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. Consequences Woodin cardinals are important in descriptive set theory. By a result of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is Lebesgue measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense sets), and the perfect set property (is either countable or contains a perfect subset). The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that is Woodin in the class of hereditarily ordinal-definable sets. is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see Θ (set theory)). Mitchell and Steel showed that assuming a Woodin cardinal exists, there is an inner model containing a Woodin cardinal in which there is a -well-ordering of the reals, ◊ holds, and the generalized continuum hypothesis holds. Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on is -saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an -dense ideal over . Hyper-Woodin cardinals A cardinal is called hyper-Woodin if there exists a normal measure on such that for every set , the set is --strong is in . is --strong if and only if for each there is a transitive class and an elementary embedding with , and . The name alludes to the classical result that a cardinal is Woodin if and only if for every set , the set is --strong is