# Why may club Berkeley cardinals not be Berkeley?

"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?

• 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. May 5, 2022 at 23:53