Periodicity in the cumulative hierarchy

Abstract

We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels $V_\alpha$ of the cumulative hierarchy are defined via iterated application of the power set operation, starting from $V_0=\emptyset$, 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 the structure of $V_\alpha$ is fundamentally different to that of $V_{\alpha+1}$. We show that $j$ is definable from parameters over $V_{\alpha+1}$ iff $\alpha+1$ is an odd ordinal. Moreover, if $\alpha+1$ is odd then $j$ is definable over $V_{\alpha+1}$ from the parameter $j`` V_{\alpha}=\{j(x)\bigm|x\in V_\alpha\}$, and uniformly so. This parameter is optimal in that $j$ is not definable from any parameter which is an element of $V_\alpha$. In the case that $\alpha=\beta+1$, we also give a characterization of such $j$ in terms of ultrapower maps via certain ultrafilters. Assuming $\lambda$ is a limit ordinal, 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; note that this last result is relevant to embeddings of much lower consistency strength than rank-to-rank. 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).