The wholeness axioms and $V=\rm HOD$

  • J. D. Hamkins, “The wholeness axioms and $V=\rm HOD$,” Arch.~Math.~Logic, vol. 40, iss. 1, pp. 1-8, 2001.  
    @article{Hamkins2001:WholenessAxiom,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {The wholeness axioms and {$V=\rm HOD$}},
    JOURNAL = {Arch.~Math.~Logic},
    FJOURNAL = {Archive for Mathematical Logic},
    VOLUME = {40},
    YEAR = {2001},
    NUMBER = {1},
    PAGES = {1--8},
    ISSN = {0933-5846},
    CODEN = {AMLOEH},
    MRCLASS = {03E35 (03E65)},
    MRNUMBER = {1816602 (2001m:03102)},
    MRREVIEWER = {Ralf-Dieter Schindler},
    DOI = {10.1007/s001530050169},
    URL = {http://dx.doi.org/10.1007/s001530050169},
    eprint = {math/9902079},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

The Wholeness Axioms, proposed by Paul Corazza, axiomatize the existence of an elementary embedding $j:V\to V$. 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.