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 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 elementaryembedding 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.

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, [http://www.math.cmu.edu/~eschimme/papers/hyperwoodin.pdf online]