Difference between revisions of "Kunen inconsistency"
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The Kunen inconsistency, the theorem showing that there can be no nontrivial elementary embedding from the universe to itself, remains a focal point of large cardinal set theory, marking a hard upper bound at the summit of the main ascent of the large cardinal hierarchy, the first outright refutation of a large cardinal axiom. On this main ascent, large cardinal axioms assert the existence of elementary embeddings $j:V\to M$ where $M$ exhibits increasing affinity with $V$ as one climbs the hierarchy. The $\theta$strong cardinals, for example, have $V_\theta\subset M$; the $\lambda$supercompact cardinals have $M^\lambda\subset M$; and the huge cardinals have $M^{j(\kappa)}\subset M$. The natural limit of this trend, first suggested by Reinhardt, is a nontrivial elementary embedding $j:V\to V$, the critical point of which is accordingly known as a Reinhardt cardinal. Shortly after this idea was introduced, however, Kunen famously proved that there are no such embeddings, and hence no Reinhardt cardinals in $\text{ZFC}$.
Since that time, the inconsistency argument has been generalized by various authors, including Harada [1](p. 320321), Hamkins, Kirmayer and Perlmutter [2], Woodin [1](p. 320321), Zapletal [3] and Suzuki [4, 5].
 There is no nontrivial elementary embedding $j:V\to V$ from the settheoretic universe to itself.
 There is no nontrivial elementary embedding $j:V[G]\to V$ of a setforcing extension of the universe to the universe, and neither is there $j:V\to V[G]$ in the converse direction.
 More generally, there is no nontrivial elementary embedding between two ground models of the universe.
 More generally still, there is no nontrivial elementary embedding $j:M\to N$ when both $M$ and $N$ are eventually stationary correct.
 There is no nontrivial elementary embedding $j:V\to \text{HOD}$, and neither is there $j:V\to M$ for a variety of other definable classes, including $\text{gHOD}$ and the $\text{HOD}^\eta$, $\text{gHOD}^\eta$.
 If $j:V\to M$ is elementary, then $V=\text{HOD}(M)$.
 There is no nontrivial elementary embedding $j:\text{HOD}\to V$.
 More generally, for any definable class $M$, there is no nontrivial elementary embedding $j:M\to V$.
 There is no nontrivial elementary embedding $j:\text{HOD}\to\text{HOD}$ that is definable in $V$ from parameters.
It is not currently known whether the Kunen inconsistency may be undertaken in ZF. Nor is it known whether one may rule out nontrivial embeddings $j:\text{HOD}\to\text{HOD}$ even in $\text{ZFC}$.
Metamathematical issues
Kunen formalized his theorem in KellyMorse set theory, but it is also possible to prove it in the weaker system of GödelBernays set theory. In each case, the embedding $j$ is a $\text{GBC}$ class, and elementary of $j$ is asserted as a $\Sigma_1$elementary embedding, which implies $\Sigma_n$elementarity when the two models have the ordinals.
References
 Kanamori, Akihiro. The higher infinite. Second, SpringerVerlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www bibtex
 Zapletal, Jindrich. A new proof of Kunen's inconsistency. Proc Amer Math Soc 124(7):22032204, 1996. www DOI MR bibtex
 Suzuki, Akira. Nonexistence of generic elementary embeddings into the ground model. Tsukuba J Math 22(2):343347, 1998. MR bibtex  Abstract
 Suzuki, Akira. No elementary embedding from $V$ into $V$ is definable from parameters. J Symbolic Logic 64(4):15911594, 1999. www DOI MR bibtex