"Large Cardinals beyond Choice" makes the following definitions:

$\delta$ is a Berkeley cardinal if for every transitive set $M$ such that $\delta \in m$ and every $\eta \lt \delta$ there exists $j \in \mathcal{E} (M)$ with $\eta \lt crit(j) \lt \delta$.

$\delta$ is a club Berkeley cardinal if $\delta$ is regular and for all clubs $C \subseteq \delta$ and all transitive sets $M$ with $\delta \in M$ there exists $j \in \mathcal{E}$ with $crit(j) \in c$.

The differences that I can see is that for Berkeley cardinals, the critical points are required to be unbounded in $\delta$ while for club Berkeley cardinals there must be critical points in every club, and that club Berkeley cardinals must be regular. Since the ordinals between $\eta$ and $\delta$, for any $\eta \lt \delta$, constitute a club, it appears that club Berkeley cardinals are Berkeley. However, the diagram in the slides lists no implication from club Berkeley cardinals to Berkeley. What am I missing?

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    $\begingroup$ As you note, club Berkeleys are always Berkeley. Probably the arrows are supposed to be "strict" implications, and maybe it's unknown whether e.g., ZF + a club Berkeley proves the consistency of ZF plus a Berkeley. $\endgroup$ May 5, 2022 at 23:53


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