Hyper-Woodin cardinal

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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)≥ δ, and

.

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.

[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