The Kunen inconsistency
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. 320-321), Hamkins, Kirmayer and Perlmutter [2], Woodin [1](p. 320-321), Zapletal [3] and Suzuki [4, 5].
- There is no nontrivial elementary embedding $j:V\to V$ from the set-theoretic universe to itself.
- There is no nontrivial elementary embedding $j:V[G]\to V$ of a set-forcing 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 Kelly-Morse set theory, but it is also possble to prove it in the weaker system of Gödel-Bernays 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.
Reinhardt cardinal
Although the existence of Reinhardt cardinals has now been refuted in $\text{ZFC}_2$ and $\text{GBC}$, the term is used in the $\text{ZF}_2$ context to refer to the critical point of a nontrivial elementary embedding $j:V\to V$ of the set-theoretic universe to itself.
Super Reinhardt cardinal
A super Reinhardt cardinal $\kappa$, is a cardinal which is the critical point of elementary embeddings $j:V\to V$, with $j(\kappa)$ as large as desired.
References
- Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, 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):2203--2204, 1996. www MR bibtex
- Suzuki, Akira. Non-existence of generic elementary embeddings into the ground model. Tsukuba J Math 22(2):343--347, 1998. MR bibtex | Abstract
- Suzuki, Akira. No elementary embedding from $V$ into $V$ is definable from parameters. J Symbolic Logic 64(4):1591--1594, 1999. www DOI MR bibtex