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First fix the following notations:‎ ‎

‎$‎AF:=‎$ ‎The ‎axiom ‎of ‎foundation‎ ‎

‎$‎ZFC^{-}‎:=‎ZFC‎\setminus ‎\left\lbrace AF ‎\right\rbrace ‎‎‎‎‎$‎‎‎‎‎ ‎

‎$‎G:=‎$ ‎The ‎proper ‎class ‎of ‎all ‎sets‎ ‎

‎$‎V:=‎$ ‎The ‎proper ‎class ‎of ‎Von ‎neumann's ‎cumulative ‎hierarchy‎ ‎

‎$‎L:=‎$ ‎The ‎proper ‎class ‎of ‎Godel's cumulative ‎hierarchy‎ ‎

‎$‎G=V:~‎‎\forall x‎~‎\exists ‎y~(ord(y) ‎‎\wedge ‎``x\in ‎V_{y}")‎$ ‎‎ ‎

‎$‎G=L:~‎‎\forall x‎~‎\exists ‎y~(ord(y) ‎‎\wedge ‎``x\in L_{y}")‎$‎‎ ‎‎ ‎‎

Almost ‎all ‎of ‎‎$‎ZFC‎$ ‎axioms ‎have a‎ ‎same "‎nature" in some sense. ‎They ‎are ‎"generating" ‎new ‎sets ‎which form the world of mathematical objects ‎$‎G‎$‎. ‎In ‎other ‎words‎ ‎they are ‎complicating ‎‎our ‎mathematical chess by increasing its playable and legitimated nuts. ‎But ‎‎$‎AF‎$‎ ‎is ‎an ‎exception ‎in ‎‎$‎ZFC‎$. ‎It ‎is "simplifying" ‎our mathematical ‎world ‎by ‎removing ‎some ‎sets which innocently are accused to be ‎"ill ‎founded". Even ‎$‎AF‎$‎ is regulating ‎$‎G‎$ ‎by ‎‎$‎V‎$ ‎and ‎says ‎‎$‎G=V‎$. So ‎‎it‎‎ ‎is "miniaturizing" the "real" size of ‎$‎G‎$ ‎by the ‎‎"very ‎small" ‎cumulative ‎hierarchy ‎‎$‎V‎$ ‎as ‎same ‎as ‎the ‎assumption ‎of ‎constructibility ‎axiom ‎‎$‎G=L‎$. ‎In ‎fact‎ ‎"minimizing" the size of mathematical universe ‎is ontological ‎"nature" ‎of ‎all ‎constructibilty ‎kind ‎of ‎axioms ‎like ‎‎$‎G=W‎$ ‎which ‎‎$‎W‎$ ‎is ‎an ‎arbitrary ‎cumulative ‎hierarchy. ‎But ‎in ‎the ‎opposite direction ‎the ‎large ‎cardinal ‎axioms ‎says a‎ ‎different ‎thing ‎about ‎‎$‎G‎$‎. We know that any large cardinal axiom stronger than "‎$‎‎0^{\sharp}$ ‎exists" ‎implies ‎‎$‎G\neq L‎$‎. This illustrates the "nature" of large cardinal axioms. They implicitly say the universe of mathematical objects is too big and is "not" reachable by cumulative hierarchies. So it is obvious that any constructibility kind axiom such as ‎$‎AF‎$, ‎imposes a‎ ‎limitation ‎on ‎the ‎height ‎of ‎large ‎cardinal ‎tree‎. ‎One ‎of ‎the‎se serious limitations is Kunen inconsistency theorem in ‎$‎‎ZFC^{-}+AF$‎.‎ ‎

Theorem (Kunen inconsistency) There is no non trivial elementary embedding ‎$‎‎j:\langle V,\in\rangle\longrightarrow ‎\langle V,\in\rangle $ ‎(or equivalently ‎$‎‎j:\langle G,\in\rangle\longrightarrow ‎\langle G,\in\rangle$‎)‎

‎‎ The ‎proof ‎has ‎two ‎main ‎steps as follows:‎

Step (1): ‎By ‎induction ‎on ‎Von ‎neumann's ‎"rank" ‎in ‎‎$‎V‎$‎ one can prove any non trivial elementary embedding ‎$‎‎j:\langle V,\in\rangle\longrightarrow ‎\langle V,\in\rangle‎$ has a critical point ‎$‎‎‎\kappa‎$‎ on ‎$‎Ord‎$‎. ‎

Step ‎(2): ‎By ‎iterating ‎‎$‎j‎$ ‎on this critical point one can find an ordinal ‎$‎‎\alpha‎$ ‎such ‎that ‎‎$‎‎j[‎\alpha‎]=‎\lbrace‎ j(‎\beta‎)~|~‎\beta ‎\in ‎‎\alpha ‎\rbrace‎‎‎ \notin V (=G)‎$ ‎which ‎is a‎ ‎contradiction.‎ ‎

Now ‎in ‎the ‎absence ‎of ‎‎$‎AF‎$ ‎we ‎must ‎notice ‎that ‎the Kunen inconsistency ‎theorem ‎splits ‎into ‎two ‎distinct ‎statements and the orginal proof fails in both of them‎‎‎‎. ‎ ‎

Statement (1):(Strong version of Kunen inconsistency) There is no non trivial elementary embedding ‎‎‎$‎‎j:\langle G,\in\rangle‎\longrightarrow ‎\langle G,\in\rangle$‎.‎

Statement (2):(Weak version of Kunen inconsistency) There is no non trivial elementary embedding ‎‎‎$‎‎j:\langle V,\in\rangle\longrightarrow \langle V,\in\rangle$‎.

‎‎ In statement ‎(1)‎, step (1) collapses because without ‎$‎AF‎$‎ we have not a "rank notion" on ‎$‎G‎$ ‎and ‎the ‎induction ‎makes ‎no ‎sense. So we can not find any critical point on ‎$‎Ord‎$ ‎for ‎‎$‎j‎$ ‎by "‎this ‎method". ‎ ‎

In statement ‎(2)‎, step (2) fails because without ‎$‎AF‎$‎ we don't know‎ $‎‎G=V$ and so $j[‎\alpha‎]\notin V‎$ ‎is ‎not a‎ ‎contradiction.‎ ‎

‎But ‎it is clear that ‎in ‎‎$‎ZFC^{-}‎$ ‎the ‎original ‎proof ‎of ‎Kunen ‎inconsistency theorem ‎works ‎for ‎both of the ‎following ‎propositions:‎ ‎

Proposition (1): There is no elementary embedding ‎‎‎$‎‎j:\langle G,\in\rangle\longrightarrow ‎\langle G,\in\rangle $‎ with a critical point on ‎$‎Ord‎$‎.

Proposition (2): Every non trivial elementary embedding ‎‎‎$‎‎j:\langle V,\in\rangle\longrightarrow \langle V,\in\rangle$‎ has a critical point on ‎$‎Ord‎$‎. ‎‎

Now ‎the ‎main ‎questions ‎are‎: ‎‎

‎‎Question ‎(1): ‎Is ‎the ‎statement ‎"‎There is a non trivial elementary embedding $‎‎j:\langle V,\in\rangle\longrightarrow ‎\langle V,\in\rangle$‎" an acceptable large cardinal axiom in ‎the absence of ‎$‎AF‎$($‎G=V‎$‎)‎‎? What about other statements by replacing ‎$‎V‎$ ‎with a‎n ‎arbitrary ‎cumulative ‎hierarchy ‎‎$‎W‎$‎?(In this case don't limit the definition of a cumulative hierarchy by condition ‎$‎‎W_{‎\alpha +1‎}\subseteq P(W_{‎\alpha‎})$‎)‎ ‎

Note that such statements are very similar to ‎the ‎statment "‎‎$‎‎0^{\sharp}$ exists" that ‎is ‎equivalent ‎to ‎‎existence of a non trivial elementary embedding $‎‎j:\langle L,\in\rangle\longrightarrow ‎\langle L,\in\rangle$ ‎and ‎could ‎be an ‎"acceptable" ‎large ‎cardinal ‎axiom ‎in ‎the ‎"absence" ‎of ‎‎$‎‎G=L$‎‎. So if the answer of the question (1) be positive, ‎we can go "beyond" weak version of Kunen inconsistency by removing ‎$‎AF‎$ ‎from ‎$‎ZFC‎$‎ and so we can find a family of "Reinhardt shape" cardinals correspond to any cumulative hierarchy ‎$‎W‎$ by a similar argument to proposition (2) dependent on "good behavior" of "rank notion" in ‎$‎W‎$‎‎. ‎

Question ‎(2): ‎Is ‎‎$‎AF‎$ ‎necessary to prove "strong" version of ‎Kunen ‎inconsistency ‎theorem? ‎In ‎the ‎other ‎words ‎is ‎the‎ ‎statement ‎"$Con(ZFC)‎\longrightarrow Con(ZFC^{-}+ ‎\exists‎$‎ a‎ ‎non ‎trivial ‎elementary ‎embedding ‎‎$‎‎j:\langle G,\in\rangle‎\longrightarrow \langle G,\in\rangle)‎‎$"‎‎ ‎true?‎

It seems to go beyond Kunen inconsistency it is not necessary to remove‎ $‎AC‎$ ‎which possibly "harms" our powerful tools and changes the "natural" behavior of objects.It simply suffices that one omit ‎$‎AF‎$‎'s limit on largeness of "Cantor's heaven" and his "set theoretic intuition". َAnyway whole of the set theory is not studying $L$, $V$ or any other cumulative hierarchy and there are many object "out" of these realms. For example ‎without limitation of ‎‎$‎G=L‎$ ‎we ‎can see more large cardinals that are "invisible" in small "scope" of $L$.‎In the same way without limitation of $AF$ we ‎can probably discover ‎more stars in the mathematical universe out of scope of $V$. Furthermore we can produce more interesting models and universes and so we can play an extremely ‎exciting ‎mathematical ‎chess beyond inconsistency, beyond imagination!

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Could you clarify how you intend to formalize the assertion $\exists j$ in questions 1 and 2? After all, this is a second-order quantifier, and it is not directly formalizable in ZFC-AF alone. –  Joel David Hamkins Jul 8 '13 at 11:01
    
Dear professor Hamkins, I really don't know. But as you mentioned in your paper "Generalizations of Kunen Inconsistency", expressing this kind of questions is a big question itself! Can you suggest a meaningful restating for these questions? Anyway I think this "meaningless" questions are so "natural" too! –  Ali Sadegh Daghighi Jul 8 '13 at 11:50
    
Although you have labeled them "strong" and "weak" formulations of the Kunen inconsistency, the answers show that there isn't actually an implication from the strong form to the weak form. –  Joel David Hamkins Jul 8 '13 at 13:39
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2 Answers

up vote 15 down vote accepted

The answer to question 2 is yes, and one can even have nontrivial automorphisms. For example, the theory $\mathit{ZFC}^-+{}$“there are two urelements (i.e., sets $x$ satisfying $x=\{x\}$) and the whole universe is obtained from them by iterated power set” is consistent relative to ZFC, and one can define uniquely in this theory an automorphism swapping the two urelements.

For a theory rich in elementary embeddings and automorphisms, Boffa’s set theory (introduced in [1]) is relatively consistent wrt ZFC. The theory proves that any class endowed with a set-like binary relation satisfying the axiom of extensionality (but not necessarily well founded) is isomorphic to $\langle T,\in\rangle$ for some transitive class $T$. (For example, such a transitive collapse of the diagonal on the universe gives you a proper class of urelements, and you can construct even weirder objects. More to the point, any ultrapower of the universe gives you an elementary embedding into a transitive class.) Also, every isomorphism $f\colon\langle t,\in\rangle\to\langle s,\in\rangle$ of transitive sets $t,s$ can be extended to an automorphism of the universe.

Boffa’s theory consists of $\mathit{ZFC}^-$ + global choice + the following axiom:

If $t$ is a transitive set, and $\langle x,e\rangle$ a structure satisfying extensionality which is an end-extension of $\langle t,\in\rangle$, then there exists a transitive set $s\supseteq t$ and an isomorphism $f\colon\langle x,e\rangle\to\langle s,\in\rangle$ identical on $t$.

[1] Maurice Boffa, Forcing et négation de l’axiome de Fondement, Académie Royale de Belgique, Classe des Sciences, Mémoires, Collection 8o, 2e Série, tome XL, fasc. 7, 1972, 52pp.

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The first example was not supposed to refer to Boffa’s theory—the point being that the relative consistency of the theory in the example wrt ZFC is much easier to verify. However, I am not convinced that the mere existence of two urelements is enough. While the condition on iterated powersets is by no means necessary (indeed, Boffa’s theory refutes it), I don’t even see a reason why any two Quine atoms should satisfy the same formulas. –  Emil Jeřábek Jul 8 '13 at 13:55
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In fact, assume that there are three distinct sets $a,b,c$ such that $a=\{a\}$, $b=\{b\}$, and $c=\{a,c\}$, and that the universe consists of the iterated power set of $a,b,c$. (This scenario is relatively consistent with $\mathit{ZFC}^-$.) Then there is no nontrivial elementary embedding of the universe into itself. –  Emil Jeřábek Jul 8 '13 at 14:00
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Very nice. (And I have now deleted my earlier comment, which was incorrect.) –  Joel David Hamkins Jul 8 '13 at 16:43
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Update. We have now written an article summarizing and extending the answers that we provided to this question.

A. S. Daghighi, M. Golshani, J. D. Hamkins, and E. Jeřábek, The role of the foundation axiom in the Kunen inconsistency (under review).

Please click through to the arxiv for the preprint.

The answer to question 1 is negative, and the answer to question 2 depends on which flavor of anti-foundation one chooses to adopt.

But let me begin by remarking that I believe that there are some serious issues of formalization involved in the Kunen inconsistency. We discuss this issues at length in the preliminary section of our paper J. D. Hamkins, G. Kirmayer, N. Perlmutter, Generalizations of the Kunen inconsistency. For example,

  • The most direct issue is that the quantifier "$\exists j$" is a second-order quantifier that is not directly formalizable in ZFC.

  • Many set theorists interpret all talk of classes in ZFC as referring to the first-order definable classes. In this formalization, the Kunen inconsistency becomes a scheme, asserting of each possible definition of $j$ that it isn't an elementary embedding of the universe. (For example, Kanamori adopts this approach.) My view is that this is not the right interpretation, however, because actually there is a much easier proof of the Kunen inconsistency, a formal logical trick not relying on AC and not using any of the combinatorics usually associated with the Kunen inconsistency proof. (See our paper for explanation.)

  • So it seems natural to want to use a second-order set theory, such as Gödel-Bernays or Kelly-Morse set theory. In GBC, we have class quantifiers for expressing "$\exists j$", but then the issue arises that it is not directly possible express the assertion "$j$ is elementary", and so set-theorists usually settle for the assertion "$j$ is $\Sigma_1$-elementary and cofinal", which implies that $j$ is $\Sigma_n$-elementary for meta-theoretical natural numbers $n$, by induction carried out in the meta-theory.

  • Kunen formalized his theorem in Kelly-Morse theory, which as a truth predicate for first-order truth and thus both "$\exists j$" and "$j$ is elementary are expressible" are expressible.

Let us suppose that we have a formalization that allows us to refer to elementary embeddings $j$.

The answer to question 1 is that the assertion is still inconsistent, even without AF. The reason is that $V$ is definable in $G$ as the class of sets that are well-founded, and every axiom of ZFC, incuding AF, is provable relativized to $V$. Thus, we may simply carry out any of the usual proofs of the Kunen inconsistency to arrive at a contradiction. For example, there still must be a critical point $\kappa$, and then the supremum of the critical sequence $\lambda=\sup_n \kappa_n$, where $\kappa_{n+1}=j(\kappa_n)$, has $j(\lambda)=\lambda$ and also $j\upharpoonright V_{\lambda+2}:V_{\lambda+2}\to V_{\lambda+2}$, which is sufficient for all the various proofs of the Kunen inconsistency of which I am aware.

Note that if $j:G\to G$ is elementary, then $j\upharpoonright V:V\to V$, since $V$ is definable in $G$. Also, if $S$ is any set in $V$, then the image $j[S]\subset V$ and so $j[S]$ is also well-founded, leading to $j[S]\in V$, which I think may resolves an issue behind your question.

For question 2, the answer depends on which anti-foundational theory you adopt. On the one hand, Emil has given an excellent answer showing that there can be nontrivial embeddings when there are Quine atoms, such as in the Boffa theory. Let me show now in contrast that we may reach the opposite answer under Aczel's anti-foundation axiom.

Theorem. Under GBC-AF+AFA, there is no nontrivial elementary embedding $j:G\to G$.

Proof. Since $V$ is definable in $G$, it follows that $j\upharpoonright V:V\to V$. For any set $x\in G$, consider the underlying directed graph $\langle \text{TC}(\{x\}),{\in}\rangle$. In AFA (and many other anti-founational set theories), equality of sets is determined by the isomorphism type of this graph. By the axiom of choice, we may well-order the nodes of this graph and thereby find a copy of this graph inside $V$. Thus, if $j(x)\neq x$, it follows that $j(d)\neq d$, where $d$ is an isomorphic copy in $V$ of the underlying graph of $x$. Thus, $j\upharpoonright V$ is also nontrivial and elementary on $V$. And so the hypothesis is refuted by the usual Kunen inconsistency applied inside $V$. QED

The argument applies just as well to any of the anti-foundational theories where equality of sets is determined by the isorphism type of the underlying $\in$-relation on the hereditary members of the set, such as by the bisimilarity type.

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