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 $V_0=\varnothing$, setting $V_{\alpha +1}=\mathcal{P}(V_{\alpha })$, and taking unions at limit stages. Assuming that

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).