The role of the axiom of foundation in the Kunen inconsistency, CUNY September 2013

This will be a talk for the CUNY Set Theory Seminar on September 20, 2013 (date tentative).

Abstract. The axiom of foundation plays an interesting role in the Kunen inconsistency, the assertion that there is no nontrivial elementary embedding of the set-theoretic universe to itself, for the truth or falsity of the Kunen assertion depends on one’s specific anti-foundational stance.  The fact of the matter is that different anti-foundational theories come to different conclusions about this assertion.  On the one hand, it is relatively consistent with ZFC without foundation that the Kunen assertion fails, for there are models of  ZFC-F  in which there are definable nontrivial elementary embeddings $j:V\to V$. Indeed, in Boffa’s anti-foundational theory BAFA, the Kunen assertion is outright refutable, and in this theory there are numerous nontrivial elementary embeddings of the universe to itself. Meanwhile, on the other hand, Aczel’s anti-foundational theory GBC-F+AFA, as well as Scott’s theory GBC-F+SAFA and other anti-foundational theories, continue to prove the Kunen assertion, ruling out the existence of a nontrivial elementary embedding $j:V\to V$.

This talk covers very recent joint work with Emil Jeřábek, Ali Sadegh Daghighi and Mohammad Golshani, based on an interaction growing out of Ali’s question on MathOverflow, which lead to our recent article, The role of the axiom of foundation in the Kunen inconsistency.

Generalizations of the Kunen inconsistency

  • J. D. Hamkins, G. Kirmayer, and N. L. Perlmutter, “Generalizations of the Kunen inconsistency,” Annals of Pure and Applied Logic, vol. 163, iss. 12, pp. 1872-1890, 2012.  
    @article{HamkinsKirmayerPerlmutter2012:GeneralizationsOfKunenInconsistency,
    title = "Generalizations of the {Kunen} inconsistency",
    journal = "Annals of Pure and Applied Logic",
    volume = "163",
    number = "12",
    pages = "1872 - 1890",
    year = "2012",
    note = "",
    issn = "0168-0072",
    doi = "10.1016/j.apal.2012.06.001",
    eprint = {1106.1951},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = "http://www.sciencedirect.com/science/article/pii/S0168007212000966",
    author = "Joel David Hamkins and Greg Kirmayer and Norman Lewis Perlmutter"
    }

We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions.

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.

Generalizations of the Kunen Inconsistency, Singapore 2011

A talk at the Prospects of Infinity: Workshop on Set Theory  at the National University of Singapore, July 18-22, 2011.

I shall present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself, including generalizations-of-generalizations previously established by Woodin and others.  For example, there is no nontrivial elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or from V to the gHOD, or conversely from gHOD to V; indeed, there can be no nontrivial elementary embedding from any definable class to V.  Other results concern generic embeddings, definable embeddings and results not requiring the axiom of choice.  I will aim for a unified presentation that weaves together previously known unpublished or folklore results along with some new contributions.  This is joint work with Greg Kirmayer and Norman Perlmutter.

SlidesArticle