In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number λ {\displaystyle \lambda } such that for all functions f : λ → λ {\displaystyle f:\lambda \to \lambda } , there exists a cardinal κ < λ {\displaystyle \kappa <\lambda } with { f ( β ) ∣ β < κ } ⊆ κ {\displaystyle \{f(\beta )\mid \beta <\kappa \}\subseteq \kappa } and an elementary embedding j : V → M {\displaystyle j:V\to M} from the Von Neumann universe V {\displaystyle V} into a transitive inner model M {\displaystyle M} with critical point κ {\displaystyle \kappa } and V j ( f ) ( κ ) ⊆ M {\displaystyle V_{j(f)(\kappa )}\subseteq M} . An equivalent definition is this: λ {\displaystyle \lambda } is Woodin if and only if λ {\displaystyle \lambda } is strongly inaccessible and for all A ⊆ V λ {\displaystyle A\subseteq V_{\lambda }} there exists a λ A < λ {\displaystyle \lambda _{A}<\lambda } which is < λ {\displaystyle <\lambda } - A {\displaystyle A} -strong.
λ A {\displaystyle \lambda _{A}} being < λ {\displaystyle <\lambda } - A {\displaystyle A} -strong means that for all ordinals α < λ {\displaystyle \alpha <\lambda } , there exist a j : V → M {\displaystyle j:V\to M} which is an elementary embedding with critical point λ A {\displaystyle \lambda _{A}} , j ( λ A ) > α {\displaystyle j(\lambda _{A})>\alpha } , V α ⊆ M {\displaystyle V_{\alpha }\subseteq M} and j ( A ) ∩ V α = A ∩ V α {\displaystyle j(A)\cap V_{\alpha }=A\cap V_{\alpha }} . (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.p. 364
Explanation The hierarchy V α {\displaystyle V_{\alpha }} (known as the von Neumann hierarchy) is defined by transfinite recursion on α {\displaystyle \alpha } :
V 0 = ∅ {\displaystyle V_{0}=\varnothing } ,
V α + 1 = P ( V α ) {\displaystyle V_{\alpha +1}={\mathcal {P}}(V_{\alpha })} ,
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