Why is ZFC preferred over other set theories?

Question

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

Answer 1

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.