Periodicity in the cumulative hierarchy

Abstract

We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in $\mathrm{ZF}$ set theory without the Axiom of Choice. Recall that the set-theoretic universe is naturally stratified by the cumulative hierarchy, whose levels $V\_\alpha$ are defined via iterated application of the power set operation, starting from $V\_0=\emptyset$, setting $V\_{\alpha+1}=\mathcal{P}(V\_\alpha)$, and taking unions at limit stages. Assuming that $$ j:V\_{\alpha+1}\to V\_{\alpha+1} $$ is a (non-trivial) elementary embedding, we show that $V\_\alpha$ is fundamentally different from $V\_{\alpha+1}$: we show that $j$ is definable from parameters over $V\_{\alpha+1}$ iff $\alpha+1$ is an odd ordinal. The definability is uniform in odd $\alpha+1$ and $j$. We also give a characterization of elementary $j:V\_{\alpha+2}\to V\_{\alpha+2}$ in terms of ultrapower maps via certain ultrafilters. For limit ordinals $\lambda$, we prove that if $j:V\_\lambda\to V\_\lambda$ is $\Sigma\_1$-elementary, then $j$ is not definable over $V\_\lambda$ from parameters, and if $\beta<\lambda$ and $j:V\_\beta\to V\_\lambda$ is fully elementary and $\in$-cofinal, then $j$ is likewise not definable. If there is a Reinhardt cardinal, then for all sufficiently large ordinals $\alpha$, there is indeed an elementary $j:V\_\alpha\to V\_\alpha$, and therefore the cumulative hierarchy is eventually periodic (with period 2).