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

Abstract

Assume ZF (without the Axiom of Choice). Let $j:V_\varepsilon\to V_\delta$ be a non-trivial $\in$-cofinal $\Sigma_1$-elementary embedding, where $\varepsilon,\delta$ are limit ordinals. We prove some restrictions on the constructibility of $j$ from $V_\delta$, mostly focusing on the case $\varepsilon=\delta$. In particular, if $\varepsilon=\delta$ and $j\in L(V_\delta)$ then $\delta$ has cofinality $\omega$. However, assuming ZFC+I$_3$, with the appropriate $\varepsilon=\delta$, one can force to get such $j\in L(V^{V[G]}_\delta)$. Assuming Dependent Choice and that $\delta$ has cofinality $\omega$ (but not assuming $V=L(V_\delta)$), and $j:V_\delta\to V_\delta$ is $\Sigma_1$-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 $j:V\to M$ with $M$ transitive. The main results rely on a development of extenders under ZF (which is most useful given such wLS cardinals).