Extenders under ZF and constructibility of rank-to-rank embeddings

Abstract

Assume ZF (without the Axiom of Choice). Let j : Vε → Vδ be a nontrivial ∈-cofinal Σ1-elementary embedding, where ε, δ are limit ordinals. We prove some restrictions on the constructibility of j from Vδ, mostly focusing on the case ε = δ. In particular, if ε = δ and j ∈ L(Vδ) then cof(δ) = ω. However, assuming ZFC + I1, with the appropriate ε = δ, there is a generic extension V [G] of V such that V [G] satisfies “there is an elementary embedding j : VδV [G] → VδV [G] with j ∈ L(VδV [G])”. Assuming Dependent Choice and cof(δ) = ω (but not assuming V = L(Vδ)), and j : Vδ → Vδ is non-trivial Σ1-elementary, we show there are “perfectly many” Σ1-elementary embeddings j : Vδ → Vδ, with none being “isolated”. Assuming a proper class of weak Löwenheim–Skolem cardinals, we also give a first-order characterization of critical points of embeddings j : V → M with M transitive. The main results rely on a development of extenders under ZF (which is most useful given such wLS cardinals).