Hyper-Woodin cardinal

In axiomatic set theory, a hyper-Woodin cardinal is a kind of large cardinal. A cardinal κ is called hyper-Woodin if and only if there exists a normal measure "U" on κ such that for every set "S", 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)&ge; δ, and

:$j(S)\; cap\; H\_delta\; =\; S\; cap\; H\_delta$.

The name alludes to the classical result that a cardinal is Woodin if and only if for every set "S", the set

:{λ < κ | λ is <κ-"S"-strong}

is a stationary set

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.

The measure "U" will contain the set of all Shelah cardinals below κ.

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]