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A cardinal is weakly Reinhardt if it is the critical point of an elementaryembedding from the universe of sets into a model that contains the doublepowerset of every ordinal. This note establishes the equiconsistency of aproper class of weakly Reinhardt cardinals with a proper class of Reinhardtcardinals in the context of second-order set theory without the Axiom ofChoice.

Reinhardt cardinals in inner models