Woodin Cardinal - Weakly Hyper-Woodin Cardinals

Weakly Hyper-Woodin Cardinals

A cardinal κ is called weakly hyper-Woodin 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

The name alludes to the classic result that a cardinal is Woodin 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.