Woodin Cardinal - Hyper-Woodin Cardinals

Hyper-Woodin Cardinals

A cardinal κ is called hyper-Woodin 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 measure U will contain the set of all Shelah cardinals below κ.