Weakly hyper-Woodin cardinal

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In axiomatic set theory, weakly hyper-Woodin cardinals are a kind of large cardinals. A cardinal κ is called weakly hyper-Woodin if and only if for every set S there exists a normal measure U on κ such that the set {λ < κ | λ is <κ-S-strong} is in U. λ is <κ-S-strong if and only if for each δ < κ there is a transitive class N and an elementary embedding j : V → N with λ = crit(j), j(λ) >= δ, and $j(S) \cap H_\delta = S \cap H_\delta$ .

The name alludes to the classic result that a cardinal is Woodin if and only if for every set S, the set {λ < κ | λ is <κ-S-strong} is stationary.

The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of U does not depend on the choice of the set S for hyper-Woodin cardinals.

[edit] References

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