AD$^+$ implies that $ω_1$ is a $Θ$-Berkeley cardinal

TL;DR

The paper investigates how determinacy axioms, particularly ${AD^+}$, constrain Berkeley-type large-cardinal phenomena at small and intermediate cardinals. It proves that under ${AD^+}$, every regular Suslin cardinal is ${ω}$-club ${Θ}$-Berkeley, implying that limits of Suslin cardinals are ${Θ}$-Berkeley (and that ${ω_2}$ is ${Θ}$-Berkeley), while also identifying optimality constraints by showing that ${ω_1}$ is not ${Θ^+}$-Berkeley in ${V=L(\mathbb{R})+AD}$. The approach combines inner-model and hod-analysis techniques, Woodin’s Derived Model Theorem, coarse tuples, and a detailed cutpoint analysis to represent Suslin cardinals as cutpoints in ${HOD}$-like models and to extract embeddings with targeted critical points. The results illuminate the interaction between determinacy, ${HOD}$ structure, and possible extensions of elementary embeddings, with implications for the feasibility of ${HOD}$-Berkeley cardinals and the boundaries of the Ultimate L program. Overall, the work advances our understanding of how determinacy frameworks shape high-level large-cardinal phenomena and embeddedness in ${HOD}$-style universes.

Abstract

Following \cite{bagaria2019large}, given cardinals $κ<λ$, we say $κ$ is a club $λ$-Berkeley cardinal if for every transitive set $N$ of size $<λ$ such that $κ\subseteq N$, there is a club $C\subseteq κ$ with the property that for every $η\in C$ there is an elementary embedding $j: N\rightarrow N$ with crit$(j)=η$. We say $κ$ is $ν$-club $λ$-Berkeley if $C\subseteq κ$ as above is a $ν$-club. We say $κ$ is $λ$-Berkeley if $C$ is unbounded in $κ$. We show that under AD$^+$, (1) every regular Suslin cardinal is $ω$-club $Θ$-Berkeley (see \rthm{main theorem}), (2) $ω_1$ is club $Θ$-Berkeley (see \rthm{main theorem lr} and \rthm{main theorem}), and (3) the ${\tildeδ}^1_{2n}$'s are $Θ$-Berkeley -- in particular, $ω_2$ is $Θ$-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 $V=L(\mathbb{R})+{\mathrm{AD}}$, $ω_1$ is not $Θ^+$-Berkeley, so the result stated in the title is optimal (see \rthm{lr optimal} and \rthm{thetareg optimal}).

Theorems & Definitions (77)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 2.1: Woodin's ${\sf{Derived\ Model\ Theorem}}$, St07DMTSt09DMT
• Theorem 3.1
• proof
• Claim 3.2
• proof
• Claim 3.3