Consistency of V = HOD With the Wholeness Axiom

Abstract. The Wholeness Axiom (WA) is an axiom schema that can be added to the axioms of ZFC in an extended language {∈,j}, and that asserts the existence of a nontrivial elementary embedding j: V → V. The well-known inconsistency proofs are avoided by omitting from the schema all instances of Replacement for j-formulas. We show that the theory ZFC + V = HOD + WA is consistent relative to the existence of an I1 embedding. This answers a question about the existence of Laver sequences for regular classes of set embeddings: Assuming there is an I1-embedding, there is a transitive model of ZFC+WA+ “there is a regular class of embeddings that admits no Laver sequence.