Does counting inaccessibles have a fixpoint?

Question

Assume the existence of a proper class of inaccessibles $I$, and let $f : \mathrm{Ord} \to I$ denote the unique strictly increasing class surjection given by Mostowski Collapse. Does $f$ have a fixpoint? More generally, is the existence of a proper class of inaccessibles needed, or is an inaccessible $\kappa$ such that $f : \kappa \to (\kappa \cap I)$ has the previously listed properties special on its own?

Answer 1

No it doesn't imply that. If we let $\kappa$ be the least fixed point of the increasing enumeration of inaccessibles, then $V_\kappa$ is a model of ZFC with a proper class of inaccessibles and no such fixed point. On the other hand, stronger large cardinal properties imply such fixed points exist... e.g. if there are any Mahlo cardinals then they have this property.

(And really it's a large cardinal property of its own, stronger in consistency strength than a proper class of inaccessibles, but much weaker than the existence of a Mahlo cardinal. Such a $\kappa$ is sometimes called $1$-inaccessible, in anticipation of a hierarchy where $2$-inaccessibles are fixed points in the enumeration of $1$-inaccessibles and so on, though the terminology concerning properties between inaccessible and Mahlo is not very standardized.)