AD$^+$ implies that $\omega_1$ is a $\Theta$-Berkeley cardinal

Abstract

Following \cite{bagaria2019large}, 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 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 AD$^+$, (1) every regular Suslin cardinal is $\omega$-club $\Theta$-Berkeley (see \rthm{main theorem}), (2) $\omega_1$ is club $\Theta$-Berkeley (see \rthm{main theorem lr} and \rthm{main theorem}), and (3) the ${\tilde\delta}^1_{2n}$'s are $\Theta$-Berkeley -- in particular, $\omega_2$ is $\Theta$-Berkeley (see \rrem{omega2}). Along the way, we represent regular Suslin cardinals in direct limits as cutpoint cardinals (see \rthm{char extenders}). This topic has been studied in \cite{MPSC} and \cite{jackson2022suslin}, albeit from a different point of view. We also show that, assuming $V=L(\mathbb{R})+{\mathrm{AD}}$, $\omega_1$ is not $\Theta^+$-Berkeley, so the result stated in the title is optimal (see \rthm{lr optimal} and \rthm{thetareg optimal}).