The Wholeness Axioms and V=HOD

Joel David Hamkins

Published 1999-02-13Version 1

The Wholeness Axioms, proposed by Paul Corazza, axiomatize the existence of an elementary embedding j:V-->V. Formalized by augmenting the usual language of set theory with an additional unary function symbol j to represent the embedding, they avoid the Kunen inconsistency by restricting the base theory ZFC to the usual language of set theory. Thus, under the Wholeness Axioms one cannot appeal to the Replacement Axiom in the language with j as Kunen does in his famous inconsistency proof. Indeed, it is easy to see that the Wholeness Axioms have a consistency strength strictly below the existence of an I_3 cardinal. In this paper, I prove that if the Wholeness Axiom WA_0 is itself consistent, then it is consistent with V=HOD. A consequence of the proof is that the various Wholeness Axioms WA_n are not all equivalent. Furthermore, the theory ZFC+WA_0 is finitely axiomatizable.