Woodin cardinal

In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number ${\displaystyle \lambda }$ such that for all functions

${\displaystyle f:\lambda \to \lambda }$

there exists a cardinal ${\displaystyle \kappa <\lambda }$ with

${\displaystyle \{f(\beta )\mid \beta <\kappa \}\subseteq \kappa }$

and an elementary embedding

${\displaystyle j:V\to M}$

from the Von Neumann universe ${\displaystyle V}$ into a transitive inner model ${\displaystyle M}$ with critical point ${\displaystyle \kappa }$ and

${\displaystyle V_{j(f)(\kappa )}\subseteq M.}$

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

${\displaystyle \lambda _{A}}$ being ${\displaystyle <\lambda }$-${\displaystyle A}$-strong means that for all ordinals ${\displaystyle \alpha <\lambda }$, there exist a ${\displaystyle j:V\to M}$ which is an elementary embedding with critical point ${\displaystyle \lambda _{A}}$, ${\displaystyle j(\lambda _{A})>\alpha }$, ${\displaystyle V_{\alpha }\subseteq M}$ and ${\displaystyle j(A)\cap V_{\alpha }=A\cap V_{\alpha }}$. (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.