In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992.
This can be continued any finite number of times, and to the extent that the model has dependent choice, transfinitely.
Thus, plausibly, this notion can be strengthened simply by asserting more dependent choice.
While all these notions are incompatible with Zermelo–Fraenkel set theory (ZFC), their
There is no known inconsistency with ZFC in asserting that, for example: For every ordinal λ, there is a transitive model of ZF + Berkeley cardinal that is closed under λ sequences.