The wholeness axioms and 𝑉 =HOD

[bibtex key=Hamkins2001:WholenessAxiom]

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 𝐼3 cardinal. In this paper, I prove that if the Wholeness Axiom 𝑊𝐴0 is itself consistent, then it is consistent with 𝑉 =𝐻𝑂𝐷. A consequence of the proof is that the various Wholeness Axioms 𝑊𝐴𝑛 are not all equivalent. Furthermore, the theory 𝑍𝐹𝐶 +𝑊𝐴0 is finitely axiomatizable.

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