The role of the foundation axiom in the Kunen inconsistency
Ali Sadegh Daghighi, Mohammad Amin Golshani, Joel David Hamkins, Emil Jeřábek · arXiv (Cornell University) · 2013
We consider the role of the foundation axiom in the Kunen inconsistency, the assertion that there is no nontrivial elementary embedding of the set-theoretic universe to itself. The truth or falsity of the Kunen assertion, we show, depends on the specific anti-foundational theory one adopts. 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, for in this theory there are numerous nontrivial elementary embeddings $j:V\to V$ 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 $j:V\to V$.