Independent sentences and consequences for set theory Disclaimer: I am not a set theorist; but I am a mathematics student with an interest in the foundations of my subject! --- Ultimately, after I blabber on, this question is a reference request for some research programs in foundations.
ZFC is one set theory that (most) mathematicians (if pressed) have decided is the foundation of mathematics. As such, one would hope that we could answer any questions we have about sets with this theory. Of course, Goedel and Cohen showed us that the continuum hypothesis CH is independent of ZFC.
One reaction (that I have) to independent sentences, such as CH, is that ZFC is not quite the axiom system that we want. That we should either (i) start again or, (ii) add axioms in to get a concrete decision on CH.
I don't think adding CH as an axiom is a good idea; for that does not (by itself) teach us anything. I also understand that any adjustments we make to ZFC which give us a decision on CH will be equivalent (with in ZFC) to adding in CH as an axiom. 
However one can imagine adding in a(some) simple/intuitive axiom(s) which, on the surface, don't look or feel like CH (nor, $\lnot$ CH) but ultimately allow us to get a concrete answer as to the truth of CH in this expanded system. 
Question: Some people must have pursued this idea, or shown that it is hopeless in some sense. Could someone point me to places to read about research on this? Or give me some keywords to search for? Or tell me why my hopes and dreams are impossible? 
Question: What would it take for ZFC to be overthrown as the foundation of mathematics? Are there any set theories that are gaining large numbers of followers? --- There is no reason why we got it right the first (I know ZFC is not the first set theory) time, so there could be a successor one day. I would think deciding CH would be a good reason to accept a new set theory. 
Finally, if this question is hopelessly naive, please do tell me!
Thanks. 
 A: No, there are no alternative set theories comparable to $\mathsf{ZFC}$ and its extensions in terms of expressive power and malleability. What I mean by this is that there are many alternative set theories (see here, for instance), but they are either too weak in terms of consistency strength or in terms of expressive power, or have other problems to consider any of them as a serious alternative. (This has nothing to do with whether they are interesting objects of study in their own right, of course.) 
For example, New Foundations ($\mathsf{NF}$) is awkward, although some people like it; but we do not even have a sketchy picture of what a model of $\mathsf{NF}$ should look like. We do not know that $\mathsf{NF}$ is consistent (in spite of recent claims), but what we expect is that in consistency strength it is much much weaker than $\mathsf{ZFC}$ (if the recent claims end up working out, this will now be a theorem rather than just an expectation), and therefore its usefulness as foundations is limited. There are variants, like $\mathsf{NFU}$, that are more flexible, and better understood, but this understanding comes from the fact that they end up being much closer to $\mathsf{ZFC}$, so they end up not being a serious alternative (more like a repackaging).
(Note that the claim that $\mathsf{NF}$ is awkward is not simply a bias of mine. Randall Holmes, one of the leading experts in this theory, is very open about the same problems, and he has readily admitted that $\mathsf{NF}$ does not feel like a viable foundational system for mathematics.)
I've heard people argue that some of these weak alternative set theories suffice to formalize “everyday mathematics”, but this is not serious. I do not want to formalize only “everyday” mathematics, nor do I feel that the term is meaningful.
There are, on the other hand, several other (non set-theoretic) foundational programs (for instance, see here). The most significant right now is based on Voevodsky's proposal (HoTT). It is true that some category-theoretic constructions are not readily interpretable within $\mathsf{ZFC}$, but this is not the issue. Similarly, several foundational programs based on or inspired by category theory feel... strange... handling certain set-theoretic constructions. I think one is missing the point if one think that this is the reason to choose one system or the other. See here.
A very nice recent essay by Penelope Maddy discusses one possible way of thinking about foundations, see here.

MR3656316. Maddy, Penelope. Set-theoretic foundations. In Foundations of mathematics, 289–322, Contemp. Math., 690, Amer. Math. Soc., Providence, RI, 2017. 

It should also be pointed out that for certain questions, $\mathsf{ZFC}$ ends up being too blunt an instrument. The program of reverse mathematics, for example, studies the question of how different theorems (“of everyday mathematics”, he added jokingly) relate to each other. Of course, any two theorems imply each other, so what one means is, working over a very weak but still reasonable basis, weak enough so that neither statement is now provable, can we now determine whether one of these statements implies the other? The appropriate setting for these questions, at least currently, is that of subsystems of second-order arithmetic. Of course all these systems are formalizable in $\mathsf{ZFC}$, but even people that do not feel (for philosophical or other reasons) that the endless infinities of $\mathsf{ZFC}$ are meaningful, may work in and appreciate reverse mathematics from the perspective of someone interested in foundations. 
(Solomon Feferman is perhaps the most recognizable logician who worked in foundations and did not feel uncountable infinities were entirely meaningful. Of course you want a different foundations of mathematics than $\mathsf{ZFC}$ if that is the case.) 
Honestly, I do not know if univalent foundations or some other approach will in the long run end up being a preferable system by the majority of mathematicians. Currently, I feel that in $\mathsf{ZFC}$ we have the forcing construction (and decades of experience with it) and the ladder of large cardinal axioms, and there is no serious alternative to these two tools in other systems to make them viable to handle the plethora of incompleteness phenomena that we know are inescapable to any foundational studies. This is not to say that only $\mathsf{ZFC}$ and its extensions can deal with incompleteness. It is really a matter of malleability, and it may very well be the case that in twenty years or so other approaches will allow us the same or greater ease.

Several times above I've mentioned “$\mathsf{ZFC}$ and its extensions”. The reason is that, as one readily notices, $\mathsf{ZFC}$ ends up being too weak a theory for most purposes (settling the continuum hypothesis is but one of them, we now have many examples of natural unprovable statements). 
Large cardinal axioms provide us with a standard way of extending $\mathsf{ZFC}$. The expectation is that any natural theory we end up considering will be (provably in a very weak system) equiconsistent with some extension of $\mathsf{ZFC}$ via large cardinals. This is a very useful guiding heuristic and in practice we have confirmed it many times (although at least one set theorist does not expect it to be true). 
Large cardinals on their own do not settle $\mathsf{CH}$ one way or the other, but several extensions of $\mathsf{ZFC}$ do, and some of these extensions are even considered natural by some. The main examples of such extensions are the so-called strong forcing axioms. An excellent paper dealing with some of these ideas is 

MR3656315. Moore, Justin Tatch. What makes the continuum $\aleph_2$. In Foundations of mathematics, 259–287, Contemp. Math., 690, Amer. Math. Soc., Providence, RI, 2017. 

There are other extensions of $\mathsf{ZFC}$, incompatible with forcing axioms, that also settle $\mathsf{CH}$, and that can be considered natural, or desirable, for a variety or reasons. For instance, Freiling's axiom of symmetry, or the existence of an extension of Lebesgue measure defined on all sets of reals. 
My personal view is that large cardinal axioms are nowadays very much accepted (even if not universally) by the mainstream set-theoretic community as part of the  “correct” formalization of set theory beyond $\mathsf{ZFC}$. Large cardinals formalize the idea that the universe of sets is tall. Forcing axioms and, more precisely, strong reflection principles, formalize the idea that the universe is wide. But I do not think there is currently near enough agreement within the community about their “correctness”. Whether forcing axioms, or another family of statements, will end up being embraced this way, it seems too early to say. But yes, the idea of looking for meaningful extensions of $\mathsf{ZFC}$ that end up settling $\mathsf{CH}$ has been seriously pursued and is still the subject of active research interest.
(And I'll stop here, although there is much more that could be said.)
A: The question is not naïve at all, and if I'm not mistaken, one of Gödel's goals was to find new "self-evident"/"intuitive" axioms for set theory that may decide (prove or refute) the continuum hypothesis. 
One of the ways of doing this is currently seen in reasearch in the area of large cardinals: large cardinal axioms are axioms asserting the existence of certain cardinals that are often very large (most if not all of them are inaccessible cardinals, meaning that they cannot be "reached" from below through power set operations or taking limits of smaller cardinals). There are also other avenues for deciding (more or less concretely) CH, such as Woodin's $\Omega$-conjecture (though admittedly I don't know anything about it), or simply looking for other kinds of intuitive axioms.
The largeness properties of the large cardinals often entail that they have very important implications, most (if not all) of them entail for instance that ZFC has a model, in symbols $C \implies Cons(ZFC)$ where $C$ is such a large cardinal axiom. By virtue of Gödel's second incompleteness theorem, this implies that one can never hope to prove that these axioms are consistent with ZFC (even assuming ZFC is consistent), so in a sense accepting them would be a "leap of faith" (unlike accepting AC, which is known to be "not more harmful than the rest of ZF"): all one can hope for is a refutation of their existence (this was the case of Reinhardt cardinals for instance, which were proven inconsistent with ZFC, or Berkeley cardinals whose existence wasn't yet refuted, but most set theorists think they are inconsistent if I'm not mistaken). But keep in mind that this leap of faith is comparable to the one we take when taking as an axiom that there is an infinite set (infinite sets compared with finite sets is very similar in some ways to large cardinals compared with smaller cardinals).
I don't know if there is much more research surrounding possibilities of settling CH, so this isn't a complete answer (I am not myself an expert)
Finally, as Hagen Von Eitzen points out in the comments, it's hopeless to try and find a reasonable set theory that would settle all questions, even all interesting ones, because of Gödel's incompleteness theorems.
A: I noticed myself being quoted to the effect that I do not consider NF to feel like a viable foundation of mathematics, and that is indeed my opinion.  I consider suitable extensions of NFU to be viable foundations, though, and I have argued to this effect in published work.  However, I think that Zermelo style set theory is better:  NFU has characteristic (but not insuperable) difficulties with particular kinds of construction, notably anything to do with indexed families of sets.
