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 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 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 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 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 ZFC and GBC, the term is used in the ZF 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
- Hamkins, Joel David and Kirmayer, Greg and Perlmutter, Norman. Generalizations of the {Kunen} inconsistency. Annals of Pure and Applied Logic 163(12):1872 - 1890, 2012. www arχiv DOI 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