Woodin Cardinal

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Woodin cardinals are a generalization of the notion of strong cardinals and they have been used to calibrate the exact proof-theoretic strength of the Axiom of Determinacy. Woodin cardinals are weaker than superstrong cardinals in consistency strength and fail to be weakly compact in general, since they are not $\Pi_1^1$ indescribable. Their exact definition has several equivalent but different characterizations, each of which is somewhat technical in nature. Nevertheless, an inner model theory encapsulating infinitely many Woodin cardinals and slightly beyond has been developed.

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