Berkeley

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A cardinal $\kappa$ is a Berkeley cardinal, if for any transitive set $M$ with $\kappa\in M$ and any ordinal $\alpha<\kappa$ there is an elementary embedding $j:M\to M$ with $\alpha<\text{crit }j<\kappa$. These cardinals are defined in the context of ZF set theory without the axiom of choice.

The Berkeley cardinals were defined by W. Hugh Woodin in about 1992 at his set-theory seminar in Berkeley, with J. D. Hamkins, A. Lewis, D. Seabold, G. Hjorth and perhaps R. Solovay in the audience, among others, issued as a challenge to refute a seemingly over-strong large cardinal axiom. Nevertheless, the existence of these cardinals remains unrefuted in ZF.

Various strengthenings of the axiom are obtained by imposing conditions on the cofinality of $\kappa$.