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 $j:Vto V$. 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 $j:Vto V$.

This is 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.  Our paper will be completed soon.