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

Assume ZF (without the Axiom of Choice). Let j:V_ε→ V_δ be a non-trivial ∈-cofinal Σ₁-elementary embedding, where ε,δ are limit ordinals. We prove some restrictions on the constructibility of j from V_δ, mostly focusing on the case ε=δ. In particular, if ε=δ and j∈ L(V_δ) then $δ$ has cofinality $ω$. However, assuming ZFC+I₃, with the appropriate ε=δ, one can force to get such j∈ L(VV[G]_δ). Assuming Dependent Choice and that $δ$ has cofinality $ω$ (but not assuming V=L(V_δ)), and j:V_δ→ V_δ is Σ₁-elementary, we show that there are "perfectly many" such j, with none being "isolated". Assuming a proper class of weak Lowenheim-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).