Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

View the Project on GitHub neugierde/cantors-attic

**Quick navigation**

The upper attic

The middle attic

The lower attic

The parlour

The playroom

The library

The cellar

**Sources**

Cantor's Attic (original site)

Joel David Hamkins blog post about the Attic

Latest working snapshot at the wayback machine

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 $\text{ZFC}$.

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.

- Kanamori, A. (2009).
*The higher infinite*(Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3 - 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 - Suzuki, A. (1998). Non-existence of generic elementary embeddings into the ground model.
*Tsukuba J. Math.*,*22*(2), 343–347. - 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