$\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $-Berkeley cardinal

Abstract Following [1], given cardinals $\kappa <\lambda $ , we say $\kappa $ is a club $\lambda $ -Berkeley cardinal if for every transitive set N of size $<\lambda $ such that $\kappa \subseteq N$ , there is a club $C\subseteq \kappa $ with the property that for every $\eta \in C$ , there is an elementary embedding $j: N\rightarrow N$ with $\mathrm {crit }(j)=\eta $ . We say $\kappa $ is $\nu $ -club $\lambda $ -Berkeley if $C\subseteq \kappa $ as above is a $\nu $ -club. We say $\kappa $ is $\lambda $ -Berkeley if C is unbounded in $\kappa $ . We show that under $\textsf {AD}^{+}$ , (1) every regular Suslin cardinal is $\omega $ -club $\Theta $ -Berkeley (see Theorem 7.1), (2) $\omega _1$ is club $\Theta $ -Berkeley (see Theorem 3.1 and Theorem 7.1), and (3) the ’s are $\Theta $ -Berkeley – in particular, $\omega _2$ is $\Theta $ -Berkeley (see Remark 7.5). Along the way, we represent regular Suslin cardinals in direct limits as cutpoint cardinals (see Theorem 5.1). This topic has been studied in [31] and [4], albeit from a different point of view. We also show that, assuming $V=L({\mathbb {R}})+{\textsf {AD}}$ , $\omega _1$ is not $\Theta ^+$ -Berkeley, so the result stated in the title is optimal (see Theorem 9.14 and Theorem 9.19).