The Kunen inconsistency

• Metamathematical issues

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 (Kanamori, 2009)(p. 320-321), Hamkins, Kirmayer and Perlmutter (missing reference), Woodin (Kanamori, 2009)(p. 320-321), Zapletal (Zapletal, 1996) and Suzuki (Suzuki, 1998; Suzuki, 1999).

• 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 possible 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.

References

1. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3
2. Zapletal, J. (1996). A new proof of Kunenś inconsistency. Proc. Amer. Math. Soc., 124(7), 2203–2204. https://doi.org/10.1090/S0002-9939-96-03281-9
3. Suzuki, A. (1998). Non-existence of generic elementary embeddings into the ground model. Tsukuba J. Math., 22(2), 343–347.
4. Suzuki, A. (1999). No elementary embedding from V into V is definable from parameters. J. Symbolic Logic, 64(4), 1591–1594. https://doi.org/10.2307/2586799

Main library