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 {∈, 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.” 1991 Mathematics Subject Classification. Primary 03E55, 03E35.