The foundation axiom and elementary self-embeddings of the universe

[bibtex key=DaghighiGolshaniHaminsJerabek2013:TheFoundationAxiomAndElementarySelfEmbeddingsOfTheUniverse]$\newcommand\ZFC{\text{ZFC}}\newcommand\ZFCf{\ZFC^{\rm-f}}\newcommand\AFA{\text{AFA}}\newcommand\BAFA{\text{BAFA}}$

Festschrift celebrating 60th birthdays of Peter Koepke and Philip Welch
In this article, we examine the role played by the axiom of foundation in the well-known Kunen inconsistency, the theorem asserting that there is no nontrivial elementary embedding of the set-theoretic universe to itself. All the standard proofs of the Kunen inconsistency make use of the axiom of foundation (see Kanamori’s books and also Generalizations of the Kunen inconsistency), and this use is essential, assuming that $\ZFC$ is consistent, because as we shall show there are models of $\ZFCf$ that admit nontrivial elementary self-embeddings and even nontrivial definable automorphisms. Meanwhile, a fragment of the Kunen inconsistency survives without foundation as the claim in $\ZFCf$ that there is no nontrivial elementary self-embedding of the class of well-founded sets. Nevertheless, some of the commonly considered anti-foundational theories, such as the Boffa theory $\BAFA$, prove outright the existence of nontrivial automorphisms of the set-theoretic universe, thereby refuting the Kunen assertion in these theories.  On the other hand, several other common anti-foundational theories, such as Aczel’s anti-foundational theory $\ZFCf+\AFA$ and Scott’s theory $\ZFCf+\text{SAFA}$, reach the opposite conclusion by proving that there are no nontrivial elementary embeddings from the set-theoretic universe to itself. Our summary conclusion, therefore, is that the resolution of the Kunen inconsistency in set theory without foundation depends on the specific nature of one’s anti-foundational stance.

This is joint work with Ali Sadegh Daghighi, Mohammad Golshani, myself and Emil Jeřábek, which grew out of our interaction on Ali’s question on MathOverflow, Is there any large cardinal beyond the Kunen inconsistency?

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.