Didn't Woodin suggest Ultimate L to incorporate all large cardinals, and it happens to imply CH?

@trb456: There's a lot about history, tradition, the fact that we are interested in both CH holding and failing since both have interesting consequences, and of course, we aim to have a minimal axiomatic setting. So once the dust settled and $\sf ZFC$ seemed like a good candidate, it's hard to add more axioms to the "hardcore of mathematics" without a good reason.

Large cardinal axioms are not accepted as "correct" either. They are popular because rich consequences can be derived from them with current methods. Before that forcing axioms were popular, some still are. Before that GCH itself was popular, etc. The reason why they weren't permanently adopted is not specific to CH: none of them are needed in most of mathematical practice.

@trb456: I don't know why my previous comment is not a good enough reason. It's a combination of tradition, the fact that adding new axioms removes interesting problems from the mathematical world, and the fact that we are interested in "less axioms" rather than "more axioms".

Your original question was completely different though: why CH isn't 'close enough' to large cardinals. It is not on the same level because it is not as intuitive as AC, doesn't provide highly useful closure properties in classical mathematics like AC, and AC is already suspicious.

I think that you're not asking the right question. Note that "natural axiom" is something which depends a lot on the context, and set theory has many different contexts. $\sf PFA$ can be very natural in some contexts, and having a real-measurable continuum can be natural in a different context. Not to mention choiceless contexts of $\sf AD$ where the continuum has properties both of $\sf CH$ and of its failure (in more than one way). This is like asking why aren't all groups abelian, or why aren't all rings commutative/PID/etc. -- because there's more to it than just this one property.

Yes, not all groups are abelian, but not all groups are not abelian either. Both things have uses and generate interest.

@zibadaw: No, that's not quite right. The continuum hypothesis and its negations have implications to operator theory and they continue from there. All the way to physics. To say that no one but set theorists care for its value is plain false; in fact I don't think that many set theorists care for the value of the continuum, and perhaps the emerging school of multiverse approach to set theory (rather than formalist or Platonist) will end up as the dominant one in the future. Precisely because there are so many conflicting interests.

@zibadawa: In most non-set theoretical areas, I think, most researchers wouldn't dare writing something like "Now, assuming the continuum hypothesis ...". But it doesn't mean there is no research towards that, or that it has no implications. People just try to circumvent these issues, or take steps closer to where $\sf CH$ has little to no effect. But things like Kaplansky conjecture, automorphisms of the Calkin algebra, and so on, end up being dependent on set theoretic assumptions, in particular $\sf CH$. And it trickles down the implications ladder, slowly but surely.

@zibadawa: As to conflicting interests, I meant that set theorists are interested both in models where $\sf CH$ holds and in models where it is violated. So there is a conflicting interest as to whether or not $\sf CH$ should even have a decided truth value, since once it has, many of these interesting problems disappear from our attention. And mathematics is inclusive rather than exclusive, at least in this aspect.

@Asaf Karagila Interesting problems related to $\neg$AC did not disappear after AC got into ZFC. And there are niche issues outside of set theory where $\neg$AC matters, coloring the plane for example. The difference between CH and AC has to be elsewhere.

@Conifold: Choice is by far more useful as an axiom, as it helps tame infinite objects. Even things which are seemingly "natural" to us (through the widely use, but hardly noticable, Principle of Dependent Choice). There are still many interesting problems regarding the axiom of choice, and people in set theory still do that. The axiom of choice, however, has a much greater effect on "working mathematics" (read: non-set theory), than the continuum hypothesis has. Which is why it was settled quite quickly into the axioms of set theory and mathematics.

@Conifold: If I had to pin down axioms which extend the usability of the axiom of choice outside of set theory, I'd not choose the continuum hypothesis. The obvious candidates would be either diamond axioms (which tame down uncountable cardinalities by ensuring that "when things happen, they happen often") or Martin's axiom which is more difficult to use, but has immediate topological implications, and it too helps to bring a little bit of structure into infinite cardinals. Between these two, I'd probably use diamonds if you'd force me to make a choice.

@Asaf Karagila Well, GCH would give us all the usability of choice along with CH and plenty more of taming infinite objects. What it lacks, like CH, is intuitive basis, and like you said, much of practical effect beyond AC. And AC already has strange enough consequences to go any further. These seem to be the main reasons and not just against CH.

@Conifold: The "strange consequences" that people like to object to so much, are in reality effects of the axiom of infinity, and all those weird things infinite objects have. People who object to the Banach-Tarski paradox clearly don't know that in the models we know that BT fails you can partition the line into more non-empty parts than points. To me, this is far more boggling than the BT paradox. Or that without the axiom of choice obvious "truths" need not be true anymore. E.g. $\Bbb N$ or $\Bbb R$ need not be Lindelof, both may seem obvious to us; or the regularity of $\omega_1$.

@Asaf Karagila But you can not do so provably. The point of conservatism is to have some core where provable things are in line with intuition as much as possible, ZF+DC is probably that classically, and ZFC is a minimal compomise for more abstract mathematics. By the way, conservatism is exactly the inclusiveness that you praised, the smaller the core the more inclusive the theory.