AD⁺ implies ω ₁ is a club Θ -Berkeley cardinal

Following [1], 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 ⊆ N , there is a club C⊆ κ with the property that for every η ∈ C , there is an elementary embedding j: N→ N with crit (j)=η . We say κ is ν -club λ -Berkeley if C⊆ 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 Theorem 7.1), (2) ω ₁ is club Θ -Berkeley (see Theorem 3.1 and Theorem 7.1), and (3) the ’s are Θ -Berkeley – in particular, ω ₂ is Θ -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(R)+AD , ω ₁ is not Θ ⁺ -Berkeley, so the result stated in the title is optimal (see Theorem 9.14 and Theorem 9.19).