Weakly hyper-Woodin cardinal

In axiomatic set theory, weakly hyper-Woodin cardinals are a kind of large cardinals. A cardinal κ is called weakly hyper-Woodin iff 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.

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.

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