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Asaf Karagila
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Are hyper-Berkeley cardinals equiconsistent with club Berkeley cardinals or with Berkeley cardinals?

Let define cardinal $\kappa$ as hyper-Berkeley if for any transitive set $M$ what $\kappa$ $\in M$ exist elementary embedding $j$ where fixed point $\lambda$ above critical and $\lambda<\kappa$.

  1. This cardinal is equiconsistent with $\sf ZF$+"club Berkeley cardinal"?
  2. Are consistent to $\sf ZF+BC$ (Berkeley cardinal)?
Alex O.
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