Claim: A Berkeley cardinal is a cardinal κ in a Zermelo–Fraenkel set theory with the property that for every transitive set M includes κ; however, its existence is inconsistent with the axiom of countable choice.
Evidence: A Berkeley cardinal is a cardinal κ in a model of Zermelo–Fraenkel set theory with the property that for every transitive set M that includes κ, there is a nontrivial elementary embedding of M into M with critical point below κ. Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals, implying that they are not compatible with the axiom of choice. In fact, the existence of Berkeley cardinals is inconsistent with the axiom of countable choice.