| مستخلص: |
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 j:V α+1 →V α+1 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:Vα+2 → Vα+2 in terms of ultrapower maps via certain ultrafilters. For limit ordinals λ, we prove that if j:Vλ→Vλ is Σ1-elementary, then j is not definable over Vλ from parameters, and if β < λ and j:Vβ → Vλ 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: Vα → Vα, and therefore the cumulative hierarchy is eventually periodic (with period 2). [ABSTRACT FROM AUTHOR] |
