In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number such that for all functions there exists a cardinal with and an elementary embedding from the Von Neumann universe into a transitive inner model with critical point and An equivalent definition is this: is Woodin if and only if is strongly inaccessible and for all there exists a which is --strong. being --strong means that for all ordinals , there exist a which is an elementary embedding with critical point , , and . (See also strong cardinal.) A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact. (Wikipedia).
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