Inaccessible cardinals

Question

Let $M[G]$ be the full Solovay model, and let HOD be the model of hereditarily ordinal definable sets in $M[G]$.

Is it possible for HOD not to have an inaccessible cardinal? Does HOD satisfy GCH?

Answer 1

Generally, "the full Solovay model" means that there was $\kappa$ which was inaccessible in $M$ and $G$ is a generic filter for $\operatorname{Coll}(\omega,<\kappa)$. Since this is a homogenous forcing, it doesn't change $\rm HOD$, so $\rm HOD$ of $M[G]$ is the same as in $M$.

If $\kappa$ is inaccessible in $M$ then it is regular and strong limit. In particular in every inner model $\kappa$ is regular, and certainly strong limit (if $\mu<\kappa$ then an inner model can have at most less subsets of $\mu$ than $M$ itself, so certainly not more than $\kappa$).

So if $M[G]$ is "the full Solovay model", then in $\rm HOD$ there is an inaccessible cardinal. Unless of course when you say "the full Solovay model" you don't mean the Levy collapse of an inaccessible cardinal, in which case you should probably state explicitly what you mean.