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In an under process joint work of Bagaria, Koellner and Woodin[1] on choice-less hierarchy of large cardinals, they introduced new kinds of Berkeley cardinals, "Proto-Berkeley", "Berkeley" and "Club Berkeley". Their definition of a "Proto-Berkeley" cardinal is as same as what is defined "Berkeley" here. Should we edit this page by changing the current definition of a Berkeley cardinal to Proto-Berkeley and adding all other variants of Berkeley cardinals?