a stationary set. The measure will contain the set of all Shelah cardinals below . Weakly hyper-Woodin cardinals A cardinal is called weakly hyper-Woodin if for every set there exists a normal measure on such that the set is --strong is in . is --strong if and only if for each there is a transitive class and an elementary embedding with , , and The name alludes to the classic result that a cardinal is Woodin if for every set , the set is --strong is stationary. The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of does not depend on the choice of the set for hyper-Woodin cardinals. Notes and references Further reading For proofs of the two results listed in consequences see Handbook of Set Theory (Eds. Foreman, Kanamori, Magidor) (to appear). Drafts of some chapters are available. Ernest Schimmerling, Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model, Proceedings of the American Mathematical Society 130/11, pp. 3385–3391, 2002, online Large cardinals Determinacy