setup of ZFC, wherein elements of sets must themselves be sets (so one can always ask "what are the elements of the elements?") In my view, the "sets" axiomatized by ZFC are so far from what most mathematicians mean by the word "set" that it's misleading to use the same word. $\endgroup$ – Tom Leinster Jan 15 '10 at 13:07

1$\begingroup$ Tom, in the early days of set theory, people had used real numbers and other mathematical objects as urelements, for precisely the reasons you mention. But it was soon realized that one doesn't need urelements for any foundational purpose, since one can build a copy of the reals using just the pure sets. We don't care what the real numbers are made out of, as long as they are the unique complete order in which the rationals are dense. And since the set theory is far more elegant without urelements, they were abandoned. $\endgroup$ – Joel David Hamkins Feb 9 '10 at 16:37
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."

$\begingroup$ "simplest" probably the most complicated to define. ZF is not finitely axiomatisable, but some category theoretical variants are. If an axiom schema only 'counts' as much as an axiom, then ZF has a shorter list of axioms+schemas, but... $\endgroup$ – David Roberts Jun 7 '10 at 6:41

$\begingroup$ @David: Maclane's categorical set/type theories are not foundationally complete, in Friedman's sense. $\endgroup$ – Charles Stewart Jun 7 '10 at 9:12

2$\begingroup$ I didn't see Friedman's sense of "foundationally complete" from the link provided (and actually, I was actively following the FOM discussions during that time). The responses to categorical/structuralist foundations (advocated by McLarty, Awodey, and others) by Friedman, Simpson, and others committed to materialist foundations were, IMO, disappointingly shallow and ridiculously emotional (hostile). The question Friedman raises about V(w+w) is very interesting however (and particularly interesting for the question of adequacy of Mac Lane's set theory!). $\endgroup$ – Todd Trimble♦ Jun 7 '10 at 16:14

$\begingroup$ @Todd: I did not find the discussion shallow, and I did not find Friedman to be hostile, although I entirely agree that Simpson framed the whole issue in a pointlessly divisive manner: the list1 vs. list2 distinction was presented almost as a matter of moral character. I agree that justice was not done to the structuralist viewpoint, but I still think that Friedman is right to say that categorical logicians (among others) have not presented a genuinely independent ontology for mathematics that achieves what modern set theory does. $\endgroup$ – Charles Stewart Jun 10 '10 at 11:27

2$\begingroup$ "Friedman is right to say that categorical logicians (among others) have not presented a genuinely independent ontology for mathematics that achieves what modern set theory does." I am afraid I don't understand what is trying to be said here. "Genuinely independent" in particular sound like weasel words to me, and the use of the word "ontology" is not very clear to me. Insofar as there are structural set theories which have the same strength as membershipbased set theories, can you clarify what precisely is lacking in structural set theories? (Only saw this after > 2.5 years; sorry.) $\endgroup$ – Todd Trimble♦ Dec 29 '12 at 20:54
There is Freiling's axiom of symmetry (AX) it is equivalent to the negation of the continuum hypothesis more on this is here:
http://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry
For a categorical foundation of mathematics look here:
ZFC is only "standard" because it (or conservative extensions of it like Universes) allows us to do pretty much whatever we want. However, recent developments in higher category theory have led some to call for a new "set theory" (mentioned above) that categorifies the classical theory of sets and cleanses the "evil" from it. (Evil, of course, in the sense of considering objects up to equality rather than isomorphism.)
It seems unlikely, to me at least, that someone will be able to develop a useful alternative set theory that isn't equivalent to or stronger than ZFC, the operative word there being useful. That is, we don't want whole edifices of mathematical thought falling into the ocean under our new set theory.

$\begingroup$ In particular, see ncatlab.org/nlab/show/ETCS . $\endgroup$ – Qiaochu Yuan Nov 14 '09 at 21:09

$\begingroup$ To expand on Qiaochu's comment: ETCS (Lawvere's Elementary Theory of the Category of Sets) is a counterexample to fpqc's feeling. That is, it's a useful alternative set theory that isn't equivalent to or stronger than ZFC. It's weaker, but still allows development of a very great deal of mathematics (say, all of a typical undergraduate programme). Compare Joel David Hamkins's answer, where he points out that "most" known theorems can be proved using only a relatively weak set theory. $\endgroup$ – Tom Leinster Nov 21 '09 at 0:54

$\begingroup$ Regarding the parenthetical remark in the first sentence, the existence of universes is not a conservative extension of ZFC, since from the existence of a universe, one can prove new assertions, even new arithmetic assertions such as Con(ZFC), that we cannot prove in ZFC assuming it is consistent. $\endgroup$ – Joel David Hamkins Nov 15 '12 at 16:53
I think, the worst thing that can happen to mathematics is to impose some dogma onto it. And the worst kind of dogmas are those concerning foundations. It is in the essence of mathematics that an object has its existence justified as soon as it is clear that this object is welldefined. And that´s really all about it. There is no usefulness or appropriateness or 'better' when it comes down to laying the foundations. For axioms are not subordinated to their implications. As for the so called 'intuition', the latter tells me that $\mathbb{Q}$ has greater cardinality than $\mathbb{N}$... In other words I don´t see a single reason why we shouldn´t have 'multiple' mathematics, each based on its foundations, each equally justified!
I am a bit uncomfortable with your wording; you say “it seems that ZF is flawed” or “ZF is the "best" set theory we can come up with?”…
I don’t think that the words “flawed” and “best” are suitable… Mathematics is what mathematicians do, just as Art is what artists do and there are lots of mathematicians doing lots of mathematics.
Some are interested in ZF others in Elliptic curves. Considering ZF as a foundation for “all of mathematics” is like considering music as a foundation of “all of art” or Physics as a foundation of “all of science”… Mathematics, Art, Science, Philosophy… have no foundation or if you prefer studying foundation*s* is just one of the many branches of a “discipline” one of many things done by human beings for pleasure.
The fact that axiom of choice is independent of ZF and the fact that one might want to consider this axiom as true is just a very interesting question about the hidden puzzles of choice and infinity precisely sort of puzzles that stimulate, motivate and inspire those who like to study them. Wanting to have a framework that settles this precise question might excite some but bore others.
Personally I think that event the “finite version of the axiom choice” needs thought…

2$\begingroup$ I would disagree with it being inappropriate to call ZF a foundation for all of mathematics. People talking about elliptic curves still do need some foundation, and it is at least convenient to have a foundation which is fairly similar to ZF, if not ZF itself. ZF(C) is important because it is a single choice of foundation which has proven to be appropriate for doing (almost?) all of mathematics. In many areas of mathematics it is not important whether you are exactly using ZFC or some variant of it, but having some foundation matters. $\endgroup$ – Eric Wofsey Nov 15 '09 at 1:11

1$\begingroup$ @Eric: I don't understand your first sentence. As for your other sentences, I don't agree that e.g. people talking about elliptic curves need some foundation. For example, I think that I personally would still be philosophically pretty comfortable if there were no good "foundation for mathematics". If, say, ZFC somehow collapsed tomorrow, I think for the most part I'd carry on just the same, as if nothing had happened. $\endgroup$ – Kevin H. Lin Nov 20 '09 at 23:03