What alternatives are there to the generalized continuum hypothesis?

Question

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.

Answer 1

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.

Answer 2

One thing to keep in mind are models where the GCH holds up to a point and then "fails." Gitik says something about the argument in "No bound for the first fixed point" requiring the GCH below the FFP but then there is no special bound on the FFP's powerset. Otherwise, I don't know if there's a generalization from $2^{\aleph_0} = 2^{\aleph_1} = \aleph_2$ but...

Now here it says that:

... Woodin showed (again assuming large cardinals) that it is possible that for all κ, 2^κ = κ⁺⁺.

But the context is singular cardinals so I'm not sure "all $\kappa$" means all cardinals whatsoever, or singular ones. Still, whatever the proof actually is, it constitutes some kind of at least partial alternative to the GCH.

Answer 3

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!