What alternatives are there to the generalized continuum hypothesis? 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.

There are many reasonable axioms that have been studied in diverse situations and all are known to contradict instances of GCH.
By far the strongest case can be made for strong forcing axioms such us Martin's maximum MM, or some of their significant consequences or close relatives (strong reflection principles). These forcing axioms imply that the continuum is $\aleph_2$ (so CH fails) and have many appealing combinatorial consequences. They also imply the singular cardinals hypothesis, which in turn implies many instances of GCH, so their effect in cardinal arithmetic is not "wild". 
But axioms rather different than these have been considered as well. For example, the existence of (atomlessly) real-valued measurable cardinals implies that the continuum is very large. The axiom is perhaps most appealing in its alternative formulation stating that there is a measure extending Lebesgue measure and defined on all sets of reals. (On the other hand, the effect of this axiom on cardinal characteristics of the continuum is rather different from the nice behavior provided by Martin's axiom.)
There are also ad hoc axioms that have been considered precisely because of their wild behavior. I think these are different from the examples above, which are reasonable assumptions for a working mathematician. Perhaps the best known here is the maximality principle, still not known to be consistent, which implies that GCH fails everywhere. In a sense, MM seems to come from an attempt to obtain a reasonable version of this principle, see the comments to this answer in Mathoverflow.  
