The consistency strength of hyperstationarity

We introduce the large-cardinal notions of ζ-greatly-Mahlo and ζ-reflection cardinals and prove (1) in the constructible universe, L, the first ζ-reflection cardinal, for ζ a successor ordinal, is strictly between the first ζ-greatly-Mahlo and the first ?ζ1-indescribable cardinals, (2) assuming the existence of a ζ-reflection cardinal κ in L, ζ a successor ordinal, there exists a forcing notion in L that preserves cardinals and forces that κ is (ζ + 1)-stationary, which implies that the consistency strength of the existence of a (ζ + 1)-stationary cardinal is strictly below a ?ζ1-indescribable cardinal. These results generalize to all successor ordinals ζ the original same result of Mekler-Shelah [A. Mekler and S. Shelah, The consistency strength of every stationary set reflects, Israel J. Math. 67(3) (1989) 353-365] about a 2-stationary cardinal, i.e. a cardinal that reflects all its stationary sets.