How badly can the GCH fail globally?

Question

It's known that we can 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 $\lambda$.

Answer 1

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 $\kappa_n, n<\omega,$ above 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.