Here is the paragraph of the faq most related to vagueness.

MathOverflow is not an encyclopedia. MO is a site for questions that have answers. MathOverflow visitors should know how to learn new things and do mathematics on their own, but we all get stuck sometimes, and this is where MO saves the day. When you're stuck, you can come to MathOverflow and say "I'm trying to do X. How can I do that? Does this work? Does anybody have a reference?" The idea being that for an expert, it should take very little effort to understand your confusion and set you on the right path. Or maybe a non-expert has come across the same sticking point and can explain how she resolved it. MathOverflow is not the appropriate place to ask somebody to write an expository article for you. If you want somebody to write an article about some subject, you should make a stub on Wikipedia or make a query block on nLab.

As you can see, the vagueness issue is not a judgement of academic or mathematical merit. The issue is that vague questions interfere with the functionality of MO.

MO can be very time consuming, partly because it is so addictive and partly because it requires time to write a thoughtful answer. Vague questions make this even worse, no matter how interesting or meritorious. Closing vague, discussiony, argumentative, and subjective questions is essential to maintain the functionality of MO. It ensures that experts like you can drop by when they have spare time, answer a question or two without having to worry about extraneous context, and then leave to go about their regular business without lingering aftertaste.

I propose the following heuristic: if somebody with vast knowledge of the subject (who would clearly know the answer if there were such a thing) is likely to waste time trying to decide on an answer to give (among many possibilities), then the question is probably vague enough that it should be closed.

Of course, a principal merit of a question is the ability to lead to very good asnwers.

This is going a bit off the topic, but I want to stress that I don't think this is much of a merit. For example, consider the question "Tell me something interesting." Ignoring the fact hat it's not a question, it can certainly lead to fantastically interesting answers. However, it's a terrible question. Those fantastic answers could not properly engage the question or the questioner: they would be better off as articles. See Andrew Stacey's excellent post on this.†

†This is meant to say that I think that particular post is excellent, not that Andrew has a unique excellent post.

There are many results in number theory that I accept as proven because they (ultimately) argue from basic properties of the standard model of the natural numbers. The arguments are often second order (e.g. involve inductions which quantify over subsets of the standard model of the natural numbers). I don't know whether these can all be rewritten in some first order way, and to what extent they can be reduced to Peano arithmetic. (For example, I don't know whether FLT is first order derivable from the Peano axioms, but I certainly accept it is as a true result about the standard model of the natural numbers --- which are the numbers I personally care about.) The point is that, in the end, these results are true because they argue from true properties of numbers; not because they argue from certain axioms. (After all, we know by Godel that the true properties of numbers can't be captured by a recursively enumerable axiom scheme.)

As for (b), just as I don't think that number theory is the same as Peano arithmetic (number theory is about the standard model of the natural numbers, which are not captured by the Peano axioms), I don't think that set theory is necessarily what is captured by ZF, or other related axiom schemes. (I certainly have less intuition here than for number theory, but to the extent that I understand Godel's position on this, I probably agree with it. So I would probably be labelled a Platonist.)

It seems to me that taking a constructivist position is nothing more than saying that one doesn't like a certain choice of axioms, and therefore that all results derived from those axioms are meaningless, regardless of the work it took to get those results. It's an inherently argumentative position.

To respond to a.) I pose the following question: Would you accept a poposition as proven if it doesn't follow from a combination of axioms and hypotheses (which are also axioms in the theory defined by adjoining the hypotheses)?

And for b.), I was using ZF to mean classical ZF on one side and constructive ZF on the other as "common ground", so to speak. AC is independent of both, and adjoining AC to CZF gives you a theory equivalent to ZFC by Diaconescu's (spelled incorrectly?) theorem.

To say that "The former is simply a statement of fact, while the latter is a statement of opinion about things about which one has no business giving opinions (namely the validity of axioms that are otherwise independent of set theory)" is to state an opinion, namely that (a) mathematics is about something described by axioms; (b) set theory is about what is described by ZF. Not every mathematician believes either (a) or (b) (personally, I believe neither). There are other views of mathematics besides the formalist view, and questions of this kind invite people to explain (albeit in a sometimes indirect way) the various views of mathematics that they hold. I think that such explanations can be valuable and interesting, and I would guess I'm not alone in that.

(+1 François. Thanks for starting this thread. I hope this system catches on.)

Questions should be closed or left open based exclusively on their own merits (or given recent events, also on the merits of the questioner), not the merits of their answers.

Mathematics is big enough for constructivists and classicists to live. There's a difference between saying "I do constructive mathematics" and "I am a constructivist". The former is simply a statement of fact, while the latter is a statement of opinion about things about which one has no business giving opinions (namely the validity of axioms that are otherwise independent of set theory).

Given Kaplansky's comment, and some sign that several people agree with it, I don't think it's unreasonable for someone to ask "what's the deal with countable choice vs. choice", which is essentially what the second question asks. (Incidentally, I don't see how it duplicates the first question, which was not in the least about the issue of countable vs. full choice.)

If this question were closed, I would immediately vote to reopen it.

I think this question should be fairly heavily edited. As it is, it is argumentative and subjective, and does not contribute more to the discussion of AC than what is already available on the other questions on the subject, including the linked question. I vote to close.

The "other question" is 22927.

(I'm not the other voter, but I want to give this system a try.)