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 $\{\in,j\}$, and that asserts the existence of a nontrivial elementary embedding $j:V\to V$. 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 $I_1$ embedding. This answers a question about the existence of Laver sequences for regular classes of set embeddings: Assuming there is an $I_1$-embedding, there is a transitive model of ZFC +WA + “there is a regular class of embeddings that admits no Laver sequence.”