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I was curious why ZFC is preferred over other set theories. Are there specific reasons why? Or is this more of historical reasons?

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Who prefers ZFC? –  Colin Tan Mar 22 '14 at 12:49
Relevant: Why use ZF over NFU? –  Henning Makholm Mar 22 '14 at 13:21

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up vote 10 down vote accepted

First of all we need to understand who prefers $\sf ZFC$. And the answer is pretty much set theorists (and their adjacent mathematical fields). And just to be clear, when I write $\sf ZFC$ I mean any theory which is a "reasonable extension" of $\sf ZFC$ (e.g. large cardinal assumptions, forcing axioms, cardinal arithmetic, and so on, as well $\sf ZF+\lnot AC$ theories).

The working mathematician doesn't usually care about the axioms of set theory, or about set theory. Some of them regard set theory as some "formal safety net" that ensures that what they do can be written in a uniform way in some foundation. Others don't even care about that.

Many people working in category theory prefer to think of other foundations that allow "easier access" to large categories, things like the uprising Homotopy Type Theory (HoTT). Others prefer theories like $\sf ETCS$ or so. There are people who work in constructive systems, which are either similar in flavor to $\sf ZFC$ (e.g. $\sf CZF$), or completely different from it (e.g. Martin-Lof type theory).

So all those people don't prefer $\sf ZFC$, and they often either don't care much for it, or that they look for foundations better suited for their mathematical work.

But what about set theorists? Well, there you also have people who prefer to work in theories like $\sf NF(U),KP$ and other set theories which are weaker or very different from $\sf ZFC$.

However, it is true that a majority of set theorists work in $\sf ZFC$. Why? Well, a renowned set theorist once told me that axioms should be natural enough so you don't feel that you're using them, but rather work with properties that you felt natural for them to be true. And the axioms of $\sf ZFC$ do have this property. Of course, writing down some of the axioms (e.g. replacement) one may wonder why this is true, but it's not difficult to accept these axioms if you think about it for a little bit -- that you want your universe to be closed under definable functions. That is a reasonable thing to ask for.

This is also a historical issue, since we developed intuition which matched those axioms over time, and the notion of set as an element of a universe of $\sf ZFC$ became more and more accepted. And as time goes on, and no contradiction is found in these axioms, it just strengthen the feeling that perhaps this is indeed how sets should behave. So the next generation is being taught that from the get go, and so their intuition is developed to match these axioms, and so on.

So this is both a historical issue, as well the fact that $\sf ZFC$ allows you to work quite naturally without checking your axioms list every time to ensure that you haven't gone outside of its scope -- as $\sf NF$ and $\sf KP$ would require you to do.

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I suspect the "real" question the OP wants to ask is closer to "Why is ZFC the set theory most commonly taught in undergrad math courses?" But that may be different enough from the question actually asked here that it should be asked separately. –  Ilmari Karonen Mar 22 '14 at 14:39
There is another important reason: Only $\mathsf{ZFC}$ and its close variants are developed enough that independence results in any field can be (routinely) set up and handled (through the set theoretic forcing method). –  Andres Caicedo Mar 22 '14 at 18:50
@Andres: True, but one can argue that if we spend enough time in other set theories we might find analogies of forcing that can be implemented there. –  Asaf Karagila Mar 22 '14 at 20:33
Excellent, excellent answer. I especially like this bit: "Axioms should be natural enough so you don't feel that you're using them, but rather work with properties that you felt natural for them to be true. And the axioms of ZFC do have this property." That is so, so true, and imo the same can be said for ETCS and SEAR. On the other hand, I'm not convinced that NFU has this property. –  goblin May 20 '14 at 4:01
@user18921: I never read about SEAR properly. I found it searing to read about type theory which is meant to be a replacement to set theory which is based on category theory (at least in the philosophical approach). I also never found ETCS natural to work with because, frankly, I never found diagrams and arrows natural to work with. –  Asaf Karagila May 20 '14 at 7:19

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