Woodin cardinal

In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number λ such that for all functions

there exists a cardinal κ < λ with

and an elementary embedding

from the Von Neumann universe V into a transitive inner model M with critical point κ and

An equivalent definition is this: λ is Woodin if and only if λ is strongly inaccessible and for all ${\displaystyle A\subseteq V_{\lambda }}$ there exists a ${\displaystyle \lambda _{A}}$ < λ which is ${\displaystyle <\lambda }$-${\displaystyle A}$-strong.

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