Discussion of McCallum’s paper on Reinhardt cardinals in ZF

Rupert McCallum has posted a new paper to the mathematics arXiv

Rupert McCallum, The choiceless cardinals are inconsistent, mathematics arXiv 2017: 1712.09678.

 

He is claiming to establish the Kunen inconsistency in ZF, without the axiom of choice, which is a long-standing open question. In particular, this would refute the Reinhardt cardinals in ZF and all the stronger ZF large cardinals that have been studied.

If correct, this result will constitute a central advance in large cardinal set theory.

I am making this post to provide a place to discuss the proof and any questions that people might have about it. Please feel free to post comments with questions or answers to other questions that have been posted. I will plan to periodically summarize things in the main body of this post as the discussion proceeds.

  • My first question concerns lemma 0.4, where he claims that $j’\upharpoonright V_{\lambda+2}^N$ is a definable class in $N$. He needs this to get the embedding into $N$, but I don’t see why the embedding should be definable here.
  • I wrote to Rupert about this concern, and he replied that it may be an issue, and that he intends to post a new version of his paper, where he may retreat to the weaker claim refuting only the super-Reinhardt cardinals.
  • The updated draft is now available. Follow the link above. It will become also available on the arXiv later this week.
  • The second January 2 draft has a new section claiming again the original refutation of Reinhardt cardinals.
  • New draft January 3. Rupert has reportedly been in communication with Matteo Viale about his result.
  • Rupert has announced (Jan 3) that he is going to take a week or so to produce a careful rewrite.
  • He has made available his new draft, January 7. It will also be posted on the arXiv.
  • January 8:  In light of the issues identified on this blog, especially the issue mentioned by Gabe, Rupert has sent me an email stating (and asking me to post here) that he is planning to think it through over the next couple of weeks and will then make some kind of statement about whether he thinks he can save the argument.  For the moment, therefore, it seems that we should consider the proof to be on hold.

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://jdh.hamkins.org/generalizationsofkuneninconsistency",
    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 = {},
    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