# Berkeley

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A cardinal $\kappa$ is a *Berkeley* cardinal, if for any transitive set $M$ with $\kappa\in M$, there is an elementary embedding $j:M\to M$ having critical point less than $\kappa$. These cardinals are defined in the context of ZF set theory without the axiom of choice.
Various strengthenings of the axiom are obtained by imposing conditions on the cofinality of $\kappa$.