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

Schlutzenberg Farmer

Abstract

Assume the Zermelo Fraenkel axioms for set theory ZF, without the Axiom of Choice. Let $j:V_\delta\to V_\delta$ be a non-trivial $\Sigma_1$-elementary embedding, where $\delta$ is a limit ordinal and $V_\delta$ is the corresponding segment of the cumulative hierarchy (the existence of such an embedding is a strong large cardinal assumption). We prove some basic restrictions on the constructibility of such embeddings from $V_\delta$; that is, we prove consequences of having such an embedding $j\in L(V_\delta)$. We show that, however, assuming an $I_3$-embedding, with the appropriate $\delta,j$, it is possible to have $j\in L(V_\delta)$. Assuming Dependent Choice and that $\delta$ has countable cofinality (but not assuming $V=L(V_\delta)$), and $j$ is as above, we show that the collection of such embeddings is of high complexity, and that there are ``perfectly many'' such embeddings. We also show that a theorem of Suzuki, that there is no elementary $j:V\to V$ which is definable from parameters, assuming ZF, actually follows from a theory weaker than ZF. The main results rely on a development of extenders under ZF, which we also give.