# How badly can the GCH fail globally?

It’s known that we have global failures of GCH—for example, where $$forall lambda(2^lambda = lambda^{++})$$—given suitable large cardinal axioms.

My question is whether we can have global failures of GCH where there is a weakly inaccessible cardinal between $$lambda$$ and $$2^lambda$$ for each $$lambda$$. Similarly, whether we can have a cardinal fixed point between $$lambda$$ and $$2^lambda$$. I’d also be interested in whether $$2^lambda$$ can be weakly inaccessible/a cardinal fixed point, for every $$lambda$$.

MathOverflow Asked by Sam Roberts on January 24, 2021

In the Foreman-Woodin model The generalized continuum hypothesis can fail everywhere. for each infinite cardinal $$kappa, 2^kappa$$ is weakly inaccessible.

This answers your last question. The answer to the first two questions can be yes as well. In the case of Foreman-Woodin model, they start with a supercompact $$kappa=kappa_0$$ and infinitely many inaccessibles $$kappa_n, n about it. They first force to get $$2^{kappa_n}=kappa_{n+1}$$ preserving $$kappa$$ supercompact, and this is reflected below for all cardinals. So if for example each $$kappa_n$$ is measurable, then what you get in the final model is that for each infinite cardinal $$lambda, 2^lambda$$ has been measurable in $$V$$, in particular there are both weakly inaccessible and cardinal fixed points between $$lambda$$ and $$2^lambda.$$

See also the paper A model in which every Boolean algebra has many subalgebras by Cummings and Shelah, where they build a model in which for each infinite cardinal $$kappa, 2^kappa$$ is weakly inaccessible and $$Pr(2^kappa)$$ holds. Here $$Pr(lambda)$$ is in some sense a large cardinal property (for example it holds if $$lambda$$ is a Ramsey cardinal). For its definition see the paper.

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