 Inaccessible limit of inaccessibles

An inaccessible cardinal $\iota$ is an inaccessible limit of inaccessibles if the inaccessible cardinals below it are unbounded in $\iota$. A rank cutoff at the first inaccessible limit of inaccessibles shows that existence of unboundedly many inaccessibles does not prove existence of such a limit, provided the former theory is consistent.