y transitive set m with k∈m and any ordinal a＜k there is an elementary embedding j:m?m with a＜ crit j ＜k.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， ...

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