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

Abstract

Assume ZF (without the Axiom of Choice). Let $j:V_{\epsilon }\to V_{\delta }$ be a non-trivial $\in$-cofinal ${\Sigma }_1$-elementary embedding, where $\epsilon ,\delta$ are limit ordinals. We prove some restrictions on the constructibility of $j$ from $V_{\delta }$, mostly focusing on the case $\epsilon =\delta$. In particular, if $\epsilon =\delta$ and $j\in L(V_{\delta })$ then $\delta$ has cofinality $\omega$. However, assuming ZFC+I${}_3$, with the appropriate $\epsilon =\delta$, one can force to get such $j\in L(V_{\delta }^{V[G]})$. 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).

32 pages. This version: Extended some results (6.9, 7.1, 7.6), and modified introduction accordingly. Corrected typo in abstract which asserted a key theorem falsely. Added URL links to bibliography. arXiv admin note: text overlap with arXiv:2002.01215