Large cardinals | Determinacy

Woodin cardinal

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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Related pages

Θ (set theory) | Set theory | If and only if | Measurable cardinal | Nowhere dense set | Mahlo cardinal | Perfect set | Ordinal number | Normal measure | Inner model | Zermelo–Fraenkel set theory | Descriptive set theory | Shelah cardinal | Perfect set property | Cardinal number | Meagre set | Inaccessible cardinal | Axiom of determinacy | Critical point (set theory) | Weakly compact cardinal | Stationary set | Strong cardinal | Von Neumann universe