The Foundation Axiom and Elementary Self-embeddings of the Universe

We consider the role of the foundation axiom and various anti-foundation axioms in connection with the nature and existence of elementary self-embeddings of the set-theoretic universe. We shall investigate the role of the foundation axiom and the various anti-foundation axioms in connection with the nature and existence of elementary self-embeddings of the set-theoretic universe. All the standard proofs of the well-known Kunen inconsistency [Kun78], for example, the theorem asserting that there is no nontrivial elementary embedding of the set-theoretic universe to itself, make use of the axiom of foundation (see [Kan04, HKP12]), and this use is essential, assuming that ZFC is consistent, because there are models of ZFC−f 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 ZFC−f 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 ZFC−f + AFA and Scott’s theory ZFC−f + SAFA, reach the opposite conclusion by proving that there are no nontrivial elementary The second author’s research has been supported by a grant from IPM (No. 91030417). The third author’s research has been supported in part by Simons Foundation grant 209252, by PSC-CUNY grant 66563-00 44 and by grant 80209-06 20 from the CUNY Collaborative Incentive Award program. The fourth author has been supported by grant IAA100190902 of GA AV ČR, Center of Excellence CEITI under the grant P202/12/G061 of GA ČR, and RVO: 67985840. This inquiry grew out of a question posed by the first author on MathOverflow [Dag13] and the subsequent exchange posted by the third and fourth authors there.