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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?

Cato
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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.

Asaf Karagila
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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
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    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). – Andrés E. Caicedo Mar 22 '14 at 18:50
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    @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 GONE May 20 '14 at 04:01
  • @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 07:19
  • Oh I fully agree. My claim is not that SEAR or ETCS are super-intuitive or anything like that. Arrows are hard for me, too. Its just that (like ZFC) they don't feel like formal systems. Rather, they feel like mathematics-as-usual. – goblin GONE May 20 '14 at 07:40
  • As for type theory... yes. I still haven't worked out how to learn it, either. I think the idea is to play around with a functional programming language a lot (like Haskell), and then with an interactive theorem-prover (like Coq). Simultaneously, you have to learn a lot of homotopy theory and/or homological algebra. Then one day, you suddenly experience "nirvana" and get it. At which point you can actually start learning from books and stuff. (But what is there to learn? Its not clear to me what we learn from type theory.) – goblin GONE May 20 '14 at 07:48
  • This answer still doesn't give a reason why people like to use the axiom of choice. It says people have so much intuition of the axioms that they don't feel like they're using them but I really don't have much of an intuition that the axiom of choice must be true. – Timothy Jan 13 '18 at 19:04
  • @Timothy: Yes, every rule has an exception, this is not a mathematical proof. And let me ask, are you currently on a path to become a mathematician? Are you studying mathematics in the structured way of taking several topics in a reasonably good order as building blocks? Or are you mainly reading here, on Wikipedia, watching YouTube, etc. on set theory? Because if it is the latter, then I'm sorry to tell you, but your intuition doesn't even count for me as far as this question and answer go. – Asaf Karagila Jan 13 '18 at 23:03
  • At the moment, I'm not working my way to be a mathematician. I got a Bachelor's degree but didn't do graduate school. I discovered Cantor's paradox by myself before I ever learned about the inconsistencies of Naive set theory like Russel's paradox then I later discovered that I can't prove the axiom of choice without having first learned that it wasn't provable. I never took a course about axiomatic set theory and think that in Modern algebra, I was supposed to write proofs in Naive set theory and hope the proof doesn't end up something like proving it from Cantor's paradox. – Timothy Jan 14 '18 at 23:43
  • @Timothy: Well, then, congratulations. You are the exception to the rule. Most people in the world would believe that if you can prove that an inductive definition of $x_n$ has some "reasonable pool of candidates" for $x_{n+1}$, then there is a sequence satisfying these properties. Even for the real numbers this is not provable without the axiom of choice. Most people would argue that the product of non-empty sets shouldn't be empty, well, because we can choose representatives to construct a choice function. These are all intuitive. You're different. Congratulations. – Asaf Karagila Jan 15 '18 at 07:16
  • (Also, I don't know how you can "discover" unprovablility without formal argument, the most you can is have an intuition that something is unprovable.) – Asaf Karagila Jan 15 '18 at 07:16