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.