Consistency of ZF with a Reinhardt cardinal

Determine whether ZF (Zermelo–Fraenkel set theory without the Axiom of Choice) together with the existence of a Reinhardt cardinal—equivalently, the existence of a non-trivial elementary embedding j: V → V—is consistent, or prove that it is inconsistent.

Background

The paper surveys large cardinal notions tied to elementary embeddings and recalls Kunen’s inconsistency result for Reinhardt embeddings in ZFC. It then considers weaker subsystems and notes that, unlike the case of ZFC−, the status of Reinhardt embeddings over full ZF remains unresolved. The author’s main result addresses the existence of cofinal Reinhardt embeddings in ZF− under strong hypotheses, but this does not settle the fundamental question for full ZF.