Extenders under ZF and constructibility of rank-to-rank embeddings
Volume 266 / 2024
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 $\mathrm{cof}(\delta )=\omega $. However, assuming ZFC + I$_1$, with the appropriate $\varepsilon =\delta $, there is a generic extension $V[G]$ of $V$ such that $V[G]$ satisfies “there is an elementary embedding $j:V_\delta ^{V[G]}\to V_\delta ^{V[G]}$ with $j\in L(V_\delta ^{V[G]})$”. Assuming Dependent Choice and $\mathrm{cof}(\delta )=\omega $ (but not assuming $V=L(V_\delta )$), and $j:V_\delta \to V_\delta $ is non-trivial $\Sigma _1$-elementary, we show there are “perfectly many” $\Sigma _1$-elementary embeddings $j:V_\delta \to V_\delta $, 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 $j:V\to M$ with $M$ transitive. The main results rely on a development of extenders under ZF (which is most useful given such wLS cardinals).
