Periodicity in the cumulative hierarchy

TL;DR

This paper explores the structure of elementary embeddings within the cumulative hierarchy of set theory without the Axiom of Choice, revealing conditions for definability and periodicity related to rank-to-rank embeddings.

Contribution

It provides a detailed analysis of elementary embeddings in the cumulative hierarchy, characterizing when such embeddings are definable and establishing the eventual periodicity under certain large cardinal assumptions.

Findings

Embeddings are definable iff the ordinal is odd.

Embeddings are not definable from elements of V_α.

Under large cardinal assumptions, the hierarchy exhibits periodicity.

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).