Consistency of V = HOD with the wholeness axiom

The Wholeness Axiom (WA) is an axiom schema that can be added to the axioms of ZFC in an extended language , and that asserts the existence of a nontrivial elementary embedding . 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 embedding. This answers a question about the existence of Laver sequences for regular classes of set embeddings: Assuming there is an -embedding, there is a transitive model of ZFC +WA + “there is a regular class of embeddings that admits no Laver sequence.” Received: 7 July 1998 / Revised version: 5 November 1998