# Periodicity in the cumulative hierarchy

### Gabriel Goldberg

UC Berkeley, USA### Farmer Schlutzenberg

Westfälische Wilhelms-Universität Münster, Germany

## Abstract

We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in $ZF$ set theory without the Axiom of Choice. Recall that the set-theoretic universe is naturally stratified by the cumulative hierarchy, whose levels $V_{α}$ are defined via iterated application of the power set operation, starting from $V_{0}=∅$, setting $V_{α+1}=P(V_{α})$, and taking unions at limit stages. Assuming that

is a (non-trivial) elementary embedding, we show that $V_{α}$ is fundamentally different from $V_{α+1}$: we show that $j$ is definable from parameters over $V_{α+1}$ iff $α+1$ is an odd ordinal. The definability is uniform in odd $α+1$ and $j$. We also give a characterization of elementary $j:V_{α+2}→V_{α+2}$ in terms of ultrapower maps via certain ultrafilters.

For limit ordinals $λ$, we prove that if $j:V_{λ}→V_{λ}$ is $Σ_{1}$-elementary, then $j$ is not definable over $V_{λ}$ from parameters, and if $β<λ$ and $j:V_{β}→V_{λ}$ is fully elementary and $∈$-cofinal, then $j$ is likewise not definable.

If there is a Reinhardt cardinal, then for all sufficiently large ordinals $α$, there is indeed an elementary $j:V_{α}→V_{α}$, and therefore the cumulative hierarchy is eventually *periodic* (with period 2).

## Cite this article

Gabriel Goldberg, Farmer Schlutzenberg, Periodicity in the cumulative hierarchy. J. Eur. Math. Soc. 26 (2024), no. 6, pp. 2091–2126

DOI 10.4171/JEMS/1318