Periodicity in the cumulative hierarchy

  • Gabriel Goldberg

    UC Berkeley, USA
  • Farmer Schlutzenberg

    Westfälische Wilhelms-Universität Münster, Germany
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We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in set theory without the Axiom of Choice. Recall that the set-theoretic universe is naturally stratified by the cumulative hierarchy, whose levels are defined via iterated application of the power set operation, starting from , setting , and taking unions at limit stages. Assuming that

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

For limit ordinals , we prove that if is -elementary, then is not definable over from parameters, and if and is fully elementary and -cofinal, then is likewise not definable.

If there is a Reinhardt cardinal, then for all sufficiently large ordinals , there is indeed an elementary , 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