- A. S. Daghighi, M. Golshani, J. Hamkins, and E. Jeřábek, “The foundation axiom and elementary self-embeddings of the universe,” in Infinity, computability, and metamathematics: Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch, S.~Geschke, B.~Löwe, and P.~Schlicht, Eds., Coll. Publ., London, 2014, vol. 23, pp. 89-112.
`@incollection {DaghighiGolshaniHaminsJerabek2013:TheFoundationAxiomAndElementarySelfEmbeddingsOfTheUniverse, AUTHOR = {Daghighi, Ali Sadegh and Golshani, Mohammad and Hamkins, Joel David and Je{\v{r}}{\'a}bek, Emil}, TITLE = {The foundation axiom and elementary self-embeddings of the universe}, BOOKTITLE = {Infinity, computability, and metamathematics: Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch}, SERIES = {Tributes}, VOLUME = {23}, PAGES = {89--112}, PUBLISHER = {Coll. Publ., London}, EDITOR = {S.~Geschke and B.~L\"owe and P.~Schlicht}, YEAR = {2014}, MRCLASS = {03E70 (03E30)}, MRNUMBER = {3307881}, eprint = {1311.0814}, archivePrefix = {arXiv}, primaryClass = {math.LO}, url = {http://jdh.hamkins.org/the-role-of-foundation-in-the-kunen-inconsistency/}, }`

$\newcommand\ZFC{\text{ZFC}}\newcommand\ZFCf{\ZFC^{\rm-f}}\newcommand\AFA{\text{AFA}}\newcommand\BAFA{\text{BAFA}}$

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?

Pingback: The role of the axiom of foundation in the Kunen inconsistency, CUNY September 2013 | Joel David Hamkins

on the axiom of foundation. I am teaching elementary set theory course, and I arrived at a small difficulty. The axiom of foundation is stated for sets, and not for classes. To prove the class schema of foundation, I need to use replacement. Some proofs (such as V=\bigcup_a V_a) apparently need the class schema of foundation, and therefore use replacement. Am I correct here?

Jindra, thanks for the comment. If you have mere separation plus the existence of transitive closures of sets, then this is enough to prove the class version of foundation, since for any nonempty class $X$, simply select any $x\in X$ and consider $X\cap \text{TC}(\{x\})$. Any $\in$-minimal element of this set will also be $\in$-minimal in $X$.

This is exactly the trouble. The existence of TC({x}) without replacement, that is what I do not know how to prove.

I don’t think that this is provable without replacement. I recall that Mathias has some very crazy models of Zermelo set theory, and in particular, I seem to recall that sets without transitive closures was one of the situations. (See his remarks at the bottom of page 1 on his article at https://www.dpmms.cam.ac.uk/~ardm/slim.pdf, where indeed he says this and provides references to Boffa and others.)

A related issue arises with finite set theory—if you just drop the infinity axiom from the usual ZFC and add its negation, you cannot prove that sets have transitive closure, by a result due to Ali Enayat.

aha, that link settles it, thanks

For the construction of a model for ZFC\Rep such that there is a set with non-set transitive closure you can look at the following reference:

F. Drake, Set Theory: An introduction to large cardinals, Chapter 4, Page 110, Exercise 1.