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.

Very interesting indeed! Does this show that the extremely large cardinals lie (in some sense outside ZFC which seem to characterize very well the notion of ‘set of something’? It certainly makes sense that Boffa’s system BAFA refutes the Kunen inconsistency but that Aczel’s and Scott’s systems do not seems strange because the point of anti-foundation axioms is to allow for sets of sets that are contained in themselves. Why is this? Might the notions of cardinal and ordinal numbers be absolute but not describable in specific formal theories? I wonder….

We’re not quite finished with the paper, but you can see part of the main idea on the mathoverflow posts. In the case of AFA, the reason is that any purported embedding would have to fix the accessible pointed graph relation underlying the non-well-founded sets (since these have a well-founded copy by the axiom of choice), and this is why it must also fix the ill-founded sets. But in Boffa’s system, there are numerous automorphisms of the universe, since the sets are not determined by the isomorphism type of the membership relation on their transitive closures.

Thanks. This is very helpful. Could one use a forcing-like argument to show that Foundation is ‘damaged’ (i.e. does not hold) if there exist nontrivial elementary embeddings j: V –> V for ZFC?

I’m looking forward to your talk on the role of foundation in the Kunen inconsistency on Friday!