TITLE: What alternatives are there to the generalized continuum hypothesis?
QUESTION [2 upvotes]: Since the generalized continuum hypothesis is independent of ZFC, we can adopt its negation as an axiom. However, this would not be a very "nice" axiom, since all it does is assert that there is at least one instance in which GCH fails.
My question is what axioms have been studied that prove the negation of the GCH, but has many more consequences than the negation of GCH. In particular, I am looking for ones that mathematicians would consider appealing axioms for mathematical foundations, not merely any statement that implies the negation of the GCH.
As an example, I would consider the Axiom of determinacy a "nice" alternative to the Axiom of choice.

REPLY [1 votes]: I'm personally a fan of the inner model hypothesis - or rather, hypotheses might be more appropriate. IMHs should be understood as codifications of the general idea that "Anything which can happen, does" in a precise sense via inner and outer model theory.
Specifically, the idea is that the set-theoretic universe $V$ should be maximally rich with respect to inner models. For example, both $\mathsf{CH}$ and $\neg\mathsf{CH}$ are "easy to achieve" via forcing, so $V$ should have inner models where $\mathsf{CH}$ holds and inner models where $\mathsf{CH}$ fails. A bit more specifically:

Anything which can happen in an inner model of an outer model of $V$, already happens in some inner model of $V$.

I personally find this idea quite attractive. The technicality of inner/outer models is definitely something which can give someone new to the field pause, but I think is actually very natural - especially in light of the role that forcing and inner model constructions already play in set theory.
Of course this is all wildly imprecise, and this is where the pluralization above comes in: there are multiple different ways one might try to formulate this principle. See here (and the papers cited therein) for more details on this point. The point is that even a mild IMH makes the continuum quite large. It also, interestingly, contradicts even the existence of a single inaccessible cardinal!
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