2
$\begingroup$

"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?

$\endgroup$
1
  • 2
    $\begingroup$ 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

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.